==== TABLE OF CONTENTS ====
Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two//counts.tex
Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two//dynamics.tex
Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two//geometry.tex
Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two//handles.tex
Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two//introduction.tex
Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two//main.tex
Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two//morse.tex
Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two//references.tex
===========================

==== BEGIN Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two//counts.tex ====
\section{The integral counts and the relative deficit}
\label{sec:counts}

Let $X_m$ be the complex nine-dimensional manifold constructed in
Proposition~\ref{geo:family}, and let $Y$ be the surface of
Proposition~\ref{dyn:surface}. Set
\[
 C_m=X_m\times\CP^1,\qquad M_m=X_m\times Y.
\]
Both products are simply connected and Kähler, of real dimensions $20$
and $22$. Our task is to compare the ordinary Morse count on $C_m$ with
the stable Morse count on $M_m$. By Theorem~\ref{morse:equality}, this
amounts to computing their quantities $\lambda$.

\subsection{The two torsion profiles}
The ranks in Lemma~\ref{geo:surfaces} give $b(S_2)=b(S_3)=8$.
Since $\CP^1\times\CP^1$ has free homology of ranks $(1,2,1)$ in degrees
$(0,2,4)$, the center $Z_3=S_3\times\CP^1\times\CP^1$ has
$b(Z_3)=32$. The integral blow-up decomposition in
\eqref{geo:family-homology} therefore yields
\begin{equation}\label{counts:rank-X}
 b(X_m)=10+m(6\cdot8+4\cdot32)=10+176m.
\end{equation}

All the homology shifts in that decomposition are even. Torsion in the
surface degrees $1$ and $2$ consequently gives two separate lists, in
degrees $1+2j$ and $2+2j$. For either list we have
\begin{equation}\label{counts:base-profiles}
 \Tor H_{1+2j}(X_m;\Z)\cong\Tor H_{2+2j}(X_m;\Z)
 \cong(\Z/2)^{ma_j}\oplus(\Z/3)^{mq_j},
\end{equation}
where the finite sequences, indexed starting at $j=1$, are
\[
 a=(2,2,2,2,2,2),\qquad q=(1,3,4,4,3,1),
\]
and all other entries are zero. To see the second sequence directly, let
$u$ record a shift by two homological degrees. The product with
$\CP^1\times\CP^1$ and the four blow-up shifts give
\[
 (u+u^2+u^3+u^4)(1+2u+u^2)
 =u+3u^2+4u^3+4u^4+3u^5+u^6.
\]
The first sequence records six blow-up shifts of two copies of $\Z/2$.
Equation~\eqref{counts:base-profiles} accounts for all torsion.

For nonnegative integers $a,q$, the primary-generator identity
\eqref{morse:primary-maximum} gives
\begin{equation}\label{counts:generator-maximum}
 d\bigl((\Z/2)^a\oplus(\Z/3)^q\bigr)=\max(a,q).
\end{equation}
In particular, generator counts must be computed in each degree
before summing over degrees.

If a factor $B$ has torsion-free homology concentrated in even degrees, put
$e_k=\rank H_{2k}(B;\Z)$ for $k\geq0$. The integral Künneth theorem replaces $a$ and $q$
by their convolutions with $e$:
\[
 (e*a)_j=\sum_{k\geq0}e_k a_{j-k},\qquad
 (e*q)_j=\sum_{k\geq0}e_k q_{j-k}.
\]
There are no additional Künneth torsion terms. For $C_m$ use $e=(1,1)$;
for $M_m$, \eqref{dyn:Y-homology} gives $e=(1,6,1)$. The resulting
profiles in either degree list, divided by $m$, are shown in
Table~\ref{tab:profiles}.

\begin{table}[tb]
\centering
\small
\begin{tabular}{@{}llr@{}}
\toprule
Product and primary part & Multiplicities in one degree list & Sum\\
\midrule
$C_m$, $2$-primary & $(2,4,4,4,4,4,2)$ & $24$\\
$C_m$, $3$-primary & $(1,4,7,8,7,4,1)$ & $32$\\
$C_m$, generators & $(2,4,7,8,7,4,2)$ & $34$\\
\midrule
$M_m$, $2$-primary & $(2,14,16,16,16,16,14,2)$ & $96$\\
$M_m$, $3$-primary & $(1,9,23,31,31,23,9,1)$ & $128$\\
$M_m$, generators & $(2,14,23,31,31,23,14,2)$ & $140$\\
\bottomrule
\end{tabular}
\caption{Primary torsion multiplicities and their entrywise maxima.
Every displayed entry is multiplied by $m$, and the same list occurs
once in odd degrees and once in even degrees. The generator row takes
the maximum within each column, as in
\eqref{counts:generator-maximum}.}
\label{tab:profiles}
\end{table}

It follows that
\begin{align}
 \sum_i t_i(C_m)&=2\cdot34m=68m,&
 \sum_i t_i(M_m)&=2\cdot140m=280m.\label{counts:torsion-totals}
\end{align}
The free ranks are $b(C_m)=2b(X_m)$ and $b(M_m)=8b(X_m)$, so
\begin{equation}\label{counts:lambda}
 \lambda(C_m)=20+488m,\qquad
 \lambda(M_m)=80+1968m.
\end{equation}
The table also explains the sign of the gap. The total multiplicity at
each individual prime is four times as large for $M_m$ as for $C_m$.
However, relative to the $3$-primary row, taking the degreewise maximum
adds $2m$ generators per parity for $C_m$ and $12m$ for $M_m$. Thus the
generator count for $M_m$ exceeds four times that for $C_m$ by
$(12-4\cdot2)m=4m$ per parity. The free-rank contributions cancel,
and the two parities and the factor $2$ in $\lambda$ give
\begin{equation}\label{counts:gap}
 \lambda(M_m)-4\lambda(C_m)
 =2(280-4\cdot68)m=16m.
\end{equation}

\subsection{Completion of the construction}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
Fix $m\geq1$. Choose any Kähler form on $X_m$ and equip
$M_m=X_m\times Y$ with its product with the invariant form on $Y$ from
Proposition~\ref{dyn:surface}. Theorem~\ref{morse:equality} applies to
$C_m$ and $M_m$, since both are closed and simply connected and their
real dimensions are at least $9$. It supplies an ordinary Morse function
$h_m$ on $C_m$ with $20+488m$ critical points, and it gives
$\SM(M_m)=80+1968m$.

Corollary~\ref{dyn:product}, applied to $X_m$ and $h_m$, produces a smooth
one-periodic Hamiltonian $H_m$ on $M_m$ with precisely
\[
 4\#\Crit(h_m)=80+1952m
\]
fixed points. They are all nondegenerate. Simple connectivity of $M_m$
makes every corresponding loop along $\phi_{H_m}^t$ contractible, so the
two fixed-point counts agree.

Finally,
\[
 \frac{\SM(M_m)-\#\Fix(\phi_{H_m}^1)}{\SM(M_m)}
 =\frac{16m}{80+1968m}
 =\frac{m}{5+123m}\geq\frac1{128},
\]
where the inequality is equivalent to $m\geq1$.
For a given $R>0$, take
$m=\max\{1,\lceil(R-80)/1968\rceil\}$. Then
$\SM(M_m)\geq R$, while the dimension and $\delta=1/128$ remain fixed.
\end{proof}

\begin{remark}\label{counts:field-bounds}
The total Betti numbers of $M_m$ over fields of characteristic $2$ and $3$
are respectively $80+1792m$ and $80+1920m$. Indeed, the universal coefficient
theorem adds twice the total primary multiplicity to the integral free
rank, and Table~\ref{tab:profiles} gives those multiplicities. Both
numbers are smaller than the fixed-point count $80+1952m$. Over any
field of characteristic different from $2$ and $3$, the total Betti
number is $80+1408m$. Thus taking a single
coefficient field does not recover the integral generator count used here:
the prime contributing the larger number of generators changes with the
degree.
\end{remark}

\begin{remark}\label{counts:integral-Floer-bound}
The integral Floer bound of Bai--Xu uses homology graded modulo twice the
minimal Chern number~\cite[Theorem~A]{BaiXu2022}; see also
\cite[Corollary~1.2]{BaiXu2026}. The minimal Chern number $N$ is the positive generator of
the nonzero subgroup $c_1(TM_m)(\pi_2(M_m))\subset\Z$.
For $M_m$ it is $1$: if $E\subset Y$ is an exceptional line, then its
tangent bundle has degree $2$ and its normal bundle has degree $-1$.
Thus $c_1(TM_m)$ evaluates to $1$ on the sphere $\{x\}\times E\subset M_m$.
In the grading modulo $2$, all odd-degree groups are combined and all
even-degree groups are combined. In either parity, Table~\ref{tab:profiles}
then gives primary torsion multiplicities $96m$ and $128m$. The minimum
number of torsion generators is therefore $128m$ in each parity. The
cyclically graded integral bound is
\[
 b(M_m)+2(128m+128m)=80+1920m,
\]
which is smaller than the fixed-point count by $32m$. The strict gap in
Theorem~\ref{thm:main} relies on keeping the integer homological degrees
separate when computing the stable Morse number.
\end{remark}
==== END Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two//counts.tex ====

