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\hypersetup{pdftitle={A PD4 group without an aspherical manifold model},pdfauthor={OpenAI}}
\title{A PD4 group without an aspherical manifold model}
\author{OpenAI}
\date{September 24, 2026}
\begin{document}
\maketitle
\begin{abstract}
A Poincar\'e duality group is a group that behaves, algebraically, like
the fundamental group of a closed aspherical manifold. Wall asked
whether every such group actually comes from a manifold. We construct
a finitely presented integral Poincar\'e duality group of dimension
four with a finite classifying space, and prove that no closed
aspherical topological four-manifold has it as fundamental group.
\end{abstract}
\section{Introduction}\label{sec:introduction}
A manifold is \emph{aspherical} if its universal cover is contractible.
For such a manifold, every homotopy invariant is read off from the
fundamental group $\Gamma = \pi_1 M$: the higher homotopy groups vanish,
the universal cover $\widetilde M$ is contractible, and the chains of
$\widetilde M$ form a finite free resolution of $\Z$ over $\Z\Gamma$.
Poincar\'e duality then forces the cohomology of $\Gamma$ with
group-ring coefficients to vanish except in the top degree:
\[
H^i(\Gamma;\Z\Gamma) = 0 \quad (i \neq n), \qquad
H^n(\Gamma;\Z\Gamma) \cong \Z \text{ with trivial action}.
\]
A group satisfying this algebraic condition is called an
\emph{oriented integral $\PD_n$ group}. More formally, an oriented
integral $\PD_n$ group has a finite resolution of the trivial module
$\Z$ by finitely generated projective modules, vanishing group-ring
cohomology off degree $n$, and $\Z$ in degree $n$ with trivial right
action.
\subsection{Wall's question}
Wall asked whether the algebraic condition is sufficient
\cite[Problem~G2]{WallProblems}: given an oriented integral $\PD_n$
group, is there a closed aspherical $n$-manifold whose fundamental
group is that group?
Two choices matter: the dimension and the category of manifold. This
paper works in dimension four, in the topological category.
For $n \geq 5$, the \emph{Borel conjecture} answers the topological
version for a large class of groups: if $G$ satisfies appropriate
surgery assembly conditions, every closed aspherical $n$-manifold with
$\pi_1 = G$ is homotopy equivalent to any other, and surgeries between
them are standard. Large group classes (hyperbolic, CAT(0), and so on)
are known to satisfy those assembly conditions. The remaining question
in high dimensions is which finitely presented PD groups are realized
at all; that is more open, but a great deal is known.
For $n = 4$ the situation is much more delicate. Freedman's
classification of closed topological four-manifolds and the failure of
the Whitney trick in the middle dimension mean that surgical arguments
need disk-embedding input, not just surgery exactness. Whether every
finitely presented integral $\PD_4$ group has a closed aspherical
topological four-manifold model has been open.
Finiteness is essential: a closed manifold has finitely presented
fundamental group. Without that hypothesis, examples already exist.
Bestvina--Brady provided finiteness counterexamples, and Davis
\cite[Theorem~C]{DavisCohomology} turned them, using reflection, into
integral $\PD_n$ groups that are not finitely presented, for every
$n \geq 4$. Those resolve Wall's question in the unqualified form.
What remains is the \emph{finitely presented} case. This is Question~3.4
of Davis's survey \cite{DavisPD} and Conjecture~7.29 of the cited
version of L\"uck's survey \cite{LuckFJ}; Kielak's 2026 ICM survey
records that no counterexamples to this formulation were known
\cite[Section~4.1]{Kielak}.
The coefficient ring matters too. Fowler \cite[Theorem~2.1]{Fowler}
constructed torsion-free finitely presented \emph{rational} Poincar\'e
duality groups that cannot be realized by closed aspherical ANR
rational homology manifolds. Those examples have no finite classifying
space. The example below satisfies \emph{integral} duality and already
has a finite four-dimensional classifying space. The obstruction is
specifically to the passage from that Poincar\'e complex to a closed
topological manifold.
\subsection{The main theorem}
\begin{theorem}\label{thm:main}
There is a finitely presented group $G$ with a finite, oriented,
aspherical four-dimensional Poincar\'e complex $Q$ such that
\[
H^i(G;\Z G) = 0 \quad (i \neq 4), \qquad
H^4(G;\Z G) \cong \Z
\]
with trivial right $G$-action on the top group, and no closed aspherical
topological four-manifold has fundamental group $G$.
In particular, $\Z$ has a finite resolution by finitely generated free
$\Z G$-modules.
\end{theorem}
The theorem answers the question quantified over all dimensions
$n \geq 3$ at once: the answer is not always yes. It does not assert a
counterexample in every dimension. The orientation qualification is
harmless: if a hypothetical aspherical four-manifold had group $G$,
its orientation character would equal the right action on the top
dualizing module, hence trivial, so such a manifold would be
orientable.
\subsection{The source of the obstruction}
Our starting point is the marked geometric and tensor obstruction of
the companion paper \emph{A marked tensor obstruction to
four-dimensional disk embedding}~\cite{TO}. That paper packages a
specific pattern of geometric data, a finite family of based loop
relations with independently chosen conjugating paths, together with
algebraic dual spheres. Its algebraic half converts a compatible cut
configuration into a finite system of power-series identities over a
ring with nilpotent parameters. A finite tensor theorem rules those
identities out.
Three kinds of input are taken from \cite{TO}: the marked bodies,
exterior duals, and their intersection calculus; the passage from
vertex data to tensor identities; and the finite tensor theorem itself.
Everything else is proved here.
The new geometric input of the present paper is a compact smooth
chamber $D$ with boundary $S$ and a specified hyperbolic basis of its
entire middle homology over the group ring. Attaching three-cells along
the basis produces a marked graph-type Poincar\'e pair $(P,S)$, with
$P \simeq BH$ for $H = \pi_1 D$ a free group. The claim is that the
boundary $S$, together with the marking $S \to BH$, admits no
\emph{topological graph-type filling}: no compact four-manifold $D'$
with $\partial D' = S$ and a homotopy equivalence $D' \to BH$
extending the boundary marking.
Reflection then closes the pair to a finite Poincar\'e complex
$Q = BG$. The group $G$ satisfies strong assembly theorems; that is
what allows us to reach a contradiction from a hypothetical
aspherical manifold model.
\subsection{Proof architecture}
The argument has four steps.
\textbf{Step 1: a chamber with no marked filling.}
Section~\ref{ch:section} constructs a smooth compact oriented
four-manifold $D$ with connected boundary $S$, free fundamental group
$H$, a framed hyperbolic basis of $\pi_2 D$ over $\Z H$, and a
homotopy model $P \simeq BH$ obtained by attaching three-cells along
the basis. The pair $(P,S)$ is an oriented Poincar\'e pair of
dimension four. The claim, \ref{ch:chamber}, is that no topological
filling of $S$ respecting the boundary marking and homotopy equivalent
to $BH$ exists.
The companion paper \cite{TO} proves this claim via a cut argument:
two vertex pieces joined by $2m$ coordinate tori, with marked loop laws
imposed at those pieces, force the tensor identities that the finite
tensor theorem forbids. We transfer that argument to arbitrary marked
\emph{topological} fillings, removing the smooth-structure assumption.
Quinn's punctured smoothing \cite[Corollary~2.2.3]{QuinnEnds} handles
the topological base, and a tailored differential-graded algebra
handles the vertex calculation.
\textbf{Step 2: reflection and duality.}
Section~\ref{rf:section} applies Davis's reflection method
\cite[Sections~13 and~15, especially Remark~15.9]{Davis1983}. Choose
a flag triangulation of $S$, take the right-angled Coxeter group $C$
on its vertices, and develop the chamber $\mathcal U(C,D)$. A
finite-index torsion-free subgroup $\Delta \leq C$ acts freely and
cocompactly, yielding a closed manifold $N$ and a finite Poincar\'e
complex $Q = \Delta \backslash \mathcal U(C,P)$. Reflection preserves
the hyperbolic pairs and shows $Q$ is the aspherical four-dimensional
Poincar\'e complex of a finitely presented group $G = \pi_1 Q =
\pi_1 N$.
A support-tree construction on the lifted Coxeter system produces a
proper cocompact CAT(0) model for $G$. This lets us apply the
Farrell--Jones conjectures via Bartels--L\"uck and Wegner.
\textbf{Step 3: marked stable realization.}
Section~\ref{st:setup} assumes a hypothetical closed aspherical
$M$ with $\pi_1 M \cong G$, and shows $M$ forces a stable geometric
realization of all the marked pairs.
First, an integral refinement of the second-Postnikov-stage argument
produces a marked homotopy equivalence $N \simeq L =
M \# b(S^2\times S^2)$. Integrality is nontrivial here: the ordinary
intersection form does not see the group-ring data, so we use
equivariant cup products to distinguish fundamental classes inside
$H_4$ of a split Postnikov stage.
Second, the Kasprowski--Land surgery result \cite{KL} turns the marked
homotopy equivalence into a marked $s$-cobordism, and Quinn's stable
product theorem \cite{QuinnStable} stabilizes that to a marked
homeomorphism $N^* \cong L^*$ preserving all sphere classes.
Lifting the stabilized data to the reflection development yields two
compatible systems in a single manifold $N^*$: the chamber hyperbolic
pairs, and standard pairs with disjoint punctured $S^2 \times S^2$
neighbourhoods.
\textbf{Step 4: controlled removal.}
Section~\ref{ct:section} produces a graph-type filling of $S$ from
those data, contradicting Step~1.
Three sub-steps. First, a complementary filling lemma
(Lemma~\ref{ct:fillings}) shows every chamber bound in the reflection
development has a simply connected filling $J$ with $H_2(J)\cong
H_2(S)$. Capping the noncentral boundary factors of a chamber union
$U_L$ gives a compact manifold $Z$ with $\pi_1 Z = H$, $\partial Z=S$,
and middle homology freely generated by the surviving standard pairs.
Second, peeling removes an inner ball of standard pairs, and a
direction-control argument using the Davis complex prepares the
surviving pairs so their upper grope branches all land in contractible
direction patches. This lets us apply the Powell--Ray--Teichner disk
embedding machinery \cite{PRT} with no group-theoretic hypothesis on
$H$. Their usual good-group null-loop step is replaced by an explicit
geometric containment.
Third, surgery on the surviving marked Lagrangian spheres produces a
compact four-manifold $V$ with $\partial V = S$, $\pi_1 V = H$, and
contractible universal cover. That is exactly the marked graph-type
filling forbidden in Step~1, and the contradiction completes the
proof.
\begin{figure}[H]
\centering
\begin{tikzpicture}[>=Latex, node distance=14mm and 25mm,
every node/.style={align=center,font=\small}]
\node (D) {$D\longrightarrow P\simeq BH$\\marked boundary $S$};
\node[right=of D] (N) {$N\longrightarrow Q\simeq BG$\\reflection};
\node[below=of N] (M) {hypothetical $M\simeq Q$\\marked stable product};
\node[below=of D] (Z) {$Z$ with boundary $S$\\cap and remove inner pairs};
\node[below=of Z] (W) {graph-type filling of $S$\\contradicts the cut obstruction};
\draw[->] (D) -- (N);
\draw[->] (N) -- (M);
\draw[->] (M) -- (Z);
\draw[->] (Z) -- node[right,text width=35mm] {direction control\\and sphere surgery} (W);
\end{tikzpicture}
\caption{The two uses of the chamber. Its middle spheres permit the
reflection and stable-surgery construction; its marked boundary
forbids the filling obtained at the end.}
\label{fig:architecture}
\end{figure}
\subsection{Relation to known work}
Davis and Hillman \cite[Theorem~D]{DavisHillman} prove relative
manifold realization for aspherical Poincar\'e pairs with nonempty
closed manifold boundary in dimensions at least five, under the $K$-
and $L$-theoretic Farrell--Jones conjectures, and in dimension four
for elementary amenable fundamental groups. Our chamber has nonabelian
free fundamental group and lies outside that elementary amenable
hypothesis; the chamber obstruction is a relative realization
counterexample in that case.
\subsection{Conventions}
All manifolds in the surgery argument are oriented, and unqualified
homology has integral coefficients. For a connected space with
fundamental group $\Gamma$, the coefficient ring $\Z\Gamma$ refers to
the homology of its universal cover. We fix lifts and paths to a
basepoint when recording sphere classes. The involution on the group
ring is $\overline{\sum n_g g} = \sum n_g g^{-1}$. A hyperbolic plane
over $\Z\Gamma$ has matrix
$\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)$. The
phrase \emph{framed hyperbolic system} of immersed spheres includes
vanishing Wall self-intersection obstructions in its framing
requirements.
All stabilizations are interior connected sums with $S^2\times S^2$
and leave boundary markings unchanged. References to Freedman--Quinn
\cite{FQ} use the January 2013 reformatted edition's numbering.
\section{A chamber with no marked graph filling}\label{ch:section}
We construct a compact chamber whose boundary is marked and cannot be
filled. The defect is not a localised disk: the entire boundary $S$
together with its map to $BH$ is unfillable in the topological category
by anything homotopy equivalent to $BH$. Every hypothetical graph-type
filling would necessarily realise a specific configuration of
geometric data that the companion paper rules out algebraically.
The argument uses three pieces of imported machinery from \cite{TO}.
First, a marked plumbing body with distinguished port meridians and
extra loops, and a family of grope bodies attached inside its boundary.
Second, the exterior algebra of that body with its dual spheres. Third,
a finite tensor theorem that any hypothetical filling would have to
satisfy, and that is impossible. Everything else is constructed here.
\begin{proposition}[The marked chamber]
\label{ch:chamber}
There are a compact connected oriented smooth four-manifold $D$, its
connected boundary $S$, a finitely generated free group $H$, and a map
$c \colon D \to BH$ with three properties.
\begin{enumerate}
\item The map $c$ induces an isomorphism on fundamental groups, and $D$
has the homotopy type of a finite graph with $2k$ two-spheres attached
at its basepoint. The module $\pi_2 D$ has a specified hyperbolic basis
over $\Z H$, realised by framed immersed spheres with vanishing
quadratic self-intersection obstructions.