==== BEGIN Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two//dynamics.tex ====
\section{From Morse functions to Hamiltonian fixed points}
\label{dyn:section}
\label{sec:dynamics}

The dynamical construction turns a Morse function on
$X\times\CP^1$ into a Hamiltonian diffeomorphism of $X\times Y$ with
four times as many fixed points.  Here $Y$ will be a K\"ahler surface
with a Hamiltonian involution whose fixed locus consists of four
projective lines.  We give the construction of
\cite[Section~4]{OpenAICompanion}, including the argument that counts
all fixed points and checks their nondegeneracy in the ambient manifold.

We use the convention
\[
  \omega(X_f,\,\cdot\,)=df
\]
for the Hamiltonian vector field of a smooth function $f$.  A fixed
point $x$ of a diffeomorphism $\Phi$ is nondegenerate when
$d_x\Phi-I$ is invertible.  The following elementary estimate will
exclude additional fixed points after a small autonomous perturbation.
It is of the same type as Yorke's lower bound on periods for Lipschitz
vector fields~\cite{Yorke1969}; only a positive lower bound is needed here.

\begin{lemma}[A lower bound on nonconstant periods]
\label{dyn:short-period}
Let $V$ be a smooth vector field on a closed smooth manifold $P$, with
flow $\psi^s$.  There is a number $\tau>0$ such that
\[
  0<T<\tau,\qquad \psi^T(x)=x
  \quad\Longrightarrow\quad V(x)=0.
\]
\end{lemma}

\begin{proof}
Choose finitely many coordinate balls covering $P$, each with closure
contained in a larger coordinate ball in the same chart.  Bounds on
the coordinate speeds, together with the positive distances from
the smaller closed balls to the boundaries of the larger balls, give
$\eta>0$ such that a trajectory starting in a smaller ball stays in
its larger ball for time $\eta$.  Choose a common positive Lipschitz
constant $L$ for the coordinate expressions of $V$ on the larger
balls; their coordinate images are convex and their closures may be
taken compact in the charts.

Suppose $\psi^T(x)=x$ with $0<T<\eta$.  In a chart containing the whole
trajectory, write its coordinates as $y:[0,T]\to\R^d$, and write $v$
for the coordinate vector field.  Put
\[
  m=\max_{0\leq t\leq T}|v(y(t))|,
\]
and choose $t_0$ where the maximum is attained.  Since $y(T)=y(0)$,
\[
  \int_0^T v(y(t))\,dt=0,
  \qquad |y(t)-y(t_0)|\leq Tm.
\]
Consequently
\[
\begin{split}
  m
  &=\left|\frac1T\int_0^T
       \bigl(v(y(t_0))-v(y(t))\bigr)\,dt\right|\\
  &\leq\frac{L}{T}\int_0^T |y(t_0)-y(t)|\,dt
   \leq LTm.
\end{split}
\]
For $T<\tau:=\min\{\eta,1/L\}$ this forces $m=0$, proving the claim.
\end{proof}

The next proposition applies the estimate to an involution.  Its
fixed components are symplectic submanifolds.  An invariant extension
of Morse functions on those components supplies the perturbation;
squaring the perturbed map will identify every one of its fixed points.

\begin{proposition}[Perturbing a Hamiltonian involution]
\label{dyn:involution}
Let $(P,\omega)$ be a closed symplectic manifold and let $g:P\to P$
be a Hamiltonian diffeomorphism with $g^2=\mathrm{id}_P$.  Write
$\Fix(g)=F_1\sqcup\cdots\sqcup F_r$ for its connected fixed
components.  For each $j$, let $h_j:F_j\to\R$ be a smooth Morse
function.  There is a smooth $g$-invariant function $f:P\to\R$
restricting to $h_j$ on $F_j$ such that, for all sufficiently small
$\epsilon>0$, its Hamiltonian flow $\psi^s$ satisfies
\begin{equation}
\label{dyn:fixed-set}
  \Fix(\psi^\epsilon\circ g)
    =\bigsqcup_{j=1}^r\Crit(h_j).
\end{equation}
Every fixed point in this equality is nondegenerate.  Moreover,
$\psi^\epsilon\circ g$ is the time-one map of a smooth one-periodic
Hamiltonian.  If $P$ is simply connected, all fixed-point loops along
the resulting Hamiltonian path are contractible.
\end{proposition}

\begin{proof}
Average a Riemannian metric over $\{\mathrm{id}_P,g\}$.  At a fixed
point $x$, the resulting exponential chart intertwines $g$ with
$d_xg$.  Thus the fixed locus is locally a smooth submanifold,
whose tangent space at $x$ is
\[
  E_x^+=\ker(d_xg-I).
\]
This local description and compactness imply that it has finitely
many connected components, each a smooth closed submanifold.
The involution gives the
decomposition
\[
  T_xP=E_x^+\oplus E_x^- ,
  \qquad E_x^-=\ker(d_xg+I).
\]
For $u\in E_x^+$ and $v\in E_x^-$, symplecticity gives
\[
  \omega_x(u,v)
  =\omega_x(d_xg(u),d_xg(v))=-\omega_x(u,v).
\]
Thus the summands are symplectically orthogonal.  Since $\omega_x$
is nondegenerate, its restriction to each summand is nondegenerate
as well.  In particular, $F_j$ is symplectic and $E_x^-$ is its
symplectic normal space.

Using disjoint tubular neighborhoods of the $F_j$ and smooth cutoff
functions, extend the $h_j$ to a smooth function $\widetilde f$ on $P$.
Set
\[
  f=\tfrac12(\widetilde f+\widetilde f\circ g).
\]
This function is $g$-invariant and restricts to $h_j$ on $F_j$.
Invariance and symplecticity imply $g_*X_f=X_f$, so $g$ commutes with
$\psi^s$.  At a point $x\in F_j$, invariance also implies
$d_xf(v)=0$ for $v\in E_x^-$.  Orthogonality of the two summands then
shows that $X_f$ is tangent to $F_j$, where it is the Hamiltonian
vector field of $h_j$ for $\omega|_{F_j}$.  Hence
\begin{equation}
\label{dyn:equilibria}
  x\in F_j:\qquad X_f(x)=0
  \quad\Longleftrightarrow\quad d_xh_j=0.
\end{equation}

Put $\Phi_\epsilon=\psi^\epsilon\circ g$.  The commutation relation
and $g^2=\mathrm{id}_P$ give
\[
  \Phi_\epsilon^2=\psi^{2\epsilon}.
\]
Let $\tau$ be given by Lemma~\ref{dyn:short-period} for $X_f$, and
take $2\epsilon<\tau$.  Any fixed point $x$ of $\Phi_\epsilon$
satisfies $\psi^{2\epsilon}(x)=x$, so the lemma gives $X_f(x)=0$.
The flow therefore fixes $x$ at every time, and
$\psi^\epsilon(g(x))=x$ implies $g(x)=x$.  Equation~\eqref{dyn:equilibria}
now places $x$ among the prescribed critical points.  Conversely,
each prescribed critical point is fixed by both $g$ and $\psi^\epsilon$.
This proves~\eqref{dyn:fixed-set}.

We next check nondegeneracy in both the tangent and the normal
directions.  At a point $x$ in the finite set~\eqref{dyn:fixed-set},
let $A_x=d_xX_f$ be the linearization at the equilibrium.  Then
\[
  d_x\psi^\epsilon=\exp(\epsilon A_x).
\]
Equivariance implies that $A_x$ commutes with $d_xg$, so it preserves
$E_x^+$ and $E_x^-$.  Denote its restrictions by $A_x^+$ and $A_x^-$.
On $E_x^+=T_xF_j$, differentiating the Hamiltonian equation at the
critical point gives
\[
  \omega_x(A_x^+u,v)=\operatorname{Hess}_x h_j(u,v).
\]
The Morse condition therefore makes $A_x^+$ invertible.  Since
\[
  \frac{\exp(\epsilon A_x^+)-I}{\epsilon}
       \longrightarrow A_x^+
       \quad\text{as }\epsilon\longrightarrow0,
\]
the tangent block of $d_x\Phi_\epsilon-I$ is invertible for all
sufficiently small positive $\epsilon$.  The normal block is
\[
  -\exp(\epsilon A_x^-)-I\longrightarrow-2I,
\]
and is likewise invertible.  There are finitely many prescribed
critical points, so a single positive upper bound for $\epsilon$
ensures nondegeneracy at all of them as well as the fixed-set equality.

Finally, choose a smooth Hamiltonian $K_s$, $0\leq s\leq1$, whose
time-one map is $g$.  Choose nonnegative smooth functions $a,b$ with
integral one and supports contained in two successive disjoint
subintervals of $(0,1)$.  Define
\[
  A(t)=\int_0^t a(u)\,du,\qquad
  H_\epsilon(t,x)=a(t)K_{A(t)}(x)+\epsilon b(t)f(x),
  \qquad 0\leq t\leq1.
\]
During the support of $a$, the flow traverses the path of $K_s$ from
the identity to $g$.  During the later support of $b$, it follows
the flow of $f$ for total time $\epsilon$.  Thus its time-one map
is $\psi^\epsilon\circ g$.  Since $H_\epsilon$ vanishes near both
endpoints of $[0,1]$, its periodic extension is smooth.  If $P$ is
simply connected, every fixed-point loop of this specified path is
contractible.
\end{proof}

We now construct the surface that gives four copies of the Morse
problem.  Blowing up each isolated fixed point of a half-turn
replaces it by a fixed projective line.

\begin{proposition}[The surface with four fixed curves]
\label{dyn:surface}
Let $Y$ be the complex blow-up of $\CP^1\times\CP^1$ at
\[
  \{0,\infty\}\times\{0,\infty\}.
\]
There is a circle-invariant K\"ahler form $\omega_Y$ for which the
lifted diagonal circle action is Hamiltonian.  Its half-turn $g_Y$
has fixed locus precisely the four exceptional curves
$E_1,\ldots,E_4$, each isomorphic to $\CP^1$.  Along these curves,
$d g_Y$ is the identity on the tangent bundle and minus the identity
on the symplectic normal bundle.  Moreover, $Y$ is simply connected
and its integral homology is
\begin{equation}
\label{dyn:Y-homology}
  H_i(Y;\Z)\cong
  \begin{cases}
    \Z,&i=0,4,\\
    \Z^6,&i=2,\\
    0,&\text{otherwise}.
  \end{cases}
\end{equation}
\end{proposition}

\begin{proof}
In affine coordinates, the diagonal circle action on
$\CP^1\times\CP^1$ is
\[
  t\cdot(z,w)=(e^{2\pi it}z,e^{2\pi it}w),
  \qquad t\in\R/\Z.
\]
It preserves the four centers and therefore lifts holomorphically
to $Y$.  Proposition~\ref{geo:blowup} shows that $Y$ is K\"ahler and
simply connected.  Average a K\"ahler form over the lifted circle
action to obtain an invariant K\"ahler form $\omega_Y$.  If $V$ is
the infinitesimal generator, Cartan's formula gives
\[
  d\bigl(\omega_Y(V,\,\cdot\,)\bigr)
     =\mathcal L_V\omega_Y=0.
\]
As $H^1(Y;\R)=0$, there is a smooth function $k:Y\to\R$ with
$dk=\omega_Y(V,\,\cdot\,)$.  Thus $V=X_k$, and the autonomous
Hamiltonian $k/2$ has time-one map $g_Y$.

The downstairs half-turn fixes exactly the four centers.  Using
the coordinate $z$ near $0$ and $1/z$ near $\infty$ in each factor,
the circle weights at the centers are
\[
  (1,1),\quad(1,-1),\quad(-1,1),\quad(-1,-1).
\]
For weights $(\alpha,\beta)$ the induced action on the exceptional
line is
\[
  [z_1:z_2]\longmapsto
  [e^{2\pi i\alpha t}z_1:e^{2\pi i\beta t}z_2].
\]
When the signs differ this action has nonzero relative weight
$\beta-\alpha=\pm2$; in all four cases its value at $t=1/2$ is the
identity.  To see the action also in the normal direction, use the
local blow-up model
\[
  \{(\ell,v):\ell\in\CP^1,\ v\in\ell\subset\C^2\}.
\]
The downstairs half-turn is $v\mapsto-v$, so its lift is
\[
  (\ell,v)\longmapsto(\ell,-v).
\]
It fixes exactly the zero section, with derivative $+I$ on its
tangent space and $-I$ on its normal line.  The symplectic normal
space is the negative eigenspace by the orthogonality argument in
Proposition~\ref{dyn:involution}.  Away from the exceptional curves,
the blow-down is an equivariant isomorphism.  There are therefore
no additional fixed points.

Finally, each point blown up has complex codimension two, so
\eqref{geo:blowup-homology} adds one free generator in degree two.
The integral homology of $\CP^1\times\CP^1$ is free with ranks
$(1,2,1)$ in degrees $(0,2,4)$, respectively.  Four blow-ups give
\eqref{dyn:Y-homology}.
\end{proof}

\begin{corollary}[Product realization]
\label{dyn:product}
Let $X$ be a connected closed simply connected K\"ahler manifold,
let $\omega_X$ be any K\"ahler form on $X$, and let $(Y,\omega_Y)$ be as in
Proposition~\ref{dyn:surface}.  Give $M=X\times Y$ the product form
$\omega_X\oplus\omega_Y$ and put $C=X\times\CP^1$.
For every smooth Morse function $h:C\to\R$, there is a smooth
one-periodic Hamiltonian $H$ on $M$ whose time-one map has only
nondegenerate fixed points and satisfies
\begin{equation}
\label{dyn:product-count}
  \#\Fix(\phi_H^1)
   =\#\Fix_0(\phi_H^1;H)
   =4\,\#\Crit(h).
\end{equation}
\end{corollary}

\begin{proof}
The pullback to $M$ of $k/2$ generates the Hamiltonian involution
$g=\mathrm{id}_X\times g_Y$ for the given product form.  Its fixed
components are exactly $X\times E_1,\ldots,X\times E_4$.  Choose
isomorphisms $E_j\cong\CP^1$ and use them to pull $h$ back to Morse
functions $h_j$ on these four components.  Each $h_j$ has
$\#\Crit(h)$ critical points.  Proposition~\ref{dyn:involution}
applies to their induced symplectic forms and produces exactly
the fixed points counted in~\eqref{dyn:product-count}, all
nondegenerate.  Finally, $M$ is simply connected, so every
fixed-point loop for the constructed path is contractible.
\end{proof}
==== END Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two//dynamics.tex ====

==== BEGIN Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two//geometry.tex ====
\section{Torsion in a fixed complex dimension}
\label{geo:section}
\label{sec:geometry}

For every integer $m\geq1$, we will construct a simply connected closed
K\"ahler manifold $X_m$ of complex dimension nine by blowing up $\CP^9$
along $2m$ disjoint centers. Two quotient surfaces supply torsion at
primes two and three. A simultaneous projection places arbitrarily many
copies of the centers in $\CP^9$, and the integral blow-up formula then
records their contributions to homology. The surface construction and
blow-up argument also appear in \cite[Section~3]{OpenAICompanion}; the
projection step is what keeps the dimension independent of $m$.