\item Attach a three-cell along each basis sphere. The resulting space
$P$ has $(P,S)$ an oriented Poincar\'e pair of formal dimension four,
and $P \simeq BH$. The boundary marking is $c|_S \colon S \to BH$.
\item There is no compact oriented topological four-manifold $D'$ with
boundary identified with $S$, and a homotopy equivalence $c'\colon D'
\to BH$ whose restriction to $S$ is homotopic to $c|_S$.
\end{enumerate}
\end{proposition}
Sphere classes and intersections below use fixed whiskers and deck
translations. A hyperbolic pair has zero squares, mutual intersection
$1$, and vanishing framing and Wall self-intersection obstructions. We
prove the proposition without using embedded surgery on $D$.
\subsection{The geometric and tensor inputs}
\label{ch:inputs}
We use the marked geometric construction of \cite[Section~2]{TO}
together with its cut obstruction. The marked bodies and exterior duals
come from \cite[Lemmas 2.3 and 2.4]{TO}. The finite tensor theorem is
\cite[Theorem 3.1]{TO}. The vertex data is
\cite[Proposition 6.1]{TO}. Edge algebraicity and formal comparison are
\cite[Theorem 5.1 and Proposition 5.2]{TO}. The cut consequences of
\cite[Proposition 2.2 and Lemmas 2.5--2.6]{TO} are the conditional
conclusions under hypothetical disk replacements, and are distinct
from the marked geometric construction itself.
\textbf{The marked body.}
Fix $m \geq 5$ and $p \geq 9$. Start with a four-dimensional
one-handlebody on $2m + p$ handles. Attach cancelling two-handles to
the first $2m$ handles with product framings, and plumb them in $m$
pairs, interchanging the two coordinate directions at each plumbing.
The resulting smooth oriented manifold $X$ has graph homotopy type.
Its cover associated to the plumbing quotient $\pi_1 X \to F_m$ is a
tree of blocks, each a one-handlebody on $p$ extra generators. Neighbouring
blocks meet in a product four-ball. The exposed boundary of a block is
\[
K_v = \#^p(S^1 \times S^2) \setminus \operatorname{int}
\nu(L_{2m}),
\]
where $L_{2m}$ is an unlink in a ball. Port meridians
$x_1, \dots, x_{2m}$ and extra loops $l_1, \dots, l_p$ form a free
basis at $v$. At each join, the meridian on one side is the longitude
on the other.
Write $\mathcal C = \{1, l_1, \dots, l_p\}$. The marked-body
construction installs, for every ordered triple of distinct ports and
every independent choice
\[
X_a = c_a x_{r_a}^{\delta_a} c_a^{-1}, \qquad
c_a \in \mathcal C, \quad \delta_a \in \{1, -1\}, \quad a = 1, 2, 3,
\]
a disjoint two-stage grope body. Its lower surface is a punctured torus.
One branch has a single cap; the other carries the upper punctured torus
with its two caps. All tip tracks, conjugating paths, and product
framings are specified.
The initial boundary word $R$ satisfies
\begin{equation}
\label{ch:laws}
R|_{X_a = 1} = 1 \quad (a = 1, 2, 3), \qquad
R \equiv [X_1, [X_2, X_3]] \pmod{\Gamma_4 F(X_1, X_2, X_3)}.
\end{equation}
Internal changes of the marked paths are allowed by these identities. The
\emph{entire finite family} of laws must be retained; it is not
reducible to a single conjugated commutator.
\textbf{The exterior with duals.}
Let $F_1, \dots, F_k$ be the lower punctured tori, with symplectic bases
$(a_j, b_j)$, where $a_j$ is the single-cap branch. Delete their open
product tubes, including the proper edge collars, to obtain $E$.
Lemma~2.4 of \cite{TO} supplies truncated single caps $f_j$ and framed
spheres $g_j$ in $E$ with
\begin{equation}
\label{ch:exterior-duals}
\lambda(f_i, g_j) = \delta_{ij}, \qquad
\lambda(g_i, g_j) = 0, \qquad
\widetilde\mu(g_j) = 0 \quad \text{over } \Z[\pi_1 E].
\end{equation}
The framing on $f_j$ is the surface-compression framing. The $g_j$ are
compressed normal-circle tori over the $b_j$ curves; their individual
representatives have trivial normal bundle. Equation
\eqref{ch:exterior-duals} holds as an identity in the group ring,
before any meridians are killed.
\textbf{The cut obstruction.}
For use in Section~\ref{ch:forbidden-filling}, we recall exactly how
the cut obstruction is applied. Suppose a compact smooth oriented
four-manifold with the marked graph type of the parity cover of $X$ has
two connected vertex pieces $A, B$, with outside faces $K_A, K_B$, and
$2m$ connected three-dimensional cuts $Y_i$, each with its prescribed
coordinate torus as its entire boundary. Orient $Y_i$ as a face of $A$,
so as a face of $B$ it inherits the opposite orientation.
Suppose the differences of restrictions give an isomorphism in degree
two, and, in degree one, an isomorphism after the extra-loop evaluations
are set to zero. Suppose further that precisely $m$ joins are active:
at an active join, the surviving coordinate meridian is at $A$,
whereas the other coordinate is the meridian at $B$. Require every
marked law \eqref{ch:laws} in its respective vertex group, and require
the free ambient background: port meridians trivial and named extra
letters independently assignable.
\textbf{Why this is impossible.}
These data cannot exist. Proposition~6.1 (Vertex data) of \cite{TO}
produces a characteristic-zero field $K$ (one may take
$\mathbb C((\sigma))$), separate coordinate blocks $h_i$ of total
dimension $N$, and algebraic power-series germs $F_i(h_i)$, $b_i(h_i)$,
with $b_i$ indexed by the active set $I$ of size $|I| = m$. The germs
are normalized $F_i(0) = b_i(0) = 0$.
\begin{equation}
\label{ch:tensor-ring}
T = K[t_i : i \in I]/(t_i^2 : i \in I), \qquad
S_0 = \sum_i F_i, \qquad
S_t = S_0 + \sum_{i \in I} t_i b_i.
\end{equation}
There are formal parametrizations
\[
H_L(x, t) \in (x, t)T[[x]]^N, \qquad
H_R(y) \in (y)K[[y]]^N,
\]
with $x$ carrying $d$ coordinates and $y$ carrying $N - d$ coordinates.
The central tangent maps $\partial_x H_L(0, 0)$ and $\partial_y H_R(0)$
are injective with complementary images in $K^N$; nilpotent constant
displacements of $H_L$ are allowed. They satisfy
\begin{equation}
\label{ch:tensor-vanishing}
S_t(H_L) = 0, \qquad S_0(H_R) = 0.
\end{equation}
For every three distinct active indices $i, j, k$,
\begin{align}
t_i t_j t_k &\in
\bigl((\partial_{h_\nu} S_t)(H_L) : \nu = 1, \dots, N\bigr)
\subset T[[x]], \label{ch:tensor-left}\\
(b_i b_j b_k)(H_R) &\in
\bigl((\partial_{h_\nu} S_0)(H_R) : \nu = 1, \dots, N\bigr)
\subset K[[y]]. \label{ch:tensor-right}
\end{align}
Theorem~3.1 (Finite tensor obstruction) of \cite{TO} rules out
equations \eqref{ch:tensor-ring}--\eqref{ch:tensor-right} for $m \geq 5$.
Two points are used repeatedly below. The memberships are in the
\emph{actual} gradient ideals, not their radicals. And $T$ retains all
mixed square-free monomials. The independent conjugators in the full
law list are precisely what supply the triple identities.
These three inputs (the marked bodies, the vertex data giving the
tensor identities, and the finite tensor theorem), together with the
marked edge potentials, algebraicity, and the formal comparison of
\cite[Proposition 4.5, Theorem 5.1, Proposition 5.2]{TO}, form the
tensor obstruction used throughout.
\subsection{The exterior with its side marking}
\label{ch:exterior}
The imported geometry gives the surfaces and their algebraic duals. We
now assemble $D$ from that data, keeping careful track of the actual
tube-side maps. This marking later identifies the boundary of a
hypothetical filling with the loop data needed for the tensor
contradiction.
We use one placement refinement of the marked-body construction. Place
its longitudinal tip annuli on disjoint parallels outside sufficiently
thin attaching solid tori in the initial one-handlebody boundary, and
put the finitely many connectors in their complement. This is possible
by the marked handlebody model: the bodies are boundary joins of the
separated annular collars. The complement of the attaching solid tori
survives in the boundary after the two-handles are attached; the
plumbing ports are on the new belt-handle portions. Thus the unpushed
lower surfaces lie in the final exposed boundary of $X$.
The independent ribbon twists matching the cap framings are made in the
separated parallel tubes. They do not pass a body across a plumbing
join. The returning tip collars alone are lengthened back to product
parallels on the attaching regions and then into the cores. The
construction and intersection calculation of
\eqref{ch:exterior-duals} are unchanged.
Consequently the $F_j$ are proper inward pushes of disjoint boundary
surfaces, with returning collars at their initial boundary curves.
Moreover $F_j \to X$ is null-homotopic. The three tip loops bound the
corresponding core caps in $X$, hence the bottom $a_j$ and $b_j$ loops
die in $\pi_1 X$. Since $X$ has graph homotopy type and $F_j$ retracts
to a graph, the whole surface map is null-homotopic.
\begin{lemma}[The marked exterior model]
\label{ch:exterior-model}
With the surface-product trivializations, the exterior and its
tube-side maps satisfy the homotopy-pushout model
\[
E \simeq X \cup_{\coprod_j F_j}
\coprod_j (F_j \times S^1),
\]
where each gluing map to $F_j \times S^1$ is the inward section. The
displayed $F_j \times S^1$ factors represent the actual tube sides.
\end{lemma}
\begin{proof}
It suffices to describe a single surface, since the constructions have
disjoint collars. Write $B = \partial X$ and $K = \partial F$. Before
rounding corners, place the pushed surface at a horizontal collar
level, with vertical walls along $K$ returning to the boundary. The
inner part of the exterior has homotopy type $X$, the outer part has
homotopy type $B \setminus K$, and their overlap has homotopy type
$B \setminus F$. Product slices, thickened by a variable positive width
off the deleted surface, give this homotopy-pushout decomposition.
In $B \setminus K$ take a two-sided neighbourhood of
$F^\circ = F \setminus K$ whose width decreases to zero near $K$.
Relative to $B \setminus F$ it is a bridge over $F^\circ$, joining the
two sides of the cut. Replace $F^\circ$ in this diagram by a slightly
shrunken copy $F_0$; this is an objectwise homotopy equivalence. The
actual deleted surface and $K$ are unchanged.
In rectangular normal coordinates to the horizontal copy, split the
tube circle over $F_0$ into its outer and inward semicircles. The outer
semicircle is the bridge. The inward semicircle lies in the inner $X$
piece and gives a specified homotopy between the bridge's endpoint maps.
Using that homotopy to identify the endpoints replaces the bridge by
$F_0 \times S^1$, attached along its inward section. Its circle is
precisely the outer bridge followed by the inward return, hence the
actual normal circuit. Finally $F_0 \to F$ and
$F_0 \times S^1 \to F \times S^1$ are homotopy equivalences. This also
accounts for the proper edge collars and confirms the assertion about
the side maps, not merely the absolute homotopy type of $E$.
\end{proof}
For each $j$, regard a solid torus $V_j$ as a compression body from
the punctured torus $F_j$ to a disk, compressing $a_j$. Define
\[
D = E \cup_{\coprod_j (F_j \times S^1)}
\coprod_j (V_j \times S^1).
\]
All gluings use the specified product framings; round corners. The
remaining disk face of $\partial V_j$ contributes a solid torus to the
new boundary. Hence $S = \partial D$ is the surface-framed surgery of
$\partial X$ on the initial grope-boundary link. In particular it is
connected.
By Lemma~\ref{ch:exterior-model},
\[
D \simeq X \cup_{\coprod_j F_j} \coprod_j (V_j \times S^1).
\]
Each $F_j \to X$ is null-homotopic. The extra summand is therefore the
homotopy cofiber of $F_j \to V_j \times S^1$. Retract the latter space
to the torus on its longitude $b_j$ and normal meridian $t_j$. The
cofiber adds one two-cell killing $b_j$ and one on the null loop $a_j$.
Cancel the $b_j$ one/two-cell pair. The torus two-cell then has null
attaching map, so the result is
\[
S^1_{t_j} \vee S^2 \vee S^2.
\]
Thus $H = \pi_1 D$ is the original free graph group with one new free
meridian $t_j$ per surface, and $D$ has the graph-and-spheres homotopy
type in Proposition~\ref{ch:chamber}.
\subsection{The middle form and the Poincar\'e pair}
\label{ch:duality}
Close $f_j$ by a meridian disk of $V_j$ at its fixed circle coordinate,
obtaining a sphere $u_j$. The two compression framings agree, so its
second Stiefel--Whitney evaluation is zero. The $g_j$ remain in $E$;
therefore \eqref{ch:exterior-duals} gives
\[
\lambda(u_i, g_j) = \delta_{ij}, \qquad
\lambda(g_i, g_j) = 0 \quad \text{over } \Z H.
\]
The hermitian matrix of the $u_i$ has even diagonal: as $H$ has no
element of order two, every such diagonal is $z + \bar z$, and one can
choose one representative from each nonidentity inverse pair and halve
the even identity coefficient. Add suitable combinations of the $g_j$
to the $u_i$, taking a triangular half off the diagonal and this half
on it. The resulting spheres, denoted $a_i$, together with $b_i = g_i$,
have hyperbolic intersection matrix.