\subsection{Two projective quotient surfaces}

Let
\[
 E=\C/(\Z+i\Z),\qquad
 \zeta_p=\exp(2\pi i/p),\qquad
 \Lambda_2=\Z+i\Z,\qquad \Lambda_3=\Z+\zeta_3\Z,
\]
and set $E_p=\C/\Lambda_p$ for $p\in\{2,3\}$. The automorphism
\[
 \sigma_p:E\times E_p\longrightarrow E\times E_p,
 \qquad \sigma_p(z,w)=(z+1/p,\zeta_p w)
\]
is well-defined because multiplication by $\zeta_p$ preserves
$\Lambda_p$. It has order $p$. Every nonidentity power translates the
first factor by $k/p$ for some $0<k<p$, and hence has no fixed point.
Thus
\[
 S_p=(E\times E_p)/\langle\sigma_p\rangle
\]
is a smooth compact connected complex surface. These are examples of
bielliptic surfaces, whose integral topology is studied in
\cite[Theorem~4.3]{Serrano1991}. The quotient presentations and homology
groups used here are recorded in \cite[Sections~2.2--2.3]{Blomme2025}.
We give the projectivity and homology calculations needed for the
construction.

\begin{lemma}
\label{geo:surfaces}
The surfaces $S_2$ and $S_3$ are projective. With
\[
 T_2=(\Z/2)^2,\qquad T_3=\Z/3,
\]
their integral homology groups are
\begin{equation}
\label{geo:surface-homology}
 H_i(S_p;\Z)\cong
 \begin{cases}
  \Z,&i=0,4,\\
  \Z^2\oplus T_p,&i=1,2,\\
  \Z^2,&i=3,\\
  0,&\text{otherwise}.
 \end{cases}
\end{equation}
In particular, each surface has total Betti number eight.
\end{lemma}

\begin{proof}
We first realize the quotient as a projective submanifold. A line bundle
of degree three on an elliptic curve has three independent sections.
Riemann--Roch applied after subtracting one or two points, allowing
multiplicity, shows that its complete linear system has no base points
and separates points and tangent directions. It therefore embeds the
curve in $\CP^2$. Taking the two curve embeddings and their Segre product
gives an embedding
\[
 \iota_p:E\times E_p\hookrightarrow\CP^8.
\]

A point of $\CP^8$ represents a linear polynomial on $(\C^9)^*$, up to
a nonzero scalar. Associate to an orbit the product of the $p$ linear
polynomials representing
\[
 \iota_p(x),\ \iota_p(\sigma_p x),\ \ldots,
 \ \iota_p(\sigma_p^{p-1}x).
\]
This gives a well-defined holomorphic map
\[
 S_p\longrightarrow\mathbb P(\operatorname{Sym}^p\C^9).
\]
Unique factorization shows that equal projective products have the same
unordered factors. Thus the map separates the orbits.

To check immersion, choose local representatives $L_0,\ldots,L_{p-1}$
of the orbit points. A tangent vector in the kernel of the derivative
would give
\[
 \sum_{j=0}^{p-1}\dot L_j\prod_{k\ne j}L_k
       =a\prod_{k=0}^{p-1}L_k
\]
for a scalar $a$. The factors are pairwise nonproportional because the
action is free. Restricting this identity to the hyperplane $L_j=0$
forces $\dot L_j$ to vanish there: the product of the remaining factors
is a nonzero polynomial on that hyperplane. Consequently $\dot L_j$ is
a scalar multiple of $L_j$ for every $j$. In particular, the projective
derivative of $\iota_p(x)$ is zero, so the original tangent vector is
zero. The quotient map is a local biholomorphism, and therefore the
induced map on $S_p$ is an injective holomorphic immersion. Compactness
makes it an embedding. This proves projectivity.

For the homology, the universal cover $\C^2\to S_p$ has deck group
generated by
\[
 a(z,w)=(z+1/p,\zeta_p w),\quad
 d(z,w)=(z+i,w),\quad
 t_\lambda(z,w)=(z,w+\lambda)\quad(\lambda\in\Lambda_p).
\]
Here $a^p$ is translation by $1$ in the first coordinate. The elements
$a,d$ commute and generate a free abelian group of rank two. Their
conjugation actions on $\Lambda_p$ are multiplication by $\zeta_p$ and
the identity, respectively. The deck group is therefore
$\Lambda_p\rtimes\Z^2$, and its abelianization is
\begin{equation}
\label{geo:abelianization}
 H_1(S_p;\Z)\cong
 \Z^2\oplus\Lambda_p/(\zeta_p-1)\Lambda_p.
\end{equation}
For $p=2$, multiplication by $\zeta_p-1$ is $-2I$. For $p=3$, its
matrix in the basis $(1,\zeta_3)$ is
\[
 \begin{pmatrix}-1&-1\\1&-2\end{pmatrix}.
\]
This matrix has determinant three and relatively prime entries, so its
Smith normal form has diagonal $(1,3)$. The finite summands in
\eqref{geo:abelianization} are exactly $T_2$ and $T_3$.

The complex orientation makes $S_p$ a closed oriented real
four-manifold. Integral Poincar\'e duality and the universal coefficient
theorem imply
\[
 \Tor H_2(S_p;\Z)\cong\Tor H^2(S_p;\Z)
       \cong\Tor H_1(S_p;\Z)=T_p,
 \qquad H_3(S_p;\Z)\cong\Z^2.
\]
Finally, the degree-$p$ covering gives
$p\chi(S_p)=\chi(E\times E_p)=0$. The ranks already computed in degrees
$0,1,3,4$ are $1,2,2,1$, so the rank in degree two is two. This proves
\eqref{geo:surface-homology}.
\end{proof}

Define the two types of blow-up center by
\begin{equation}
\label{geo:centers}
 Z_2=S_2,\qquad Z_3=S_3\times\CP^1\times\CP^1.
\end{equation}
They are smooth connected projective varieties of complex dimensions
two and four; projectivity of $Z_3$ follows from the Segre embedding.
The projective-line factors distribute the $3$-primary torsion among
shifts by $0,2,4$, with multiplicities $1,2,1$. Together with the
different blow-up codimensions, this will make which prime has the larger
multiplicity depend on the degree; Section~\ref{sec:counts}
computes the resulting generator counts.

\subsection{Simultaneous projection of the centers}

The next lemma shows that the number of centers does not force the
ambient dimension to increase. We use the classical secant-and-tangent
projection argument; compare \cite[Chapter~IV, Exercise~3.11(a)]{Hartshorne1977}.
For a disjoint union, secants joining different components must be
avoided as well as secants within one component.

\begin{lemma}[Simultaneous projective embedding]
\label{geo:projection}
Let $d\geq0$ be an integer, and let $V_1,\ldots,V_s$ be a finite family
of smooth complex projective varieties of complex dimension at most
$d$. Their disjoint union admits a holomorphic embedding into
$\CP^{2d+1}$.
\end{lemma}

\begin{proof}
Splitting disconnected varieties into their finitely many connected
components, we may assume each $V_\alpha$ is connected.
Choose projective embeddings $V_\alpha\hookrightarrow\mathbb P(W_\alpha)$
and place their images in the pairwise disjoint linear subspaces
$\mathbb P(W_\alpha)\subset\mathbb P(\bigoplus_\alpha W_\alpha)$.
Adding unused coordinates if necessary, we obtain a disjoint embedded
union $U\subset\CP^D$ with $D>2d+1$. We prove that projection from a
suitable point of $\CP^D$ embeds this entire union in $\CP^{D-1}$.
Iteration will give the assertion.