The usual framed representative adjustments give zero Wall
self-intersections: choose normal Euler number zero using the evenness,
and then use
$\lambda(a_i, a_i) = \mu(a_i) + \overline{\mu(a_i)}$
and the absence of order-two elements. This is the ordinary
immersed-sphere calculus of \cite{FQ}; no disk-embedding assertion
enters it.
These $2k$ classes form a basis. Indeed, $\pi_2 D$ is already known
to be free of rank $2k$. Its square matrix of the displayed classes
has a left inverse from their nonsingular pairing. To see directly
that it also has a right inverse, pass to each finite quotient of $H$.
The resulting matrices act on finite-dimensional complex vector spaces,
where a left inverse is a right inverse. Residual finiteness of the free
group separates the finite support of any nonzero group-ring element, so
the right-inverse identity holds over $\Z H$ itself. We henceforth use
this specified basis.
Attach a three-cell on every $a_i$ and $b_i$ to form $P$. A map from
the graph with $2k$ two-spheres to $D$ realising this basis is a
homotopy equivalence: it induces an isomorphism on $\pi_1$ and on the
homology of simply connected covers. Thus $P \simeq BH$.
We check Poincar\'e duality for the pair. Set $\Lambda = \Z H$. In
the duality of $(D, S)$, the middle sphere module maps isomorphically
onto a split summand of $H_2(D, S; \Lambda)$ by its nonsingular form.
Attaching the three-cells quotients both absolute and relative
homology by these middle summands. The attaching map is injective on
them, so no new third homology appears. On cohomology, the pullbacks
identify the new groups with the annihilators of the sphere summands.
If $f \colon (D, S) \to (P, S)$ is the natural cell-attachment
inclusion and $[P, S] = f_* [D, S]$, cap-product naturality reads
\[
z \frown [P, S] = f_*\bigl(f^* z \frown [D, S]\bigr).
\]
Hence the old duality isomorphisms induce exactly the new ones on these
annihilators and quotients. The spaces have finite free chain complexes
over $\Lambda$, so the regular-coefficient cap-product criterion gives
Poincar\'e duality with arbitrary local coefficients. This proves
parts (1) and (2) of Proposition~\ref{ch:chamber}.
\begin{corollary}
\label{ch:boundary-homology}
The map $\pi_1 S \to H$ is onto. If $r = \operatorname{rank} H$, then
$H_1(S; \Z) \to H_1(P; \Z)$ is an isomorphism and $H_2(S; \Z) \cong
\Z^r$.
\end{corollary}
\begin{proof}
Duality and the graph type of $P$ give
$H_1(P, S; \Z H) = H_0(P, S; \Z H) = 0$. Therefore
$H_0(S; \Z H) \to H_0(P; \Z H)$ is an isomorphism. The former is free
abelian on the cosets of the image of $\pi_1 S$, and the latter is
$\Z$; there can be only one coset.
With ordinary coefficients, the pair sequence and duality give
$H_2(P, S) = H_1(P, S) = 0$ and $H_3(P, S) \cong H^1(P) \cong \Z^r$.
The assertions follow.
\end{proof}
\subsection{Topological fillings and the cut obstruction}
\label{ch:forbidden-filling}
Parts (1) and (2) of Proposition~\ref{ch:chamber} are established. The
remaining part (3) is that no topological filling of $S$ extends the
marking. We show that any such filling would produce exactly the cut
configuration excluded in Section~\ref{ch:inputs}.
\textbf{The filling as a manifold with disk boundaries.}
Suppose a marked filling $D'$ as in part (3) of
Proposition~\ref{ch:chamber} existed. Attach the dual two-handles that
undo the surgery on the initial link. Denote the result by $W$, so
$\partial W = \partial X$.
Each attaching circle represents its prescribed new free generator
$t_j$ under the marking $D' \simeq BH$. The corresponding one/two-cell
pair cancels in the homotopy type. Thus $W$ has the graph type of $X$,
with the original boundary-to-graph map. Equality of the marked
fundamental-group maps gives equality up to homotopy because the graph
is aspherical.
The two-handle cocores are disjoint proper locally flat disks bounded
by the original initial link. In particular, all of the original marked
words are now represented by disk boundaries.
\textbf{The marked tree cut.}
Use the marked quotient $\pi_1 W \to F_m$ and its associated tree cover.
Prescribe the equivariant map to be constant at the appropriate vertex
on each outside face $K_v$ and each lifted disk neighbourhood, and to
cross the appropriate edge on each boundary torus collar. These
prescriptions agree where they meet.
A section of the associated bundle with contractible tree fibre
extends them to the rest of $W$. The free boundary extra loops, with
whiskers, generate the fundamental group of this tree cover, just as for
the marked graph model of $X$.
\textbf{Handling a topological base.}
We have allowed $W$ to be topological. By Quinn's punctured smoothing
\cite[Corollary~2.2.3]{QuinnEnds}, delete an interior point away from
the disks and extend the given boundary smoothing to the complement.
Prescribe the map to be constant near this point before extending the
section. Approximate it smoothly over the open middle intervals of the
finitely many graph edges, away from the disks and the puncture, and
choose regular values. The cuts are compact smooth three-manifolds,
disjoint from the disks. Their filled complementary pieces are
topological four-manifolds, smooth away from the finite set of lifted
punctures in each finite quotient.
\textbf{Connectedness of the cuts.}
Connectedness is not automatic from a regular-level construction.
Form the component graph of the cuts in the tree cover. Its fundamental
group is a quotient of that of the cover, by joining successive
crossings through connected complementary pieces. Every extra boundary
generator lies in one piece, so maps trivially to this graph. It follows
that the component graph is a tree.
The outside faces and torus collars give an embedded principal copy of
the original plumbing tree: different vertex labels lie in different
components of the complement of the target midpoints. Any remaining
branch has a unique nearest principal vertex. Its stabilizer is
trivial, since it fixes that vertex. The quotient has finitely many
components; hence these branches have finite valence and uniformly
bounded depth, and each is finite.
Ignore their closed cuts and merge them into their principal vertices.
This leaves connected vertex pieces and one connected cut with each
prescribed torus as its entire boundary. The disk prescriptions have
not changed.
This is precisely the argument of Lemma~2.5 (Connected tree cuts) of
\cite{TO}, now applied to the marked $W$ rather than requiring a
homeomorphism $W = X$.
\textbf{The parity quotient.}
Pass to the parity quotient, sending every plumbing generator to the
nonidentity element of $\Z/2$. Its filled ambient space $W_2$ has the
homotopy type of the graph with two vertices and $2m$ edges, and $p$
extra circles at each vertex. Write $A, B$ for the filled vertex pieces
and $Y_1, \dots, Y_{2m}$ for their cuts. Over any characteristic-zero
field $K$, Mayer--Vietoris gives
\begin{equation}
\label{ch:restriction-two}
H^2(A; K) \oplus H^2(B; K) \xrightarrow{\ \cong\ }
\bigoplus_{i=1}^{2m} H^2(Y_i; K),
\end{equation}
because $H^2(W_2; K) = H^3(W_2; K) = 0$.
In degree one, the difference of restrictions is onto. Its kernel is
the image of $H^1(W_2; K)$ modulo the $2m - 1$ graph classes, and
evaluation on the $p$ named extra loops at each vertex identifies that
kernel with $K^{2p}$. Thus the restriction map becomes an isomorphism
on the subspace where these extra evaluations vanish.
\textbf{Active edges from half-lives/half-dies.}
On a cut $Y_i$, half-lives/half-dies says that the image of
$H_1(\partial Y_i; K)$ in $H_1(Y_i; K)$ is a line. If both original
coordinate loops had nonzero image, their images would be nonzero
multiples. A class from either vertex annihilates the longitude
bounding in its outside face, and would hence annihilate both
coordinates. The degree-one surjectivity just established would then
force every class on $Y_i$ to do so, a contradiction.
Therefore one original coordinate $\beta_i$ dies and the other
$\alpha_i$ survives. The plus endpoint is the side at which $\beta_i$
is a longitude. The parity involution pairs the two lifts of each join
and exchanges their plus endpoints, giving exactly $m$ active edges
at $A$.
\textbf{Laws at the vertex pieces.}
The prescribed disks lie in their own vertex pieces. They therefore
give every law \eqref{ch:laws} at those vertices, with all the
independently chosen conjugators and signs.
The marked ambient free group still has trivial port meridians and
freely assignable named extra letters. In particular it supplies the
common flat background used in the Vertex data Proposition: assign
eight extra letters the eight nonidentity matrices among
$\exp(a \sigma e) \exp(b \sigma f)$, $a, b \in \{0, 1, 2\}$, for the
standard nilpotent generators $e, f$ of $\mathfrak{sl}_2$. Together
with the identity, their adjoint operators span all endomorphisms over
$\mathbb C((\sigma))$, as in \cite[Proposition 6.1]{TO}.
Thus the geometric and marked hypotheses of the tensor interface in
Section~\ref{ch:inputs} have all been established.
\textbf{Punctured smoothness of the vertices.}
It remains to justify that the vertex calculation works for the
punctured smooth structures we actually have. We give this step rather
than assuming that the filled vertices admit smooth structures.
\begin{lemma}[Forms on a punctured vertex]
\label{ch:puncture-forms}
Let $V$ be one of the filled compact vertex pieces and let $Z$ be its
finite set of punctures. In degree zero take smooth functions on
$V \setminus Z$ separately constant near each puncture, and in positive
degrees take smooth forms vanishing near the punctures. This
differential graded algebra computes $H^*(V; \mathbb C)$, naturally on
the unpunctured faces. It supports the vertex calculation in the Vertex
data Proposition of \cite{TO}, with the filled cohomology groups and
with the same Stokes identities.
\end{lemma}
\begin{proof}
Regard these forms as a sheaf on $V$. Its stalk at a puncture is
$\mathbb C$ in degree zero and zero in higher degrees; elsewhere it is
the ordinary de Rham resolution. Partitions of unity can be chosen
constant on smaller neighbourhoods of the finitely many punctures. The
complex is therefore a fine resolution of the constant sheaf, so
computes the filled cohomology. A based condition in degree zero
removes only $H^0$.
Linear splittings onto cohomology and contracting homotopies may
consequently be chosen inside this complex, exactly as for the ordinary
vertex de Rham complex. Wedges, Lie brackets, the curvature recursions,
and differentiation in the formal parameters preserve this algebra.
Each fixed positive-degree coefficient vanishes near $Z$, so
integration and Stokes take place on a compact smooth subdomain
containing its support; the additional end boundaries contribute zero.
An entire formal series need not have one common vanishing
neighbourhood, but every calculation at a fixed coefficient uses only
finitely many terms. The source's formal lifts and divisions likewise
use finite linear combinations coefficientwise, and extensions from
face collars can be supported away from $Z$.
Removing interior points in dimension four does not change the
fundamental group. At each finite nilpotent order the flat systems in
question vanish near the punctures in their local trivializations;
gauges between them can be separately constant at those ends. Thus the
based holonomy and exact-gauge arguments use the same marked group as
the filled vertex. Allowing separate constants at distinct ends is
important here. These are precisely the uses of vertex smoothness in
the proof of Proposition~6.1 of \cite{TO}.
The edge-algebraicity theorem of \cite{TO},
\cite[Theorem 5.1]{TO}, is still applied only to the unchanged compact
smooth three-manifolds $Y_i$.
\end{proof}
\textbf{Endgame.}
Apply Lemma~\ref{ch:puncture-forms} to the two vertex pieces.
Equation~\eqref{ch:restriction-two} and the degree-one calculations
refer to their filled cohomology, which the lemma computes. The Vertex
data Proposition, \cite[Proposition 6.1]{TO}, now produces exactly
equations \eqref{ch:tensor-ring}--\eqref{ch:tensor-right}. The Finite
tensor obstruction contradicts their existence. Therefore $D'$ cannot
exist, completing the proof of Proposition~\ref{ch:chamber}.
\section{Reflection and the duality group}
\label{rf:section}
We now close the chamber without losing its labelled middle-dimensional
classes. The tool is Davis's reflection method
\cite[Sections~13 and~15, especially Remark~15.9]{Davis1983}.
A second construction will give the resulting group a proper cocompact
action on a CAT(0) space. The two developments are different: the first
retains the four-dimensional chambers, whereas the second is used only
to access assembly theorems.
The input from Proposition~\ref{ch:chamber} is a compact connected
oriented smooth four-manifold $D$, with connected boundary $S$ and free
fundamental group $H$, together with a map of pairs
\[
(D, S) \longrightarrow (P, S).
\]
Here $D$ has the homotopy type of a finite graph with finitely many
two-spheres, its $\Z H$-intersection form has a specified framed
hyperbolic basis of $k$ planes, and $P$ is obtained by attaching
three-cells along all $2k$ basis spheres. The pair $(P, S)$ is an
oriented Poincar\'e pair of dimension four, and $P \simeq BH$. The map
on fundamental groups is the given identification with $H$. We use a
collar on $S$ in both spaces, inserting a mapping cylinder on the $P$
side if necessary.
By Corollary~\ref{ch:boundary-homology}, the boundary map
$\pi_1(S) \to H$ is surjective. In particular its associated cover of
$S$ is connected.
\subsection{Panels and finite developments}
\label{rf:panels}
Choose a finite flag triangulation $\mathcal L$ of $S$, for example a
barycentric subdivision of a smooth triangulation. Write $V$ for its
vertex set and $S_s$ for the closed dual block of $s \in V$ in the
barycentric subdivision. These blocks are three-dimensional panels.
For nonempty $T \subseteq V$, the intersection $S_T = \bigcap_{s \in T} S_s$
is nonempty exactly when $T$ spans a simplex of $\mathcal L$; in that
case it is a ball of dimension $4 - |T|$. Collar neighbourhoods of these
intersections have the usual orthant structure.
Let
\[
C = \bigl\langle s \in V : s^2 = 1, \ st = ts \text{ whenever }
\{s, t\} \in \mathcal L \bigr\rangle
\]
be the right-angled Coxeter group. For a collared space $B$ with these
panels in its boundary, its development is
\[
\mathcal U(C, B) = (C \times B)/\sim, \qquad
(c, x) \sim (c', x) \Longleftrightarrow c^{-1} c' \in C_{T(x)},
\]
where $T(x) = \{s : x \in S_s\}$ is empty off the boundary and $C_T$
is the subgroup generated by $T$. Each chamber embeds, since the
equivalence relation never identifies different values of $x$. For
nonempty $T(x)$, the subgroup $C_{T(x)}$ is $(\Z/2)^{|T(x)|}$, and its
sectors fill the corresponding orthant neighbourhood.
We use three elementary Coxeter word facts: special subgroups embed;
a finite special coset has a unique shortest element; and the right
descent set $\operatorname{Des}(c) = \{s \in V : |cs| < |c|\}$ is a
clique. These follow from the right-angled reduced-word rule of
commuting adjacent commuting letters and cancelling equal consecutive
letters; see also \cite[Chapters~4--5]{Davis}.
\begin{lemma}[Labelled finite developments]
\label{rf:shelling}
Put $Y = \mathcal U(C, D)$ and let $D_c$ be its chamber with label $c$.
Order $C$ by nondecreasing word length, breaking ties arbitrarily.
Every finite initial segment containing the identity is, after rounding
its collar corners, an iterated boundary connected sum of its chambers.
In particular,
\[
U_L = \bigcup_{|c| \leq L} D_c
\]
is a compact manifold with boundary the connected sum of the
corresponding copies of $S$. Its fundamental group is the free product
of their chamber groups. The same statements hold if any chambers have
been stabilized in their interiors.
\end{lemma}
\begin{proof}
The part of a new chamber $D_c$ already present is exactly
\begin{equation}\label{rf:incoming}
A_c = \bigcup_{s \in \operatorname{Des}(c)} S_s.
\end{equation}
For possible equal-length overlaps, consider a point with panel set
$T$. Its incident chambers form the special coset $c C_T$. Let $a$ be
its shortest element. Its members have the form $a \prod_{s \in J} s$
with $J \subseteq T$, and length $|a| + |J|$. If a distinct incident
chamber occurs no later than $c$, then $c$ is not the shortest member,
so it has a descent in $T$. This places the point in \eqref{rf:incoming}.
The converse follows from the shorter adjacent chamber across each
descent panel.
The set in \eqref{rf:incoming} is a three-ball. More generally, a union
of dual panels indexed by a nonempty simplex is a ball: add the panels
one at a time; a new panel is a ball, and its intersection with the
preceding panels is the corresponding union of dual panels in its
boundary link. Induction on dimension makes that intersection a ball
of one lower dimension. The dual-block coordinates supply the product
collars needed to glue the two balls.
It remains to check that $A_c$ is a ball in the \emph{boundary} of the
preceding union. Contractibility alone would not establish the claim.
At a face incident to $q$ panels, label the $2^q$ local orthants by
subsets of those panels, starting at the shortest incident chamber.
The already present orthants form a Boolean downset $I$: every proper
subset of a present subset has smaller word length. This remains true
even with arbitrary choices within a length tie.
When $I$ is nonempty and proper, its frontier is a PL hypersurface.
For an explicit description, write local coordinates as
$x = u + t(1, \dots, 1)$ with $\sum u_i = 0$, and label an orthant by
its positive coordinates. Its union is described by
\[
t \leq f(u), \qquad
f(u) = \max_{A \text{ maximal in } I}\ \min_{i \notin A} (-u_i).
\]
The function $f$ is finite, continuous, and piecewise linear, so the
frontier is locally a graph. Along the incoming faces some sectors
are present and the new chamber sector is absent, so these faces lie
on this frontier. The panel identifications are PL, and the embedded
incoming three-ball has locally flat boundary in this three-manifold
frontier. Gluing the new chamber along it is therefore a boundary
connected sum after rounding.
Induction proves the manifold and boundary assertions. Van Kampen gives
the labelled free product, and interior stabilizations do not affect
any collar argument.
\end{proof}
\subsection{The universal development}
\label{rf:universal}
Let
\[
\Delta = \ker\bigl(C \longrightarrow (\Z/2)^V\bigr),
\]
where each generator maps to its own coordinate vector, and define
\[
N = \Delta\backslash\mathcal U(C, D), \qquad
Q = \Delta\backslash\mathcal U(C, P).
\]
The parity map is injective on every finite special subgroup. Thus
$\Delta$ has trivial intersection with every point stabilizer in either
development. The quotient has finitely many chambers.
For $D$ the orthant charts show that $N$ is a closed four-manifold.
Give chamber $c$ the orientation $(-1)^{|c|}$ times the chosen chamber
orientation. Adjacent chamber orientations fit across a panel, and
$\Delta$ preserves them. Thus $N$ is oriented. The same convention
orients the reflected Poincar\'e pairs.
Let $\widehat S \to S$ be the connected cover determined by
$\pi_1(S) \to H$, with lifted triangulation $\widehat{\mathcal L}$. This
is a flag triangulation: the vertices of a lifted clique project to
distinct pairwise adjacent vertices downstairs; they span a simplex
there; lifts of its edges and triangular relations place all the
vertices in one lift of that simplex. Denote by $\widehat C$ the
right-angled Coxeter group on $\widehat{\mathcal L}$. The deck action
of $H$ permutes its generators. Forgetting the lift gives an
$H$-invariant homomorphism $q \colon \widehat C \to C$, and hence a
surjection
\begin{equation}
\label{rf:group}
\rho \colon \widehat C \rtimes H \longrightarrow C, \qquad
\rho(\widehat c, h) = q(\widehat c), \qquad
G = \rho^{-1}(\Delta).
\end{equation}
Let $\widetilde P$ be the universal cover of $P$, with boundary preimage
$\widehat S$ and its lifted panels. On
$\widehat Q = \mathcal U(\widehat C, \widetilde P)$, let
$(\widehat c, h)$ act by
\[
(\widehat c, h)[\widehat d, x] = [\widehat c \, h(\widehat d), h x].
\]
The map $[\widehat d, x] \mapsto [q(\widehat d), \overline x]$ descends
to the ordinary development. On a chamber interior it is the usual
covering map. At a boundary point it is also a covering map: the lifted
panel set maps bijectively to its downstairs panel set, and the
corresponding finite special groups map isomorphically. These local
charts identify $\widehat Q / G$ with $Q$.
The action of $G$ is free. If an element fixes a point, its
$H$-component fixes the chamber coordinate and is therefore trivial
after conjugating the chamber label; the remaining stabilizer is a
lifted finite special group, on which $\rho$ is injective and whose
image misses $\Delta$.
The space $\widehat Q$ is contractible. Filter its chambers by word
length in $\widehat C$. The first chamber $\widetilde P$ is
contractible. Every further chamber meets the shorter ones in the
nonempty ball of its descent panels, by the argument of
Lemma~\ref{rf:shelling}. Its attachment is therefore a homotopy
equivalence. In an infinite word-length layer, new interiors are
disjoint and their mutual intersections already lie in shorter layers.
All these attachments may be performed simultaneously. The CW direct
limit is weakly contractible and hence contractible. In particular
$Q$ is a finite $K(G, 1)$.
\begin{proposition}[The reflected pair]
\label{rf:development}
The space $Q$ is a finite oriented Poincar\'e complex of dimension
four. The natural map $f \colon N \to Q$ has degree one and induces an
isomorphism on fundamental groups, identifying both with the group $G$
of \eqref{rf:group}. If $b$ is the total number of chamber hyperbolic
planes in $N$, then
\[
\pi_2(N) \cong (\Z G)^{2b}
\]
has the specified orthogonal hyperbolic basis, and
$\widetilde H_i(\widetilde N; \Z) = 0$ for $i \neq 2$. The group $G$
is finitely presented, is torsion-free, and satisfies
\begin{equation}
\label{rf:pd}
H^i(G; \Z G) = 0 \quad (i \neq 4), \qquad
H^4(G; \Z G) \cong \Z
\end{equation}
with trivial right $G$-action.
\end{proposition}
\begin{proof}
Use the same universal development with $\widetilde D$ in place of
$\widetilde P$. The covering argument is unchanged. The first chamber
is simply connected; attaching subsequent simply connected chambers
along balls shows that this development is the universal cover
$\widetilde N$. Its reduced homology is the direct sum of the chamber
reduced homologies, each concentrated in degree two.
There are finitely many chamber orbits in $N$. For each one, induction
from the chamber subgroup gives
\[
\Z G \otimes_{\Z H} \pi_2(D) \cong (\Z G)^{2k}.
\]
The spheres lie in chamber interiors, so different chambers are
orthogonal. Within a chamber the form is its given $\Z H$-form,
extended to $\Z G$. Reversing one vector in a plane absorbs any chamber
orientation sign. This yields the asserted basis and form.
The construction of $P$ attaches exactly the indicated three-cells
away from the boundary. Consequently $Q$ is obtained from $N$ by
attaching three-cells on this entire middle basis. It follows at once
that $f$ induces the displayed identification of fundamental groups.
These attachments split off a nonsingular middle summand in both
absolute and relative duality: on homology they quotient by the sphere
summand and on cohomology they restrict to its annihilator. With
$[Q] = f_* [N]$, naturality of the cap product identifies the surviving
duality maps with those of $Q$. The finite free chain criterion
therefore gives Poincar\'e duality for $Q$, just as for the chamber
pair. Equivalently one may glue the chamber duality maps across the
manifold collar sectors, where the orientations cancel in pairs. The
top fundamental class is unchanged, so $f$ has degree one.
For clarity, the regular-coefficient calculation is particularly short.
In the cohomology sequence for the three-cell attachments, the only
potentially new map is the evaluation
\[
H^2(N; \Z G) \longrightarrow
\operatorname{Hom}_{\Z G}(\pi_2(N), \Z G).
\]
Under duality it is the adjoint of the nonsingular hyperbolic form, and
hence an isomorphism. The middle cohomology thus disappears, whereas
$H^4(Q; \Z G) = H^4(N; \Z G) = \Z$ and all other degrees vanish. The
right action on the top group is trivial because the orientation chosen
above is preserved by $G$.
Finally, finiteness and asphericity of $Q$ give a finite presentation
and a finite free resolution of $\Z$ from the chains of $\widehat Q$.
They also give finite integral cohomological dimension, which excludes
torsion: restriction to a nontrivial finite cyclic subgroup would give
a finite projective resolution for that group, contrary to its
nonvanishing cohomology in arbitrarily high degrees. This proves all
the assertions, including \eqref{rf:pd}.
\end{proof}
\subsection{A proper cocompact CAT(0) model}
\label{rf:assembly}
The lifted nerve has infinitely many vertices when $H$ is infinite, so
its Davis complex alone need not give a cocompact action of the
semidirect product. The following support construction fixes this.
\begin{lemma}
\label{rf:cat0}
The group $\widehat C \rtimes H$, and hence its finite-index subgroup
$G$, acts properly and cocompactly by isometries on a proper CAT(0)
cube complex of dimension at most five.
\end{lemma}
\begin{proof}
Choose a locally finite tree $T$ on which the free group $H$ acts
freely and cocompactly, without inversions. For the trivial group one
may take a single vertex. Assign to each lifted-nerve vertex $s$ a
point $a_s \in T^{(0)}$, equivariantly under $H$. There are finitely
many vertex and edge orbits in the lifted nerve, so the distances
$d_T(a_s, a_t)$ for adjacent $s, t$ have a common finite bound. Choose
an integer $r$ large enough that the closed radius-$r$ subtrees $T_s$
about $a_s$ intersect for every such pair.
For a vertex $v$ and an edge $e$ of $T$, put
\[
A_v = \{s : v \in T_s\}, \qquad
A_e = \{s : e \subset T_s\}.
\]
These sets have uniformly bounded finite size: there are finitely many
generator orbits, and a fixed-radius ball in the free cocompact tree
contains uniformly finitely many orbit points. Let $W_v$ and $W_e$ be
the special subgroups of $\widehat C$ generated by $A_v$ and $A_e$.
For $e = [v, w]$ we have $A_e = A_v \cap A_w$, so the edge maps are
special subgroup inclusions.
The graph of groups over the tree $T$ has colimit $\widehat C$.
Occurrences of a generator $s$ are identified precisely over its
connected support $T_s$. Every defining commutation of $\widehat C$
occurs in some vertex group, because the two supports intersect at a
vertex. Conversely, each vertex group uses the induced subgraph of
the lifted nerve, and so introduces no additional relation. The
resulting presentation is exactly the right-angled presentation of
$\widehat C$. All edge maps are injective, as are their extensions
to inclusions of special subgroups in $\widehat C$.
Let $\Sigma_v$ and $\Sigma_e$ be the Davis cube complexes of these
finite-generator right-angled groups. They are CAT(0), and the special
subcomplexes $\Sigma_e \subset \Sigma_v$ are convex in the standard
piecewise Euclidean cube metric. The CAT(0) assertion is
\cite[Theorem~12.2.1(i)]{Davis}; special-subgroup metric convexity is
\cite[Appendix, Proposition~A.2]{Swiatkowski}.
Form a tree of spaces with vertex spaces
$\widehat C \times_{W_v} \Sigma_v$ and edge cylinders
$\widehat C \times_{W_e} (\Sigma_e \times [0, 1])$. Its underlying
tree is the Bass--Serre tree, whose vertices and edges are the cosets
$\widehat c W_v$ and $\widehat c W_e$, with their indicated labels in
$T$; it is not simply the original support tree. Glue each cylinder at
its endpoints by the special-subcomplex maps. Convex gluing gives
CAT(0) over every finite subtree, preserving the factors as convex
isometric subspaces
\cite[Theorem~II.11.1 and Remark~II.11.2]{BH}. Apply the gluing first
over finite subtrees and then pass to the union; the local finiteness
verified next ensures completeness.