For each ordered pair of components, consider pairs of distinct points
$(x,y)\in V_\alpha\times V_\beta$, deleting the diagonal when
$\alpha=\beta$. The two tautological lines representing $x$ and $y$
span a rank-two vector bundle over this pair space. Its projectivization
is a $\CP^1$-bundle whose evaluation map to $\CP^D$ sweeps out exactly
the secant lines joining those pairs. Its complex dimension is
\[
 \dim_{\C}V_\alpha+\dim_{\C}V_\beta+1\leq2d+1<D.
\]
The image has measure zero by Sard's theorem. The pair space need not
be compact; applying Sard in its countably many coordinate charts
still gives the same conclusion.

We also exclude every embedded projective tangent space. For a
component of complex dimension $e$, a local homogeneous lift and its
first derivatives span an $(e+1)$-dimensional vector space at each
point. These spaces are independent of the lift and coordinates and
form a vector bundle over that component. The evaluation map from its
projectivization sweeps out the embedded projective tangent spaces.
Its domain has complex dimension $2e\leq2d<D$, so this image too has
measure zero. The union $U$ itself has measure zero.

There are only finitely many components and component pairs. We may
therefore choose a point $q\in\CP^D$ outside $U$, all the secant lines
just described, and all the projective tangent spaces. Projection from
$q$ is holomorphic on $\CP^D\setminus\{q\}$. Two distinct points of
$U$ could have the same projection only if their secant line contained
$q$; hence the restricted projection is injective. The linear quotient
by the line representing $q$ is injective on each of the
$(e+1)$-dimensional tangent spans described above. Its projectivization
therefore has injective derivative on each component. The union is
compact, so this injective
holomorphic immersion is an embedding, with disjoint component images.
The same argument applies at each subsequent projection step, down to
$\CP^{2d+1}$.
\end{proof}

\subsection{The integral blow-up formula}

We now explain how the torsion of the centers passes to the ambient
manifold. An integral direct-sum formula is essential here: rational
cohomology alone would discard the information used in the final
comparison. The following classical formula holds without any
freeness assumption on the homology of the center; see
\cite[Theorem~4.1]{DuanLi2009}.

\begin{proposition}
\label{geo:blowup}
Let $A$ be a compact connected K\"ahler manifold and
$Z\subset A$ a nonempty closed connected complex submanifold of complex
codimension $c\geq2$. The complex blow-up
$\pi:\widehat A\to A$ is compact, connected, and K\"ahler.
If $A$ is simply connected, then $\widehat A$ is simply connected.
For every integer $i$,
\begin{equation}
\label{geo:blowup-homology}
 H_i(\widehat A;\Z)\cong
 H_i(A;\Z)\oplus\bigoplus_{j=1}^{c-1}H_{i-2j}(Z;\Z),
\end{equation}
where homology in negative degrees is zero.
\end{proposition}

\begin{proof}
Write $\nu$ for the complex normal bundle of $Z$ in $A$. Locally the
blow-up replaces a normal vector by a line together with a vector on
that line. It is a smooth complex manifold, and $\pi$ is a proper
holomorphic map that is an isomorphism away from $Z$. The exceptional
divisor is $E_Z=\mathbb P(\nu)$, with projection $\rho:E_Z\to Z$;
its normal line bundle is the tautological bundle $\mathcal O_{E_Z}(-1)$.
The local description gives connectedness, and properness over compact
$A$ gives compactness.

For K\"ahlerness, let $\omega_A$ be a K\"ahler form on $A$. A Hermitian
metric on $\nu$ induces a metric on $\mathcal O_{E_Z}(1)$ whose Chern
curvature is positive on each projective fiber. Extend this metric to
$\mathcal O(-E_Z)$, whose restriction to $E_Z$ is
$\mathcal O_{E_Z}(1)$, and let $\eta$ be its real Chern curvature form.
The form $\pi^*\omega_A$ is semipositive. It is positive away from
$E_Z$, and its kernel at a point of $E_Z$ consists precisely of the
tangent directions to the projective fiber. The form $\eta$ is positive
on those directions. On the unit tangent bundle for any auxiliary
metric, it remains positive in a neighborhood of the kernel directions;
on the compact complement, $\pi^*\omega_A$ has a positive lower bound
and $\eta$ is bounded. It follows that
\[
 \pi^*\omega_A+\varepsilon\eta
\]
is positive for all sufficiently small $\varepsilon>0$. It is closed
and of type $(1,1)$, so it is a K\"ahler form.

If $A$ is simply connected, general position in real codimension
$2c\geq4$ shows that $A\setminus Z$ is simply connected: paths and
contracting disks can be perturbed to avoid $Z$. Every loop in
$\widehat A$ can be perturbed off the real codimension two submanifold
$E_Z$, and then contracts in
$\widehat A\setminus E_Z\cong A\setminus Z$. Thus $\widehat A$ is
simply connected. No assumption on the fundamental group of the center
is needed.

We prove the splitting first in integral cohomology. Let
$\iota:E_Z\hookrightarrow\widehat A$ be the inclusion and put
$h=c_1(\mathcal O_{E_Z}(1))$. The integral Leray--Hirsch theorem gives
\begin{equation}
\label{geo:exceptional-cohomology}
 H^k(E_Z;\Z)=
 \bigoplus_{j=0}^{c-1}h^j\rho^*H^{k-2j}(Z;\Z).
\end{equation}
The classes $1,h,\ldots,h^{c-1}$ restrict to an integral basis on each
fiber; thus this statement does not require the base cohomology to be
free. The Gysin map $\iota_*$ raises degree by two and satisfies the
self-intersection identity
\[
 \iota^*\iota_*(v)=c_1(\mathcal O_{E_Z}(-1))v=-hv.
\]
Consequently the maps
\[
 s_j:H^{k-2j}(Z;\Z)\longrightarrow H^k(\widehat A;\Z),
 \qquad s_j(u)=-\iota_*(h^{j-1}\rho^*u),
 \qquad 1\leq j\leq c-1,
\]
lift all the positive powers of $h$ in
\eqref{geo:exceptional-cohomology}.

It remains to identify the classes whose restriction to $E_Z$ has
only an $h^0$ term. The map of pairs
$(\widehat A,E_Z)\to(A,Z)$ induces an isomorphism on relative
cohomology: collapsing the indicated subspaces gives a homeomorphism
$\widehat A/E_Z\cong A/Z$. Denote this relative cohomology isomorphism
by $\pi_{\mathrm{rel}}^*$, and let $\delta_A$ and $\delta_{\widehat A}$
be the connecting maps of the two pairs. If
$v\in H^k(\widehat A;\Z)$ satisfies $\iota^*v=\rho^*z$ for
$z\in H^k(Z;\Z)$, naturality and exactness give
\[
 \pi_{\mathrm{rel}}^*\delta_Az
 =\delta_{\widehat A}\rho^*z
 =\delta_{\widehat A}\iota^*v=0.
\]
The relative isomorphism implies $\delta_Az=0$, so $z$ extends to
$u\in H^k(A;\Z)$. The difference $v-\pi^*u$ restricts to zero on
$E_Z$, and hence comes from $H^k(\widehat A,E_Z;\Z)$. Its relative
preimage corresponds under $\pi_{\mathrm{rel}}^*$ to a class in
$H^k(A,Z;\Z)$. The absolute image of that class, added to $u$, pulls
back to $v$. Thus $v$ belongs to $\pi^*H^k(A;\Z)$.