There is no local-finiteness problem from potentially infinite
Bass--Serre valence. In a fixed vertex Davis complex, for any one
incident support edge, translates of its special subcomplex have
vertex sets equal to the cosets of $W_e$ in $W_v$. These cosets
partition the Cayley vertices. A finite subcomplex therefore meets
only finitely many of these translates. Each support vertex has only
finitely many incident edges. Since each $\Sigma_v$ is locally finite,
the whole tree of cube complexes is locally finite. Its cubes have
unit edge lengths and uniformly bounded dimension, so the induced
length metric is proper and complete.
Left multiplication gives the $\widehat C$-action. Equivariance of
the supports extends this to $\widehat C \rtimes H$. An element
stabilising a vertex space projects to an element of $H$ fixing its
support vertex, and that element is trivial. Within the vertex space
its point stabiliser is therefore a finite Coxeter stabiliser. The
same reasoning applies to edge interiors. The cellular action on this
locally finite complex is proper.
There are finitely many $H$-orbits of support vertices and edges, and
each corresponding Davis complex has compact quotient by its special
group. Hence the full action is cocompact. Each clique of the lifted
nerve has at most four vertices; the vertex spaces have dimension at
most four and the edge cylinders dimension at most five. Finally $G$
has finite index by \eqref{rf:group}, so its restricted action has all
the same properties.
\end{proof}
It follows that $G$ satisfies the $K$- and $L$-theoretic
Farrell--Jones conjectures with coefficients: use Bartels--L\"uck
\cite[Theorem~B(ii)]{BL} and Wegner
\cite[Theorems~1.1 and~3.4]{Wegner}. In particular, because $G$ is
torsion-free, the Whitehead group, reduced $K_0(\Z G)$, and negative
$K$-groups vanish, and the simple $L$-theory assembly map is an
isomorphism in every degree; see \cite[Proposition~0.3(i)]{BL}. These
are the assembly consequences used below. No four-dimensional rigidity
or disk-embedding conclusion is being inferred from CAT(0) geometry.
\subsection{The gallery map}
We record the geometric map needed in the truncation argument. The
preceding CAT(0) model supplies assembly; the map here instead records
positions and directions in the ordinary chamber development. Let
$\Sigma_C$ denote the ordinary Davis cube complex of $C$. Its vertices
are the chamber labels $C$, and the link of the identity vertex is
$\mathcal L$, whose realisation is $S$.
\begin{lemma}
\label{rf:davis-map}
There is a $C$-equivariant continuous map
\[
\eta \colon Y \longrightarrow \Sigma_C
\]
with the following properties. It sends the inner core of each chamber
to its label, respects the panels, and has uniformly bounded image
diameter on each chamber. On the boundary collar of the central
chamber its radial directions at the identity vertex give the
dual-block identification with $S$. After interior chamber
stabilizations, the same construction gives a continuous map with the
same panel, core, and diameter properties; it is equivariant under any
subgroup preserving the stabilized pattern.
\end{lemma}
\begin{proof}
The Davis chamber is the cone on the barycentric subdivision of
$\mathcal L$, with its usual dual panels on the base. Identify that
base panelwise with $S$. Map a boundary collar of $D$ to this cone by
that identification on its outer boundary and by coning to the apex on
its inner boundary; map the remainder of $D$ to the apex. The maps
agree across reflected panels and hence define $\eta$.
Every chamber image lies in the corresponding compact Davis chamber,
whose diameter is independent of its label. At the identity apex,
radial projection of the cone base to a sufficiently small metric
sphere gives exactly its link identification with $S$. Stabilization
changes only the collapsed inner core, so does not affect the
construction.
\end{proof}
\section{Marked stable realization}\label{st:setup}
We now suppose, for a contradiction, that the group constructed above
is the fundamental group of a closed aspherical topological
four-manifold $M$. The preceding construction gives a finite oriented
Poincar\'e four-complex $Q = BG$ and a closed oriented manifold $N$.
Write $q$ for the degree-one map of Proposition~\ref{rf:development}:
\[
q \colon N \longrightarrow Q.
\]
The map $q$ is an isomorphism on fundamental groups. The only reduced
homology of the universal cover of $N$ is its second homology. With
$\Lambda = \Z G$, this is the free module
\begin{equation}\label{st:module}
A = \pi_2(N) \cong \Lambda^{2b}, \qquad
\lambda_N \cong \bigoplus_{a=1}^{b}
\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}.
\end{equation}
The displayed basis consists of the prescribed chamber sphere pairs.
All identifications include whiskers and orientations. Moreover, $G$
is torsion-free, is an integral $\mathrm{PD}_4$ group, and satisfies
the $K$- and $L$-theoretic Farrell--Jones conjectures by the CAT(0)
construction. These are conclusions of the construction, not
additional assumptions on a putative realization.
Both $Q$ and $M$ are $K(G, 1)$ spaces. Their equivalence identifies
the orientation character with the action on the top dualizing module.
Consequently $M$ is orientable. Choose its orientation and its
fundamental-group marking so that its classifying map
$c_M \colon M \to Q$ has degree one, and put
\[
L = M \mathbin{\#} b(S^2 \times S^2).
\]
Collapsing the simply connected summands gives a degree-one classifying
map $c_L \colon L \to Q$. Identify $\pi_2(L)$ with $A$ by sending each
specified chamber pair in \eqref{st:module} to a specified standard pair
in $L$. This is an isometry of equivariant forms, and we must retain
this particular isometry throughout.
\subsection{An integral marked homotopy equivalence}
We first realize the prescribed module isometry by a homotopy
equivalence. The comparison takes place in a common second Postnikov
stage: group duality splits that stage, and integral cup-product tests
identify the two fundamental classes there. These are the inputs to
the lifting argument below.
\begin{lemma}\label{st:second-stage}
The second Postnikov stages of $N$ and $L$, with the preceding
markings, identify over $Q$ with the split stage
\[
B = EG \times_G K(A, 2).
\]
There are maps $u_N \colon N \to B$ and $u_L \colon L \to B$ inducing
the specified identifications of fundamental and second homotopy
groups. If $\Gamma(A) = H_4(K(A, 2); \Z)$, then
\begin{equation}\label{st:homology-extension}
0 \longrightarrow \Gamma(A)_G \longrightarrow H_4(B; \Z)
\longrightarrow H_4(Q; \Z) \longrightarrow 0
\end{equation}
is a split exact sequence.
\end{lemma}
\begin{proof}
Since $A$ is finite free over $\Lambda$, integral group duality gives
\begin{equation}\label{st:cohomology-vanishing}
H^2(G; A) = H^3(G; A) = 0.
\end{equation}
The second Postnikov invariant lies in the latter group, so it
vanishes. The former removes the degree-two ambiguity in making the
identification with the split stage for the prescribed module marking.
We use a functorial Eilenberg--Mac Lane model for $K(A, 2)$, so the
given action on $A$ defines the displayed homotopy quotient and its
section.
Separately, freeness gives $H_i(G; A) = 0$ for $i > 0$. In total
degree four, the homology spectral sequence of
\[
K(A, 2) \longrightarrow B \longrightarrow Q
\]
therefore has only $H_4(G; \Z)$ and $H_0(G; \Gamma(A))$ as possibly
nonzero terms: the intervening term $H_2(G; A)$ is zero, and the fibre
has no homology in degrees one and three. The possible incoming
differentials to $H_0(G; \Gamma(A))$ have sources $H_3(G; A) = 0$ and
$H_5(G; \Z) = 0$. The section preserves the base term. This proves
\eqref{st:homology-extension}, including its integral, rather than
merely rational, exactness.
\end{proof}
The next elementary check is useful because the middle classes are
specified over the group ring. Ordinary intersection forms would lose
the relative translates needed here.
\begin{lemma}\label{st:quadratic-detection}
With the identifications above,
\begin{equation}\label{st:fundamental-equality}
(u_N)_*[N] = (u_L)_*[L] \quad \text{in } H_4(B; \Z).
\end{equation}
\end{lemma}
\begin{proof}
Write $e_1, \dots, e_{2b}$ for the chosen $\Lambda$-basis of $A$ and
$\phi_i \colon A \to \Lambda$ for the coordinate functionals. The
low-degree cohomology sequence of the Postnikov fibration, with regular
coefficients, contains
\[
0 \longrightarrow H^2(G; \Lambda) \longrightarrow H^2(B; \Lambda)
\longrightarrow \operatorname{Hom}_\Lambda(A, \Lambda)
\longrightarrow H^3(G; \Lambda).
\]
Both outside groups vanish. Thus each $\phi_i$ is the restriction of
a unique class $x_i \in H^2(B; \Lambda)$.
Their cup products may be evaluated without multiplying away either
coefficient factor:
\begin{equation}\label{st:cup-tests}
z \longmapsto \langle x_i \smile x_j, z \rangle
\in (\Lambda \otimes_{\Z} \Lambda)_G,
\qquad z \in H_4(B; \Z).
\end{equation}
The tensor product has the diagonal left action. Its coinvariants
identify, as an abelian group, with $\Lambda$ via
$g \otimes h \mapsto g^{-1} h$. For a closed oriented manifold with
the indicated middle module, evaluations in \eqref{st:cup-tests} are
the equivariant cup form on the coordinate functionals. By
Poincar\'e duality this form is dual to the equivariant intersection
pairing. The prescribed isometry therefore makes these evaluations
equal on $(u_N)_*[N]$ and $(u_L)_*[L]$.
For completeness, these tests detect the fibre term integrally. The
underlying abelian group of $A$ is free on the elements $g e_i$. The
group $\Gamma(A)$ has a basis consisting of one square generator for
each such element and one cross generator for each unordered pair of
distinct such elements. The action of $G$ permutes this basis. Hence
its coinvariants are free abelian on the corresponding orbits. A
nonidentity element stabilizing an unordered pair would exchange its
two members and have order two; torsion-freeness rules this out.
The square orbit represented by $e_i$ evaluates to $1$ in the $(i, i)$
test. For $i < j$, the cross orbit represented by $\{e_i, t e_j\}$
evaluates to $t$ in the $(i, j)$ test. Finally, for $t \neq 1$, the
orbit represented by $\{e_i, t e_i\}$ evaluates to $t + t^{-1}$ in the
$(i, i)$ test. These last orbits are indexed by $t$ modulo inversion.
Since $G$ has no elements of order two, their supports are disjoint
two-element subsets of $G \setminus \{1\}$. The displayed tests are
therefore jointly injective on $\Gamma(A)_G$. In particular, no
integral square or hidden two-torsion is discarded.
The two fundamental classes have the same image $[Q]$ in
$H_4(Q; \Z)$. By Lemma~\ref{st:second-stage}, their difference lies in
$\Gamma(A)_G$. All its cup tests vanish, so the difference is zero.
\end{proof}
\begin{proposition}\label{st:marked-equivalence}
There is an orientation-preserving homotopy equivalence
\[
f \colon N \longrightarrow L
\]
inducing the specified isomorphism on $\pi_1$ and the prescribed
isometry $A \to \pi_2(L)$.
\end{proposition}
\begin{proof}
Replace $u_L \colon L \to B$ by a fibration. Its fibre is
two-connected, with third homotopy group $\pi_3(L)$ carrying the usual
$G$-action. The first obstruction to lifting $u_N$ is consequently the
pullback of a universal class
\[
\kappa \in H^4(B; \pi_3(L)).
\]
The pullback $u_L^* \kappa$ is zero, since $u_L$ lifts to its own
fibration replacement. Naturality of the cap product and
Lemma~\ref{st:quadratic-detection} give
\begin{align*}
(u_N)_*\bigl((u_N^* \kappa) \mathbin{\cap} [N]\bigr)
&= \kappa \mathbin{\cap} (u_N)_*[N]\\
&= \kappa \mathbin{\cap} (u_L)_*[L] = 0
\quad \text{in } H_0(B; \pi_3(L)).
\end{align*}
The map $u_N$ induces an isomorphism on fundamental groups, and hence
on these degree-zero coefficient groups. Poincar\'e duality identifies
\[
H^4(N; \pi_3(L)) \xrightarrow{\ \cap [N]\ }
H_0(N; \pi_3(L)) \cong \pi_3(L)_G.
\]
Thus the lifting obstruction vanishes.
The manifold $N$ has a finite four-dimensional CW structure from the
smooth chamber construction and the panel collars. There are no higher
obstructions on this source. No triangulation or smooth structure on
the hypothetical $M$ is required. The resulting lift $f$ has the
stipulated maps on $\pi_1$ and $\pi_2$. The universal covers of $N$
and $L$ have reduced homology only in degree two, where the lift is an
isomorphism. These manifolds have CW homotopy type
\cite[Corollary~1]{MilnorCW}. The homological Whitehead theorem on
their simply connected universal covers, followed by Whitehead's
theorem, shows that $f$ is a homotopy equivalence
\cite[Corollary~4.33 and Theorem~4.5]{Hatcher}.
Finally $c_L f \simeq q$ and both classifying maps have degree one, so
$f_*[N] = [L]$.
\end{proof}
\subsection{From the marked equivalence to a marked stable product}
The marked homotopy equivalence is now available. The next two steps
pass through an $s$-cobordism and then a stabilized product, retaining
the same identification of every original sphere class. The product
trace also specifies the classes introduced by stabilization.
\begin{lemma}\label{st:marked-bordism}
There is an $s$-cobordism $(W; N, L)$ whose two end inclusions
identify $\pi_1$ and $\pi_2$ by the marking of
Proposition~\ref{st:marked-equivalence}.
\end{lemma}
\begin{proof}
The universal intersection forms are even, so $N$ and $L$ are almost
spin. Indeed, evaluation of $w_2$ on every spherical class is its
self-intersection modulo two, and on the simply connected cover these
evaluations detect $w_2$.
We apply the proof of Kasprowski--Land's Theorem~1.1, specifically
Lemma~2.3, Lemma~3.3 and Theorem~3.1 of~\cite{KL}. All hypotheses
hold: $Q$ is an aspherical Poincar\'e four-complex; $G$ satisfies both
Farrell--Jones conjectures; the two manifolds have the same orientation
character and degree-one classifying maps; and they are almost spin.
Their normal-invariant calculation, using assembly injectivity and the
Atiyah--Hirzebruch spectral sequence, identifies the normal invariant
of this particular $f$ with a class in
\[
\ker\bigl(H_2(L; \Z/2) \longrightarrow H_2(Q; \Z/2)\bigr).
\]
The Serre spectral sequence for the universal-cover fibration shows
that this kernel is represented by spherical classes. Since $L$ is
almost spin, their $w_2$ evaluations vanish. Lemma~3.3 of~\cite{KL}
therefore supplies a target self-equivalence $\phi$ with the same
normal invariant as $f$, of the form
\[
L \longrightarrow L \vee S^4
\xrightarrow{\operatorname{id} \vee \eta^2} L \vee S^2
\xrightarrow{\operatorname{id} + a} L.
\]
Here $a \colon S^2 \to L$ represents the spherical class. This pinch
acts identically on $\pi_1$ and $\pi_2$, because its extra part
factors through the four-sphere. Thus $\phi^{-1} \circ f$ has trivial
normal invariant and the same marking as $f$. Surgery of its normal
bordism over $L$, as in Theorem~3.1 of~\cite{KL}, gives the required
$s$-cobordism. Neither this conclusion nor the cited theorem requires
$G$ to be a good group. We have not asserted that $W$ itself is a
product.
\end{proof}
We use the stable product theorem for this particular bordism.
Quinn's Theorem~1.1 in~\cite{QuinnStable}, in the form stated in
\cite[Remark~3.7]{KPR}, says that a five-dimensional topological
$s$-cobordism becomes a product relative to one end after finitely
many connected sums along arcs with $(S^2 \times S^2) \times [0, 1]$.
There is no restriction on its fundamental group. The relative
product statement is important: an arbitrary stable homeomorphism of
the ends would not, by itself, retain the specified sphere classes.
\begin{proposition}\label{st:stable-product}
For some integer $k \geq 0$ there is a homeomorphism
\[
h \colon N \mathbin{\#} k(S^2 \times S^2)
\longrightarrow L \mathbin{\#} k(S^2 \times S^2)
\]
which induces the prescribed original marking and matches every
added standard pair with its specified added standard pair. In
particular, on the left, all the prescribed original hyperbolic pairs
and all new standard pairs have simultaneous geometric representatives
with pairwise disjoint punctured $S^2 \times S^2$ neighborhoods.
\end{proposition}
\begin{proof}
Apply the stable product theorem to $W$ from
Lemma~\ref{st:marked-bordism}. Denote its stabilized ends by $N^*$ and
$L^*$ and the stabilized bordism by $W^*$. We verify its marking,
including the new summands.
The operation removes disjoint normal $D^4 \times [0, 1]$
neighborhoods of arcs between the ends, and glues
\[
\bigl((S^2 \times S^2) \setminus \operatorname{int} D^4\bigr)
\times [0, 1]
\]
along the resulting $S^3 \times [0, 1]$ interfaces. Removing these
codimension-four arcs changes neither $\pi_1$ nor the second homology
of the universal cover. Mayer--Vietoris across the lifted interfaces
therefore gives a specified decomposition
\begin{equation}\label{st:bordism-splitting}
\pi_2(W^*) \cong \pi_2(W) \oplus \Lambda^{2k}.
\end{equation}
The second summand is represented by the standard spheres in the added
product pieces. Choose their whiskers along the stabilization arcs.
The two end inclusions $j_N \colon N^* \to W^*$ and
$j_L \colon L^* \to W^*$ carry each new pair to this same pair, while
their old summands map to $\pi_2(W)$ with exactly the original bordism
identification.
A product structure $F \colon N^* \times [0, 1] \to W^*$ which is the
identity on the incoming end restricts at the other end to a
homeomorphism $h \colon N^* \to L^*$. Use the product trace for the
outgoing basepoint path. Any discrepancy from the original bordism
path can be corrected by pushing the outgoing basepoint around a loop
in $L^*$, since $j_L$ is an isomorphism on fundamental groups. The
resulting based homotopy given by $F$ implies
\[
(j_L)_* h_* = (j_N)_*.
\]
Both inclusions are isomorphisms on $\pi_1$ and $\pi_2$. Thus
$h_* = (j_L)_*^{-1} (j_N)_*$, with the same base paths. In the
decomposition \eqref{st:bordism-splitting} it preserves the original
marking and identifies the new standard basis element by element.
There is no arbitrary automorphism mixing the old and new summands.
All $b + k$ standard pairs on
$L^* = M \mathbin{\#} (b + k)(S^2 \times S^2)$ have disjoint punctured
$S^2 \times S^2$ neighborhoods. Give the two spheres of each pair the
common whisker to their intersection used for its module marking, and
pull the whole based neighborhood back by $h$. A pair has its one
prescribed transverse intersection; different pairs and their
neighborhoods are disjoint. The marking just proved identifies their
based homotopy classes with the original chamber basis and the added
standard basis, respectively.
\end{proof}
\subsection{Periodically lifted geometric data}
Let $C$ be the right-angled Coxeter group used in the reflection
construction, and let $\Delta \leq C$ be the finite-index torsion-free
normal subgroup used to obtain $N$ from its ordinary reflection
development. Place the finitely many stabilization balls on the
$N$-side inside chamber interiors. This may be done by moving their
endpoints along paths before performing the arc connected sums. Track
the new whiskers along those paths. There is no requirement that
different chambers receive equal numbers of stabilizations.
Lift the resulting $N^*$ to the ordinary reflection development and
call the lifted manifold $Y$. Its chambers, indexed by $v \in C$, are
stabilized copies $D_v$ of $D$. The numbers of stabilizations are
$\Delta$-periodic. Gallery distance is the word metric on the chamber
indices for the standard Coxeter generators.
\begin{proposition}\label{st:periodic-data}
The manifold $Y$ has two $\Delta$-periodic systems of sphere pairs,
with the same chamber labels.
\begin{enumerate}
\item Framed immersed hyperbolic pairs in the interiors of the
chambers $D_v$, including the added standard pairs, with their
equivariant products already correct over the individual chamber group.
\item Geometrically standard pairs with pairwise disjoint punctured
$S^2 \times S^2$ neighborhoods, based-homotopic sphere by sphere to
the correspondingly labelled pairs in the first system.
\end{enumerate}
There is an integer $d_0$ such that the drawing of each pair, its
standard neighborhood, and the homotopies relating the two systems
are contained in the chambers at gallery distance at most $d_0$ from
that pair's label. Basings in this statement are local basings
transported with the chamber; no bound is asserted for whiskers drawn
from one global base point to all chambers.
\end{proposition}
\begin{proof}
The original immersed representatives stay in their chamber
interiors, and the stabilization spheres can be placed in their
interior connected summands. Stabilization adds orthogonal standard
planes and changes neither the chamber fundamental group nor the old
pairings.
Proposition~\ref{st:stable-product} supplies a finite collection of
geometric pairs on the compact quotient $N^*$, representing the
specified based classes. Choose a based homotopy for each sphere
between its two representatives. Lift the neighborhoods, sphere
drawings, and homotopies to $Y$, using the prescribed marking to
choose the lift for each chamber label. Covering maps preserve $\pi_2$
and lift these based homotopies, so they end at precisely the labelled
geometric representatives. Disjoint embedded neighborhoods lift to
disjoint embedded neighborhoods.
There are only finitely many labelled data on the quotient. Each
chosen lift of a drawing, closed standard neighborhood, local basing,
or homotopy has compact image. Local finiteness of the development
implies that it meets only finitely many chambers. Take $d_0$ to be
the maximum gallery distance from its assigned label over this finite
list of chosen lifts. All remaining data are translates by $\Delta$,
whose action preserves gallery distance. The same $d_0$ therefore
works everywhere.
\end{proof}
This is the stable information needed later. It gives actual geometric
summands with their original chamber labels and compactly supported
homotopies to the chamber systems. It makes no claim of controlled
isotopy, and no claim that the unstabilized chamber obstruction has
disappeared.
\section{Controlled removal of the middle homology}\label{ct:section}
Suppose, towards a contradiction, that the group constructed above is
the fundamental group of a closed aspherical topological four-manifold.
We use the marked systems supplied by
Proposition~\ref{st:periodic-data}. Thus, after finitely many
stabilizations on the compact quotient, the ordinary reflection
development is a manifold
\[
Y = \bigcup_{v \in C} D_v.
\]
Here $D_v$ is a stabilized copy of the chamber, its boundary is $S$,
and its group is the free group $H$. Boundary identifications are
understood with the signs induced by the orientation of $Y$; signs in
hyperbolic planes are absorbed by changing one basis vector. Two
labelled systems live in $Y$: the chamber-wise immersed hyperbolic
bases, denoted $\mathcal A$, and disjoint standard hyperbolic pairs,
denoted $\mathcal B$. Each pair of $\mathcal B$ has an embedded
neighbourhood homeomorphic to
$(S^2 \times S^2) \setminus \operatorname{int} D^4$. Corresponding
spheres in the two systems are based-homotopic. All of these data,
including the homotopies, are lifts of finitely many compact data on
the quotient.
We will remove an expanding finite part of $\mathcal B$ and use the
remaining part of $\mathcal A$ to perform surgery. The latter requires
geometric duals. We obtain them by applying the proof of the disk
embedding theorem to a specially constructed null-loop input; we do
not assume $H$ is a good group.
There are three tasks. First, cap the noncentral boundary factors and
remove the inner standard pairs, leaving a compact manifold with the
original boundary marking. Next, prepare the remaining sphere systems
and place their upper grope branches over contractible direction
patches. This supplies the null-loop input. Finally, the tower
construction and sphere surgery remove the remaining middle homology
and produce a graph-type filling.
\subsection{Complementary fillings and the peeling operation}
Write $r = \operatorname{rank} H$. A stabilization of $D$ has homotopy
type
\[
\bigvee^r S^1 \vee \bigvee^{2k} S^2
\]
for some $k$, and its middle module has the specified hyperbolic
basis. Neither the boundary nor its map to $BH$ changes under
stabilization.
\begin{lemma}[Complementary fillings]\label{ct:fillings}
For every such stabilized chamber there is a simply connected compact
oriented topological four-manifold $J$ with boundary $-S$ such that
\[
H_2(S; \Z) \longrightarrow H_2(J; \Z)
\quad \text{is an isomorphism}, \qquad
H_3(J; \Z) = 0.
\]
Moreover, $D \cup_S J$ is simply connected, and inclusion of $D$
identifies its second homology and intersection form with those of
$D \cup_S J$.
\end{lemma}
\begin{proof}
First recall the consequences of the graph-type Poincar\'e pair
$(P, S)$ recorded in Corollary~\ref{ch:boundary-homology}. Duality and
its ordinary homology sequence give
\begin{equation}\label{ct:boundary-homology}
H_1(S; \Z) \cong H_1(P; \Z) \cong \Z^r, \qquad
H_2(S; \Z) \cong H^1(P; \Z) \cong \Z^r.
\end{equation}
The same corollary gives surjectivity of $\pi_1(S) \to H$.
Start with the oppositely oriented copy of $D$. Represent a free basis
of its fundamental group by disjoint embedded circles. Their normal
bundles are trivial, and general position makes the circles disjoint
from the finite sphere system. Interior surgery on these circles kills
the free generators. The old sphere classes retain their hyperbolic
intersection and framing data after augmentation to $\Z$. In the now
simply connected manifold, the sphere embedding theorem with geometric
duals realises a Lagrangian by disjoint framed embedded spheres.
Surgery on these spheres preserves simple connectivity and removes the
old hyperbolic planes. This use of the disc theorem is in the simply
connected case only; see \cite[Theorem~B]{PRT}.
Let $J$ be the result. Each circle surgery increases Euler
characteristic by two, and each sphere surgery decreases it by two.
Hence
\[
\chi(J) = (1 - r + 2k) + 2r - 2k = 1 + r.
\]
The boundary is connected and $J$ is simply connected. Thus
$H^1(J, S; \Z) = 0$ by the cohomology sequence of the pair, and
Poincar\'e--Lefschetz duality gives $H_3(J; \Z) = 0$. Since $J$ has
nonempty boundary, $H_4(J; \Z) = 0$, and therefore $b_2(J) = r$.
There is no integral index ambiguity in this calculation. The
relevant part of the duality sequence is
\begin{equation}\label{ct:cap-sequence}
0 \longrightarrow H_2(S; \Z) \longrightarrow H_2(J; \Z)
\longrightarrow H^2(J; \Z) \longrightarrow H_1(S; \Z)
\longrightarrow 0.
\end{equation}
Here $H_3(J, S; \Z) = H^1(J; \Z) = 0$ accounts for the initial zero.
The universal coefficient theorem makes $H^2(J; \Z)$ free, because
$H_1(J; \Z) = 0$. Every torsion element of $H_2(J; \Z)$ would
consequently belong to the image of the injected free group
$H_2(S; \Z)$, so there is no such element. The first two nonzero
groups in \eqref{ct:cap-sequence} now have the same rank $r$. The
middle map has rank zero and takes values in a free group, hence is
zero. Exactness proves that the first map is an integral isomorphism.
The map $H_2(S; \Z) \to H_2(D; \Z)$ is zero: the next map
$H_2(D; \Z) \to H_2(D, S; \Z)$ is the nonsingular hyperbolic
intersection map. Also $H_1(S; \Z) \to H_1(D; \Z)$ is the isomorphism
in \eqref{ct:boundary-homology}. Mayer--Vietoris, together with the
isomorphism just proved for $J$, therefore identifies
$H_2(D \cup_S J; \Z)$ with $H_2(D; \Z)$. All products are unchanged,
since their representatives lie in the interior of $D$. Finally, the
surjectivity of $\pi_1(S) \to H$ and van Kampen show that
$D \cup_S J$ is simply connected.