Subtracting the lifts $s_j$ from an arbitrary cohomology class now shows
that
\[
 \pi^*+\sum_{j=1}^{c-1}s_j:
 H^k(A;\Z)\oplus\bigoplus_{j=1}^{c-1}H^{k-2j}(Z;\Z)
 \longrightarrow H^k(\widehat A;\Z)
\]
is surjective. It is injective as well. Restriction to $E_Z$ and
\eqref{geo:exceptional-cohomology} first force all coefficients of the
positive powers of $h$ to vanish. The remaining summand is detected
by $\pi^*$, which is injective over $\Z$: the map $\pi$ has degree one,
and the pushforward defined using integral Poincar\'e duality is a left
inverse to $\pi^*$. We have proved
\begin{equation}
\label{geo:blowup-cohomology}
 H^k(\widehat A;\Z)\cong
 H^k(A;\Z)\oplus\bigoplus_{j=1}^{c-1}H^{k-2j}(Z;\Z).
\end{equation}
If $\dim_{\C}A=n$, duality in degree $2n-i$ turns the center summands
into $H_{i-2c+2j}(Z;\Z)$. Replacing $j$ by $c-j$ gives
\eqref{geo:blowup-homology}. All maps and splittings were integral, so
the formula includes torsion as direct summands.
\end{proof}

\subsection{The family of ninefolds}

We apply the preceding results to the centers $Z_2$ and $Z_3$ from
\eqref{geo:centers}.

\begin{proposition}
\label{geo:family}
For every integer $m\geq1$ there is a simply connected closed K\"ahler
manifold $X_m$ of complex dimension nine, obtained by blowing up
$\CP^9$ along $m$ disjoint copies of each of $Z_2$ and $Z_3$. Its
integral homology is
\begin{equation}
\label{geo:family-homology}
\begin{split}
 H_i(X_m;\Z)\cong{}&H_i(\CP^9;\Z)\\
 &\oplus\left(\bigoplus_{r=1}^{6}H_{i-2r}(Z_2;\Z)\right)^{\oplus m}\\
 &\oplus\left(\bigoplus_{r=1}^{4}H_{i-2r}(Z_3;\Z)\right)^{\oplus m}.
\end{split}
\end{equation}
\end{proposition}

\begin{proof}
Apply Lemma~\ref{geo:projection} with $d=4$ to the disjoint union of
$m$ copies of $Z_2$ and $m$ copies of $Z_3$. It embeds all $2m$ centers
simultaneously in $\CP^9$. Blow up the centers successively. Because
they are disjoint, each remaining center and a neighborhood of it are
unchanged by the preceding blow-ups. Their complex codimensions are
always seven and five. Proposition~\ref{geo:blowup} preserves
compactness, connectedness, K\"ahlerness, and simple connectivity at
every step, starting from $\CP^9$; the complex dimension remains nine.
Applying \eqref{geo:blowup-homology} once for each center gives
\eqref{geo:family-homology}.
\end{proof}
==== END Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two//geometry.tex ====

==== BEGIN Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two//handles.tex ====
\section{Attaining the integral bound by handle cancellation}
\label{handle:appendix}\label{app:handles}

We prove the existence assertion in Theorem~\ref{morse:equality}.
Throughout, $W$ is a simply connected closed smooth $D$-manifold with
$D\geq9$.  The argument is the classical handle proof of Smale's theorem
\cite{Smale1963}, presented here with the low-index cancellation made
explicit, as in \cite[Appendix~A]{OpenAICompanion}.

We use the correspondence between Morse functions and ordered handle
decompositions, and the standard operations of handle slides, insertion
of cancelling pairs, and cancellation.  A handle of index $i$ is a copy
of $D^i\times D^{D-i}$ attached along
$S^{i-1}\times D^{D-i}$; its belt sphere in the new boundary is
$\{0\}\times S^{D-i-1}$.  A slide changes the integral basis of the
handles of a fixed index by an elementary addition.  Consecutive
handles of indices $i$ and $i+1$ cancel when the attaching sphere of the
latter meets the belt sphere of the former transversely in one point.
Attaching maps of subsequent handles are transported by these
operations, so the represented manifold is unchanged.
For the Morse correspondence, see \cite{Milnor1963}; for cancellation
and integral basis changes, see
\cite[Theorems~5.4 and~7.6]{Milnor1965}.

\subsection{Eliminating handles of index one and coindex one}

Start with an ordered handle decomposition.  Connectedness permits the
cancellation of all but one $0$-handle along a tree of $1$-handles.
The dual operation leaves one $D$-handle.  We now eliminate the remaining
$1$-handles, allowing insertion of cancelling $2/3$ pairs, as in
\cite[Theorem~8.1]{Milnor1965}.

Let $P$ be the union of the $0$- and $1$-handles.  It retracts onto a
graph, whose free generators correspond to the $1$-handles.  The map
$\pi_1(\partial P)\to\pi_1(P)$ is an isomorphism: the handles of $P$
relative to its boundary, read in reverse order, have indices $D-1$
and $D$.  Since $\pi_1(W)=0$ and handles of index at least three do
not change fundamental groups, the attaching words of the $2$-handles
normally generate this free group.

Fix a generator.  Insert a cancelling $2/3$ pair whose $2$-handle
initially attaches along a trivial circle in a small ball in $\partial P$.
Express the chosen generator as a product of conjugates of the old
attaching words and their inverses.  Sliding the new $2$-handle over
the old ones realizes this product: a slide is a band sum, the band
path gives the conjugation, and its orientation gives the choice of
word or inverse.  General position allows the bands to avoid the other
attaching circles; their tubular attaching regions may be shrunk.

The new attaching circle is now homotopic to a standard circle that
runs once over the chosen $1$-handle and avoids the others.  Choose the
standard circle disjoint from the other attaching circles.  In the
$(D-1)$-dimensional boundary, a generic homotopy between these circles
is an isotopy and avoids the other attaching circles: the expected
dimensions for self-intersections or intersections with another circle
in a one-parameter family are at most $3-(D-1)<0$.
Transport the framing along this isotopy.  The new $2$-handle now
cancels the chosen $1$-handle, since its attaching circle meets that
handle's belt sphere once.  Repetition removes all $1$-handles.

Apply the same construction to the reversed decomposition to eliminate
the $(D-1)$-handles.  Its inserted pairs have original indices
$D-3,D-2$, so no $1$-handles are reintroduced.  We have obtained one
handle in each of indices $0,D$, none in indices $1,D-1$, and all
remaining handles in indices $2,\ldots,D-2$.

\subsection{Locating an algebraically cancellable pair}

The manifold $W$ is orientable.  Choose orientations and let $C_*$ be
the integral handle complex.  Its boundary coefficients are algebraic
intersection numbers between attaching spheres and belt spheres in the
level between consecutive indices.  With
$B_i=\operatorname{im}\partial_{i+1}$, equation
\eqref{morse:chain-rank} gives
\begin{equation}\label{handle:excess}
 \sum_i\operatorname{rank}C_i-\lambda(W)
 =2\sum_i\bigl(\operatorname{rank}B_i-t_i(W)\bigr).
\end{equation}
Each summand on the right is nonnegative.  If the total number of
handles exceeds $\lambda(W)$, then
$\operatorname{rank}B_i>t_i(W)$ for some $2\leq i\leq D-3$;
the vanishing chain groups in degrees $1,D-1$ exclude other indices.

Because $C_i/Z_i\cong B_{i-1}$ is free, $Z_i=\ker\partial_i$ is
a direct summand of $C_i$.  Thus the nonzero Smith invariant factors
of $\partial_{i+1}$ are those of $B_i\subset Z_i$.  Exactly $t_i(W)$
of them are nonunits.  The strict inequality above supplies a unit
invariant factor.  Row and column operations, realized by handle slides,
relabeling, and orientation changes, therefore produce a boundary
coefficient equal to $1$.

These slides occur in connected levels.  Indeed, let $P_i$ be the union
of handles of index at most $i$, and put $L=\partial P_i$.
The space $P_i$ is simply connected because it has no $1$-handles.
Its reversed handles relative to $L$ have indices at least $D-i\geq3$,
so $L\to P_i$ induces isomorphisms on components and fundamental groups.
The same argument applies before the $i$-handles are attached.
Slide paths can avoid other attaching spheres, since those spheres have
codimension at least two; again their tubular regions may be shrunk.

We have found an attaching sphere $A\cong S^i$ and a belt sphere
$B\cong S^{D-1-i}$ in the simply connected level $L$, with algebraic
intersection number $1$.  We next arrange that they have exactly one
geometric intersection, which is the condition for handle cancellation.