\end{proof}
Let $|v|$ denote Coxeter word length. Lemma~\ref{rf:shelling}
identifies the finite chamber union
\[
U_L = \bigcup_{|v| \leq L} D_v
\]
with an iterated boundary connected sum, with the natural markings on
its boundary factors. Fill every noncentral boundary factor by the
corresponding $J_v$ of Lemma~\ref{ct:fillings}. The resulting manifold
$U_L^+$ has boundary $S$ and admits the description
\begin{equation}\label{ct:filled-sum}
U_L^+ \cong D_e \mathbin{\#}
\mathop{\#}_{0 < |v| \leq L}(D_v \cup_S J_v).
\end{equation}
To see this description, retain the connecting three-balls in the
boundary-sum construction. Gluing a punctured filling to a noncentral
factor closes that factor away from its connecting ball and turns the
connecting boundary neck into an interior connected-sum neck.
Repeating this one factor at a time gives \eqref{ct:filled-sum}.
It follows that $\pi_1(U_L^+) = H$, with the central boundary marking.
On $\pi_1(U_L)$ the inclusion is the projection of the free product of
the chamber groups onto its central factor. The universal cover of
$U_L^+$ has reduced homology only in degree two, freely based over
$\Z H$ by all the chamber sphere systems, including stabilizations.
Its intersection form is their orthogonal hyperbolic sum. These facts
also follow directly by lifting the connected-sum necks in
\eqref{ct:filled-sum}; all the closed noncentral factors are simply
connected and have precisely the middle classes computed above.
For an integer $R$, replace every standard neighborhood in $\mathcal B$
whose label belongs to a chamber $|v| \leq R$ by a four-ball. Call
this operation \emph{peeling}. The common boundary is a standard
$S^3$. There is a continuous pinch map from each old neighborhood to
the new ball, fixed on that boundary, and these maps combine with the
identity outside to a global pinch map. Both pieces and the common
boundary are simply connected, so peeling preserves the fundamental
group. The relative homology sequence deletes exactly the hyperbolic
pair carried by that neighborhood and introduces no third homology.
We first perform this operation in the unbounded $Y$, obtaining $Y_R$.
For $L$ large enough to contain the selected neighborhoods and all
their marking homotopies, it also defines a compact manifold
\[
Z = Z_{R, L}
\]
by peeling $U_L^+$. The order of filling and peeling is immaterial
because the standard neighborhoods are in the interior. The included
homotopies identify the removed summands with exactly the chamber pairs
labelled by $|v| \leq R$. We have therefore proved
\begin{equation}\label{ct:Z-homology}
\partial Z = S, \qquad \pi_1 Z = H, \qquad
\widetilde H_i(\widetilde Z; \Z) = 0 \quad (i \neq 2),
\end{equation}
and $H_2(\widetilde Z; \Z)$ is freely based by the projected chamber
pairs with $R < |v| \leq L$, with their orthogonal hyperbolic form.
\subsection{Uniform finite preparation}
A compact object is \emph{carried within $b$ galleries of $v$} if its
image is contained in chambers whose labels are within word distance
$b$ of $v$. Replacement balls are assigned the carriers of their old
standard neighborhoods. We use this terminology for sphere maps,
paths, homotopies, and grope systems, including all their caps. A
uniform propagation bound is one value of $b$ valid for the whole
labelled family. It is stronger than a bound after an unspecified
permutation of the labels.
\begin{lemma}[Finite preparation]\label{ct:preparation}
Fix a positive integer $h$. In $Y_R$, the projected pairs of
$\mathcal A$ labelled by $|v| > R$ admit the following preparation,
with propagation bounded independently of $R$.
\begin{enumerate}
\item The two halves are represented by framed immersed spheres
$a_i, b_i$ with $a_i \cap b_j$ consisting of the designated single
point for $i = j$ and empty for $i \neq j$.
\item The remaining intersections among the $a_i$ are paired by
framed Whitney discs. Each Whitney disc is replaced by a properly
immersed capped grope of height at least $h$ with the same framed
attaching region.
\item These gropes have the Clifford dual capped surfaces used in
\cite[Definition~3.1 and Lemma~4.1]{PRT}. In particular, their caps
miss the \emph{entire} dual capped surfaces, and the bodies of the
latter lie in a boundary collar of the complement of the $a_i$.
\end{enumerate}
The construction uses no complementary filling $J_v$. Beyond a fixed
gallery distance from the peeling region, it can be carried out
separately inside individual chambers.
\end{lemma}
\begin{proof}
First project the outer immersed systems through the pinch and put
them in general position. The deleted and retained blocks were
orthogonal, since their classes agree with the correspondingly
labelled standard pairs. The retained forms and quadratic
self-intersection data are therefore still hyperbolic over
$\Z \pi_1(Y_R)$. Choose framed immersed representatives and cancel
the algebraic intersection pairs. The framing corrections and, where
necessary, local cusp changes can be chosen from finitely many local
drawings. The group in question is a free product of free chamber
groups, hence is torsion-free; there is no order-two ambiguity in
these quadratic data.
\textbf{Uniformity of the Whitney discs.}
We justify carefully the uniformity required at this and subsequent
finite stages. The initial lifted data have finitely many types under
a cocompact group of translations of $Y$. A gallery ball of fixed
radius contains a uniformly bounded number of chambers and of pieces
of those data. There are thus finitely many local templates, even
after specifying which of the standard neighborhoods meeting that
ball have been peeled. Paths on participating spheres are taken
inside their bounded carriers. Common long whiskers conjugate an
intersection label but need not be included in the geometric Whitney
circuit.
If a circuit enters a replacement ball, move its portions there to
chosen arcs on the boundary three-sphere. This gives a circuit in the
unpeeled space. Nullity is unchanged, since both the deleted
punctured $S^2 \times S^2$ and the replacement ball are simply
connected and have simply connected boundary. For every null circuit
among the finitely many resulting templates, choose one compact
filling in $Y$. There is a finite maximum of their propagation bounds.
Translate the chosen filling to the location of the circuit and apply
the actual global pinch map. This constructs a filling in $Y_R$.
A filling may encounter additional peeled neighborhoods. No further
choice of fillings is then needed: the global pinch is already defined
on them, and each such neighborhood has the same bounded carrier.
Consequently this procedure is uniform in $R$. It is a finite-template
argument for the specified circuits, not a bound for arbitrary null
loops. Repeating a fixed finite number of stages preserves the
argument: enlarge the carrier radius and refine the finite template
list at each stage.
\textbf{Geometric Casson lemma.}
Apply the geometric Casson lemma to the two halves of the sphere
system, removing paired mixed intersections by regular homotopies.
There is no iterative repair cascade here. Use
\cite[Lemma~3.3]{PRT} with the two surface arguments taken to be the
entire possibly disconnected families. Preselect all framed pairing
Whitney discs with disjoint boundaries; their interiors miss the other
Whitney boundary arcs by $2 + 1 < 4$ general position. First push the
first family off all these interiors by fingers along arcs in the
Whitney discs. Choose the arcs away from their finitely many
intersections with other Whitney discs and the second family. The
finger supports then leave the other Whitney discs and attaching
boundaries fixed. Next Whitney-move the second family using the same
discs. The new intersections in each phase are only within that
phase's family; intersections between Whitney interiors cause only
allowed second-family intersections. Thus the whole batch stays in the
original bounded Whitney neighborhoods. No later disc is transported
or chosen again through an expanding sequence of carriers. Local
finiteness permits these bounded batches on the unbounded $Y_R$.
This produces the geometric duality in (1), while permitting
self-intersections within either half. The remaining double points of
the first half admit framed Whitney discs. Tube their interiors off
the first-half spheres using the geometric duals. All the Whitney
circuits involved are the null circuits just discussed.
\textbf{Clifford caps: algebraic input.}
For completeness, we verify the algebraic input to the grope
preparation in the complement $M_0$ of small plumbed neighborhoods of
the $a_i$. The punctured geometric dual $b_i$ bounds an $a_i$-meridian
in $M_0$. Since these meridians normally generate the kernel,
inclusion induces $\pi_1 M_0 \cong \pi_1 Y_R$. Thus the calculation
retains all intersection labels in the required complement group ring.
At one point of every paired double point take its Clifford torus.
Cap its two basis curves by standard meridional discs and tube each
meridional disc off its incident $a_i$-sheet into a punctured parallel
of $b_i$, with the product framing. Denote these caps by $C_\ell$, and
let $i(\ell)$ be the index of the dual used. Their equivariant
intersection calculation is the Clifford-cap calculation in
\cite[Section~5]{PRT}. The local meridional pieces have disjoint
interiors. Choose the thin tubing tracks away from the finitely many
other designated dual points; local changes of tracks introduce only
oppositely signed points with the same group label. The remaining
intersections are those between the corresponding based parallels of
$b_{i(\ell)}$. Hence their sums are obtained from the pairings of the
$b_i$ by the basing units and orientation signs. Likewise, a cap's
self-intersection sum is that of its framed $b_i$ parallel, together
with cancelling local terms. Since the mixed Casson regular homotopies
preserved $\lambda(b_i, b_j) = 0$ and $\mu(b_i) = 0$, this proves
\[
\lambda(C_\ell, C_m) = 0, \qquad \mu(C_\ell) = 0
\quad \text{for every } \ell, m,
\]
including distinct parallel caps using the same dual sphere.
\textbf{Clifford bodies and height raising.}
The Clifford bodies lie in a boundary collar of $M_0$. Following
\cite[Section~5]{PRT}, contract temporary parallel copies and push the
Whitney discs off those contractions before using them as geometric
dual spheres. At the remaining intersections with Whitney discs, tube
the Clifford caps into parallel copies of these temporary duals.
Lemma~4.1 of \cite{PRT} now applies with its full cap-pairing
hypothesis. Its two separation batches and height raising provide the
prepared Clifford copies, whose entire capped surfaces are
geometrically dual in the sense of \cite[Definition~3.1]{PRT}. These
prepared copies, including all their caps, are the protected surfaces
used from this point onward.
The preparation in \cite[Lemma~4.1]{PRT} produces height-two Whitney
gropes, separates their caps from the protected Clifford caps, and
raises the height to $h$. The collar levels and the full dual-cap
disjointness are part of this preparation, not an assertion made after
the caps have been changed. There are exactly two cap-family
separation batches before height raising. Each uses the same
no-new-mixed-pairs invariant, with its protected bodies at fixed
collar levels.
\textbf{Local stages.}
Each operation before height raising involves only a fixed number of
kinds of tubing, push-off, Whitney-disc choice, and regular homotopy.
Their simultaneous use does not require traversing the whole list of
sphere components. At each fixed stage, bounded carriers and finite
drawings bound the number of other carriers meeting a given one.
Divide the operations into finitely many classes with disjoint
enlarged carriers and perform one class at a time. This gives a
number of local stages independent of $R$. New null circuits are
handled by the finite-template procedure above.
\textbf{Height estimate.}
For height raising there is also an explicit size estimate:
\cite[Section~2.8.3]{FQ}, in the January 2013 numbering, bounds the
increase of component-image diameter by a factor of seven at each
step. A fixed requested height therefore has a fixed propagation
bound. One can apply this estimate to the gallery map into $\Sigma_C$:
extend that map over each replacement ball by coning its boundary
image inside a metric ball of uniformly bounded radius. This is
possible by CAT(0) convexity. A point in a chamber is mapped within a
fixed distance of its label, and properness of the Coxeter action
turns a fixed metric bound into a fixed word-distance bound. The
construction uses parallel transverse gropes and neighborhood
operations; it preserves the protected Clifford capped surfaces. All
these operations can be made locally finite in the unbounded
development.
\textbf{Away from the peeling region.}
Finally, outside a fixed buffer of the peeling region the input is
the original system in an individual chamber. Its intersection
cancellations hold already over that chamber's group ring. The same
preparation, including the null discs, can consequently be chosen in
that chamber. Only a bounded buffer of the switch between the peeled
and unpeeled data requires the preceding translated templates. This
proves the last assertion and permits later truncation without
cutting off an unfinished preparation.
\end{proof}
\subsection{Directions at infinity}
Let $\Sigma_C$ be the ordinary Davis complex and let $o$ be its vertex
labelled by $e \in C$. The gallery map of Lemma~\ref{rf:davis-map}
sends the inner part of $D_v$ to $v$, interpolating across the dual
collars. The link of $o$ is $S$. A small metric sphere of radius $t$
about $o$ is consequently identified with $S$ by radial scaling in its
conical star. Write $\rho$ for radial projection onto this sphere,
defined outside the open ball of radius $t$.
\begin{lemma}[Small directions and the central marking]
\label{ct:directions}
Fix a propagation bound $b$. For objects carried within $b$ galleries
of labels $v$ with $|v| \to \infty$, their direction images in $S$
have diameter tending uniformly to zero. This assertion continues to
hold after peeling for the whole region of bounded propagation around
the outer systems, if $R$ is sufficiently large.
Let $\beta \colon S \to BH$ be the central boundary marking. On that
region, the homomorphism induced by the direction map followed by
$\beta$ agrees, up to the common change of base path, with inclusion
into $Z$ followed by its classifying map. In particular, every loop
in an image system whose entire direction image lies in a
contractible patch of $S$ is null-homotopic in $Z$.
\end{lemma}
\begin{proof}
Here is the radial estimate that supplies the first assertion. If
$x, y \in \Sigma_C$ have distance at least $s > t$ from $o$, put them
first on the same metric sphere, of radius the smaller of their two
radii. The distance between the resulting points is at most $2d(x, y)$,
by the triangle inequality and $|d(o, x) - d(o, y)| \leq d(x, y)$.
CAT(0) comparison on the two radial segments then gives
\begin{equation}\label{ct:radial-estimate}
d(\rho(x), \rho(y)) \leq \frac{2t}{s} d(x, y).
\end{equation}
The gallery map takes each chamber into a set of uniformly bounded
diameter. Thus a bounded gallery carrier has bounded image diameter.