\subsection{Removing opposite-sign intersections}

Suppose first that $i\geq3$ and $D-1-i\geq3$.
Join two opposite-sign intersections of $A$ and $B$ by an arc in each
sphere, avoiding the other intersection points.  The resulting Whitney
circle bounds a disk because $L$ is simply connected.
The high-dimensional Whitney trick removes the pair
\cite[Theorem~6.6]{Milnor1965}.
Both sphere dimensions are at least three, so general position keeps
the interior of the Whitney disk away from all attaching spheres of
the $(i+1)$-handles and all belt spheres of the $i$-handles.
The move preserves every other intersection at this level.
Repeating leaves just one intersection of $A$ and $B$.

When one sphere has dimension two, reverse the decomposition if
necessary so that $i=2$.  The level $L$ is then the boundary after
attaching the $2$-handles to the single $0$-handle.  Denote all their
belt spheres by $B_1,\ldots,B_h$, and set
\[
 V=L\setminus\bigcup_{j=1}^h B_j.
\]
We will remove a pair of intersections by moving a small disk in $A$
into $V$.  The special geometry of this complement makes that possible.

The boundary $L$ is obtained from $S^{D-1}$ by surgery on the attaching
circles.  Each surgery replaces $S^1\times D^{D-2}$ by
$D^2\times S^{D-3}$; removing its belt sphere leaves
$(D^2\setminus\{0\})\times S^{D-3}$.
This piece retracts onto its attaching boundary
$S^1\times S^{D-3}$.  Therefore $V$ has the homotopy type of the
exterior of the original attaching circles in $S^{D-1}$.
That exterior is simply connected: every loop bounds a disk in the
sphere, and general position makes the disk disjoint from the circles.
The complement of the circles retracts onto their exterior by radial
retraction in tubular neighborhoods.  We have proved
\begin{equation}\label{handle:complement-pi1}
 \pi_1(V)=\pi_1(L)=0.
\end{equation}

Choose two opposite-sign intersections of the attaching sphere
$A\cong S^2$ with one chosen belt sphere.  A thin disk
$\Delta\subset A$ surrounding an arc between them contains just these
two intersections and misses all other belt spheres.  Its boundary,
and an exterior collar in $A$, lie in $V$.
The oriented normal bundle of each belt sphere has rank two.
Excision and the Thom isomorphism give
\begin{equation}\label{handle:intersection-group}
 H_2(L,V;\Z)\cong\bigoplus_{j=1}^h\Z,
\end{equation}
with coordinates given by intersection numbers with the $B_j$.
All coordinates of $[\Delta]$ vanish, so $[\Delta]=0$.

This homological vanishing supplies a relative homotopy, without any
assumption on $\pi_2(L)$.  Indeed, \eqref{handle:complement-pi1} and
the exact sequences of the pair identify
\begin{align*}
 \pi_2(L,V)&\cong
 \operatorname{coker}\bigl(\pi_2(V)\longrightarrow\pi_2(L)\bigr),\\
 H_2(L,V;\Z)&\cong
 \operatorname{coker}\bigl(H_2(V;\Z)\longrightarrow H_2(L;\Z)\bigr).
\end{align*}
The degree-two Hurewicz isomorphisms for $V$ and $L$ identify these
cokernels.  Hence $\Delta$ can be homotoped into $V$ through maps whose
boundary stays in $V$.  Extend this boundary motion across the exterior
collar, fixing its outer edge.  We obtain a homotopy of $A$ supported
in the disk and collar; at the endpoint it removes the chosen pair and
preserves all other intersections with the belt spheres.

Finally this homotopy may be chosen to be an isotopy of the attaching
sphere.  Smooth it relative to the initial sphere and its stationary
part, and apply parametric general position.  For a one-parameter
family of maps of a $2$-sphere into the $(D-1)$-manifold $L$, the
expected dimension of self-intersections, or intersections with another
attaching $2$-sphere, is at most
\[
 2+2+1-(D-1)<0.
\]
The same general-position argument makes every map in the family an
immersion.  Since $D-1\geq8$, the relative version applies with ample
dimension to spare.  At the endpoint, the altered disk and collar form
a compact subset of the open set $V$, so a sufficiently small
approximation preserves their avoidance of all belt spheres.
Keep neighborhoods of the other transverse intersections stationary.
Intermediate intersections with belt spheres impose no restriction on
an isotopy of an attaching sphere.  Isotopy extension transports the
framing, while the other attaching tubes can be shrunk to remain
disjoint.  Repetition leaves exactly one intersection with the chosen
belt sphere.

\subsection{Finishing the count}

The chosen $i$-handle may now be ordered last among the $i$-handles,
and the chosen $(i+1)$-handle first among the $(i+1)$-handles.
They are consecutive and cancel.  Intersections with other handles do
not obstruct this operation; their attaching maps are carried along
by the cancellation diffeomorphism.
No handles of indices $1,D-1$ are introduced.

Each cancellation decreases the total count by two.  Whenever that
count exceeds $\lambda(W)$, equation~\eqref{handle:excess} supplies
another pair to which the preceding argument applies.  The process
therefore ends with exactly $\lambda(W)$ handles.  Equality in
\eqref{handle:excess} forces
$\operatorname{rank}B_i=t_i(W)$ for every $i$; substituting in
\eqref{morse:chain-rank} gives
\[
 \operatorname{rank}C_i=r_i(W)+t_i(W)+t_{i-1}(W).
\]
The Morse function associated with this decomposition has precisely
these index counts, proving Theorem~\ref{morse:equality}.
==== END Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two//handles.tex ====

==== BEGIN Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two//introduction.tex ====
\section{Introduction}
\label{sec:introduction}

The Arnold fixed-point problem asks how much of the critical-point theory of
a small autonomous Hamiltonian persists for an arbitrary Hamiltonian path.
One proposed lower bound is the stable Morse number: the least number of
critical points after allowing auxiliary quadratic variables. We construct a
family for which this bound fails by a fixed proportion, while the ambient
manifolds remain simply connected and Kähler of real dimension twenty-two.

Let $W$ be a connected closed smooth manifold. Its \emph{stable Morse number}
$\SM(W)$ is the minimum of $\#\Crit(F)$ over integers $k\geq0$ and smooth
Morse functions $F:W\times\R^k\to\R$ such that, for $k>0$,
\[
 F(x,z)=Q(z)\qquad\text{for all sufficiently large }|z|,
\]
where $Q$ is a nondegenerate quadratic form of arbitrary signature. The
radius is uniform in $x$. For $k=0$ there is no condition at infinity; the
minimum over this subclass is the ordinary Morse number $\Morse(W)$.
In particular, $\SM(W)\leq\Morse(W)$.

For a connected closed symplectic manifold $(M,\omega)$ and
$H\in C^\infty((\R/\Z)\times M,\R)$, write $\phi_H^t$ for the Hamiltonian
flow, with convention $\omega(X_{H_t},\cdot)=dH_t$. A fixed point of
$\phi_H^1$ is \emph{nondegenerate} if its derivative has no eigenvalue $1$.
We write
\[
 \Fix_0(\phi_H^1;H)
 =\{x\in\Fix(\phi_H^1):t\mapsto\phi_H^t(x)
          \text{ is contractible in }M\}.
\]
This is a set of initial points; choices of capping disks do not change its
cardinality. The unrestricted stable Morse formulation asks whether
$\#\Fix_0(\phi_H^1;H)\geq\SM(M)$ whenever all fixed points are
nondegenerate.

\begin{theorem}\label{thm:main}
For every integer $m\geq1$, there exist a simply connected closed Kähler
manifold $(M_m,\omega_m)$ of real dimension $22$ and a smooth one-periodic
Hamiltonian $H_m$ such that all fixed points of $\phi_{H_m}^1$ are
nondegenerate and
\begin{align*}
 \SM(M_m)&=80+1968m,\\
 \#\Fix(\phi_{H_m}^1)
 &=\#\Fix_0(\phi_{H_m}^1;H_m)=80+1952m.
\end{align*}
Consequently, with $\delta=1/128$, for every $R>0$ some member of this
family satisfies
\[
 \SM(M_m)\geq R,\qquad
 \#\Fix(\phi_{H_m}^1)\leq(1-\delta)\SM(M_m).
\]
\end{theorem}

\subsection*{Context and earlier work}
Arnold's questions connect Hamiltonian dynamics to the topology of smooth
functions~\cite{Arnold1986}. Conley and Zehnder established the torus case by
variational methods~\cite{ConleyZehnder1983}. Floer's construction of a
chain complex from pseudoholomorphic curves established a new approach to
homological fixed-point bounds~\cite{Floer1989}. Later extensions include the
weakly monotone theory of Hofer--Salamon and Ono and the virtual-moduli
methods of Fukaya--Ono, Liu--Tian, and Ruan
\cite{HoferSalamon1995,Ono1995,FukayaOno1999,LiuTian1998,Ruan1999}.
The resulting rational homological lower bound and the geometric Morse
bounds are different assertions: integral torsion can force more critical
points than the total rational Betti number.