Properness of the Davis complex, together with its cocompact Coxeter
action, makes its distance from $o$ tend to infinity with the word
length of its centre. Equation \eqref{ct:radial-estimate} applies.
We only need the direction map outside the central core. Consider a
peeled standard neighborhood that meets an outer object or any of its
fixed enlarged carriers. Both the object and the whole standard
neighborhood have bounded propagation. The label of this neighborhood
therefore has length at least $R$ minus a fixed constant. Its entire
boundary three-sphere, not merely the arcs encountered by the object,
has direction image of arbitrarily small diameter for large $R$.
Choose a slightly larger coordinate ball of $S$ containing this image.
Contracting that ball extends the boundary map across the replacement
four-ball, with image still in the chosen ball. These extensions agree
with the unchanged direction map on all their boundaries.
The near-central peeled neighborhoods need not admit such small
extensions. They are disjoint from all the carriers under
consideration once $R$ is large, and no direction map on them is
used. This proves the asserted far-out extension and its uniform
smallness.
Before peeling, the composition with $\beta$ extends across the
central chamber by its given map to $BH$. Its restriction to the
central boundary agrees with that map: this is the
boundary-direction identification in Lemma~\ref{rf:davis-map}. In a
noncentral chamber the gallery map collapses its inner part to the
noncentral vertex, so every loop of its chamber group has constant
direction there. The shelling description makes the chamber groups the
free factors of the fundamental group of a finite ball of chambers.
The induced homomorphism is thus exactly the projection retaining the
central factor. By \eqref{ct:filled-sum} this is also the homomorphism
induced by passage to $U_L^+$.
Replacing a simply connected piece by a simply connected ball does
not alter this calculation. Loops through the replacement can be
moved to the boundary of the piece, and the direction extension
agrees there with the old map. The comparison therefore holds on the
required region in $Z$. If a whole image system maps into a
contractible patch, every one of its loops, including loops passing
through arbitrarily many intersections, has trivial image under
$\beta$. Since $BH$ is a classifying space and $\pi_1 Z = H$, these
loops are nullhomotopic in $Z$.
\end{proof}
\subsection{Upper grope separation}
We state explicitly how the controlled data supply the input used after
the good-group step of the disc embedding proof. The units to be
separated are the upper capped subgropes attached to individual tips of
the retained stages. Separating only whole Whitney gropes would not
suffice to make the future cap discs mutually disjoint.
\begin{lemma}[Null upper replacements]\label{ct:upper-replacement}
Suppose a finite prepared Whitney-grope system as in
Lemma~\ref{ct:preparation} has at least eight surface stages. Retain
its first five stages. Suppose the upper capped subgropes, of height at
least three, have uniformly small direction images in the region of
Lemma~\ref{ct:directions}. If their diameters are sufficiently small,
depending only on a fixed finite coordinate cover of $S$, the upper
subgropes can be replaced by mutually disjoint framed immersed cap
discs. Their self-intersection loops are nullhomotopic in $Z$. The
retained stages and the entire protected Clifford capped surfaces
remain unchanged.
\end{lemma}
\begin{proof}
We first specify the relative setting. Remove a sufficiently small
regular neighborhood of the five retained stages, truncating each
upper attaching collar at its boundary. The upper components are now
properly immersed disk-like capped gropes of height at least three. By
the duality condition in Lemma~\ref{ct:preparation}, their entire
images miss the protected Clifford capped surfaces. Choose a
neighborhood of the upper system disjoint from those surfaces and from
the retained stages except at the prescribed attaching regions. All
following modifications are neighborhood operations in this relative
manifold. They therefore leave the lower stages, attaching framings,
and whole Clifford systems unchanged, including the unused
boundary-collar levels needed later.
We use the following precise content of Freedman--Quinn's controlled
separation lemma, \cite[Lemma~2.10.1]{FQ}. For a map from a
four-manifold to a locally compact metric space, a properly immersed
union of disk-like capped gropes of height at least three, with
sufficiently small component images, admits new caps for the
\emph{same bodies} such that distinct components are disjoint and
their images remain small. Its compact-case proof separates finitely
many cover collections before separating the members of each
collection, \cite[Section~2.10.2]{FQ}. We retain this two-phase
construction because we need a bound on gallery propagation only in
its first phase.
Choose compact sets $K_1, \ldots, K_q$ covering $S$, each contained in
an open coordinate ball $V_j$. Choose $\epsilon > 0$ so that the
closed $\epsilon$-neighborhood of every $K_j$ lies in $V_j$. Assign
each upper component to a $K_j$ met by its direction image. If every
initial image has diameter less than $\delta$, that component lies
within distance $\delta$ of its assigned $K_j$.
For the first phase, regard all components assigned to $K_j$ as one
collection. The construction in \cite[Section~2.10.2]{FQ} raises
height by one, contracts the added stages of the first collection and
pushes the later collections off, then continues through the
collections in order. The original bodies are retained. Its
component-diameter estimate is $3^{q-1} 7^2 \delta$. A point of each
retained body stays within $\delta$ of $K_j$, so the entire resulting
$j$th collection lies in the $3^{q+3} \delta$-neighborhood of $K_j$.
Choose $\delta$ with $3^{q+3} \delta < \epsilon$, leaving room for a
small regular-neighborhood thickening. The number of height-raising
and contraction passes depends only on $q$. The same size estimates
apply to the gallery map into $\Sigma_C$, extended over the
replacement balls as in Lemma~\ref{ct:preparation}. Since each
original body is retained, its image anchors the resulting component
within a bounded metric distance of the original label. Properness of
the Coxeter action converts this into a gallery bound independent of
the number of components and of $R$.
The collections now have pairwise disjoint images. Choose disjoint
regular neighborhoods of their entire images, with the $j$th
neighborhood still mapping into $V_j$. Inside each such neighborhood,
apply the collection-separation procedure again, this time taking
each individual upper component as a collection. There may be
arbitrarily many of them. The height is raised once more inside that
neighborhood before contracting the added stages, so this second
phase again retains the original bodies. No estimate depending on
their number is needed: all operations stay in the already chosen
neighborhood. The resulting upper components have pairwise disjoint
neighborhoods. Totally contract each in its own neighborhood to
obtain the claimed mutually disjoint framed immersed cap discs.
Every loop in a resulting cap image lies in its colour neighborhood,
whose entire direction image is contained in the contractible $V_j$.
Lemma~\ref{ct:directions} therefore makes that loop nullhomotopic in
$Z$. The second phase need not bound the gallery diameter of an
individual cap. It preserves instead the containment of its whole
image in a direction patch, which is exactly the condition needed
for nullity.
\end{proof}
\Needspace{7\baselineskip}
\subsection{Choice of parameters and the tower construction}
Fix the coordinate cover in Lemma~\ref{ct:upper-replacement}. Request
height eight in Lemma~\ref{ct:preparation}. Its uniform propagation
bound, together with the bounds for the standard neighborhoods and
marking homotopies, controls the entire preparation near the peeling
boundary. The first phase of upper separation adds only the fixed
height extension and $q$ collection passes described above. Let $b$
bound the gallery propagation of all these operations, independently
of $R$. Choose $R$ so large that the direction estimates of
Lemma~\ref{ct:directions}, including the replacement-ball extensions,
meet the chosen cover margins for every such carrier.
We now truncate the prepared family, before carrying out the
collection-separation operations. Choose $L$ large enough to contain
every peeled standard neighborhood and its marking homotopies, and
the whole $b$-buffer around the transition between peeled and
unpeeled data. Beyond that buffer, Lemma~\ref{ct:preparation}
constructs the sphere and Whitney-grope data entirely inside their own
chambers, using their chamber group-ring pairings. Retain precisely
the planes whose labels satisfy $R < |v| \le L$, together with their
complete preparations. The preparations at the outer boundary lie in
chamber interiors, so none has a partner or an attaching region
outside $U_L$. This gives a finite family in the peeled copy of $U_L$,
before any complementary filling is used.
Cap its noncentral boundary factors to form $Z = Z_{R, L}$. Apply the
first phase of Lemma~\ref{ct:upper-replacement} to this finite family.
All of its operations can be made in arbitrarily small neighborhoods
of the retained upper system inside that peeled region; the bound $b$
supplies the direction margins independently of the truncation. Apply
the second phase inside the resulting disjoint colour neighborhoods.
Neither phase requires an enlargement of $L$ or uses the complementary
fillings. The second phase supplies the immersed null-loop caps
without a uniform gallery bound on their individual images.
Write
\[
M' = Z \setminus \bigcup_i \operatorname{int} \nu(a_i)
\]
for the complement used in the Whitney construction. The spheres $a_i$
may still be immersed; the notation denotes their standard plumbed
neighborhoods. The prepared geometric dual spheres imply
\begin{equation}\label{ct:complement-group}
\pi_1(M') \xrightarrow{\ \cong\ } \pi_1(Z).
\end{equation}
Indeed the kernel of the inclusion map is normally generated by the
meridians of the $a_i$, and the punctured dual spheres bound these
meridians in the complement. General position supplies surjectivity.
This is the usual $\pi_1$-negligibility provided by geometric duality.
The new caps therefore have null double-point loops in $M'$, not just
in $Z$. They are mutually disjoint, although each may have
self-intersections. Direction control has now supplied the needed
fundamental-group condition. The remaining constructions require that
condition and the protected geometric duals, but no bound on the
diameter of the null homotopies.
\textbf{Reduction to a height-four grope.}
Symmetrically contract the fifth retained surface stage. The result is
a height-four capped Whitney grope system, still with mutually
disjoint caps. Its cap self-intersections occur in algebraically
cancelling pairs, and their loops are parallel copies of the previous
null loops. This is exactly the intermediate input in the proof of
\cite[Lemma~4.2]{PRT}, after that proof has used goodness. The
retained Clifford capped surfaces are still geometrically dual in the
stronger sense of \cite[Definition~3.1]{PRT}.
\textbf{No good-group hypothesis.}
We explain why the remainder of that proof applies without any
assumption on $H$. Cap the null double-point loops by immersed discs in
$M'$. The bodies of the Clifford capped surfaces lie at the protected
boundary-collar levels. The attaching loops of the new discs miss that
collar, so the null homotopies can be chosen to miss the Clifford
\emph{bodies} by pushing them out of the collar. They may meet the
Clifford caps and may meet the Whitney gropes arbitrarily. Correct
their boundary framings, push their intersections with the gropes down,
and tube into geometrically dual spheres obtained by contracting
suitable parallel copies of the Clifford systems. These operations
produce the tower caps. At the next collar level, contract the
remaining Clifford capped surfaces and push the tower caps off those
contractions. The resulting spheres are geometric duals to the
one-storey capped towers. These are the successive operations in the
remainder of the proof of \cite[Lemma~4.2]{PRT}; no subsequent step
uses goodness.
One-storey capped towers with four surface stages contain the embedded
discs required for the Whitney moves, with the same framed attaching
regions. Applying the final steps of \cite[Section~5]{PRT}, first move
the original sphere duals off the towers by tubing into the tower
duals. Then use the embedded Whitney discs to remove the intersections
of the $a_i$. We obtain disjoint framed embedded representatives of
their original Lagrangian classes, together with geometric transverse
spheres. The changes of upper caps do not change the retained Whitney
attaching data or the corrected sphere classes. In particular, the
symmetric-contraction independence of cap choices in
\cite[Lemma~3.4]{PRT} applies at the retained stages.
There is deliberately no assertion that the null homotopies, the tower
caps, or the final embedded Whitney discs have small gallery diameter.
From \eqref{ct:complement-group} onwards they may range throughout
$Z$. Only their existence, their relative attaching data, and the
geometric duals are needed.
\Needspace{7\baselineskip}
\begin{proposition}[Removal of the remaining planes]\label{ct:removal}
The compact manifold $Z$ admits interior surgeries producing a compact
oriented topological four-manifold $V$ with boundary $S$, fundamental
group $H$, and contractible universal cover. The boundary map to $BH$
is the original central marking.
\end{proposition}
\begin{proof}
Perform surgery on the disjoint framed Lagrangian spheres just
constructed. Their geometric duals kill the meridians in their
complement, so the surgeries preserve the fundamental group. All
changes are in the interior; the boundary and its marking are
unchanged.
For clarity, let there be $m$ remaining planes and let $W$ be the
five-dimensional surgery trace. With regular $\Z H$ coefficients, the
relative chain complex of $(W, Z)$ is concentrated in degree three,
with module $(\Z H)^m$. Its boundary map to $H_2(Z; \Z H)$ sends the
basis to the Lagrangian basis. By \eqref{ct:Z-homology} this is a
split injection. Consequently
\[
H_3(W; \Z H) = 0, \qquad
H_2(W; \Z H) \cong (\Z H)^m,
\]
the latter module represented by the dual half, and the other
positive-degree homology groups vanish. Read the same trace from $V$.
Its relative chain complex is concentrated in degree two, again with
module $(\Z H)^m$. The map
\[
H_2(W; \Z H) \longrightarrow H_2(W, V; \Z H)
\]
is the pairing of the surviving dual classes with the surgered
Lagrangian, hence is an isomorphism. The sequence of $(W, V)$
therefore gives $H_i(V; \Z H) = 0$ for every $i > 0$.
The universal cover of $V$ is simply connected and acyclic. A
topological manifold has the homotopy type of a CW
complex~\cite[Corollary~1]{MilnorCW}; in the boundary case one may
first append an exterior open collar, which does not change homotopy
type. The homological Whitehead theorem therefore makes this universal
cover contractible \cite[Corollary~4.33]{Hatcher}. Thus $V$ is a
$K(H, 1)$ and is homotopy equivalent to the finite graph $BH$. Under
this equivalence its boundary map is the retained central marking, as
asserted.
\end{proof}
The manifold $V$ is the marked graph-type filling excluded by
Proposition~\ref{ch:chamber}. This contradiction rules out the
hypothetical closed aspherical topological four-manifold and completes
the proof of Theorem~\ref{thm:main}.
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\end{document}