More recent work gives lower bounds over arbitrary fields and over the
integers~\cite{AbouzaidBlumberg2021,BaiXu2022,BaiXu2026}. The integral
Floer bound uses homology with a cyclic grading determined by the first
Chern class. The stable Morse number retains the integer grading, and
these yield different bounds for the examples below; the comparison is
computed in Remark~\ref{counts:integral-Floer-bound}.

The stable Morse formulation is stated explicitly in
\cite[Conjecture~1.2]{Golovko2020}. Damian showed that ordinary and stable
Morse numbers need not agree~\cite{Damian2002}. Positive results for the
stable bound include the theorem of Dimitroglou Rizell and Golovko for
generic Hamiltonians when
both the symplectic area class and the first Chern class vanish on
$\pi_2(M)$~\cite[Corollary~1.4]{DRG2017}; P\"oder Balkest\aa hl obtained an
independent proof using the simple homotopy type of the Hamiltonian Floer
complex~\cite[Corollary~1.2]{PoderBalkestahl2021}. These hypotheses do not
hold for the projective blow-ups used here.
The almost-quadratic convention used in these comparisons gives the same
stable Morse number as our exact condition at infinity; see
Remark~\ref{morse:infinity-conventions}.

The companion manuscript~\cite{OpenAICompanion} gives a simply connected
Kähler counterexample using mixed-prime torsion and a Hamiltonian
involution. The present construction develops that mechanism in a fixed
dimension and produces an arbitrarily large stable Morse number with a
uniform relative deficit. We reproduce the geometric and dynamical
arguments needed for the construction. The principal extra step is to
embed arbitrarily many disjoint copies of suitable centers in $\CP^9$:
the number of centers may grow without increasing their dimensions or the
dimension of the ambient projective space.

\subsection*{Mechanism and proof structure}
All homology groups below have integral coefficients unless another ring
is displayed. For a finite abelian group $T$, let $d(T)$ denote its minimum
number of generators. Put
\begin{equation}\label{intro:lambda}
\begin{gathered}
 b(W)=\sum_i\rank H_i(W;\Z),\qquad
 t_i(W)=d(\Tor H_i(W;\Z)),\\
 \lambda(W)=b(W)+2\sum_i t_i(W).
\end{gathered}
\end{equation}
The geometric Morse inequalities give $\SM(W)\geq\lambda(W)$ even with
auxiliary quadratic variables. For simply connected manifolds in the
dimensions used here, Smale's handle theorem supplies an ordinary Morse
function attaining this bound~\cite[Theorem~2.4 and the following
paragraph]{Smale1963}. Section~\ref{sec:morse} proves the stable lower
bound and states the equality; Appendix~\ref{app:handles} recalls its
handle-theoretic proof.

The generator count combines different primes by a maximum rather than
a sum. For example,
\[
 d\bigl((\Z/2)^a\oplus(\Z/3)^q\bigr)=\max(a,q).
\]
We will arrange that the prime contributing the larger multiplicity
changes with the homological degree. Redistributing those degrees by
taking products can then change the generator count in a way that the
free ranks do not detect.

To arrange the needed homology, take two projective surfaces whose torsion
groups are respectively $(\Z/2)^2$ and $\Z/3$ in degrees $1$ and $2$.
The blow-up centers are the first surface and the product of the second
with $\CP^1\times\CP^1$, of complex dimensions $2$ and $4$.
Section~\ref{sec:geometry} embeds $m$ copies of each disjointly into
$\CP^9$ and blows them up to obtain a simply connected Kähler manifold
$X_m$. The projection argument applies more generally to every finite
collection of smooth projective varieties of dimension at most $d$, using
one ambient $\CP^{2d+1}$.

The dynamical construction uses the Kähler surface $Y$ obtained by blowing
up the four coordinate corners of $\CP^1\times\CP^1$. The half-turn of the
diagonal circle action fixes exactly the four exceptional projective lines.
Thus its product with the identity on $X_m$ has four fixed components on
$M_m=X_m\times Y$, each diffeomorphic to $C_m=X_m\times\CP^1$.
Section~\ref{sec:dynamics} perturbs this involution so that each component
contributes precisely the critical points of a chosen Morse function on
$C_m$. A short-period argument excludes every other fixed point. The
construction therefore realizes $4\lambda(C_m)$ nondegenerate fixed points.

Products with $\CP^1$ and $Y$ combine the torsion in nearby degrees with
different multiplicities. Taking the maximum of the two primary
multiplicities in each degree then gives different savings in the number
of generators. Section~\ref{sec:counts} computes
\[
 \lambda(X_m\times Y)-4\lambda(X_m\times\CP^1)=16m
\]
and completes the proof. This separates the stable Morse count from the
Hamiltonian count without any distinction between ordinary and stable
Morse theory on the simply connected manifolds in the construction.
==== END Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two//introduction.tex ====

==== BEGIN Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two//main.tex ====
\documentclass[11pt]{amsart}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage[expansion=false]{microtype}
\setlength{\emergencystretch}{2em}
\usepackage{booktabs,array}
\usepackage{xcolor}
\usepackage[a4paper,margin=30mm]{geometry}
\usepackage{hyperref}
\hypersetup{colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,
  urlcolor=blue!45!black,pdftitle={Hamiltonian Fixed Points Below the Stable Morse Number in Dimension Twenty-Two},pdfauthor={OpenAI}}
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\CP}{\mathbf{CP}}
\newcommand{\Z}{\mathbf Z}
\newcommand{\R}{\mathbf R}
\newcommand{\C}{\mathbf C}
\DeclareMathOperator{\Tor}{Tor}
\DeclareMathOperator{\SM}{SM}
\DeclareMathOperator{\Morse}{Morse}
\DeclareMathOperator{\Crit}{Crit}
\DeclareMathOperator{\Fix}{Fix}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\coker}{coker}
\newcommand{\id}{\mathrm{id}}
\title[Hamiltonian fixed points in dimension twenty-two]{Hamiltonian Fixed Points Below the Stable Morse Number in Dimension Twenty-Two}
\author{OpenAI}
\date{October 5, 2026}
\subjclass[2020]{Primary 53D40; Secondary 32Q15, 57R70, 57R17}
\keywords{Hamiltonian fixed points, stable Morse number, Kähler manifolds, blow-ups, integral homology}
\begin{document}
\begin{abstract}
We disprove the stable Morse lower bound for nondegenerate Hamiltonian
fixed points with examples in fixed real dimension twenty-two. For every
integer $m\geq1$, we construct a simply connected closed Kähler manifold
with stable Morse number $80+1968m$ and a Hamiltonian diffeomorphism with
exactly $80+1952m$ fixed points, all nondegenerate and with contractible
orbit loops. The deficit is therefore unbounded, and the fixed-point count
is at most $127/128$ of the stable Morse number.
\end{abstract}
\maketitle
\input{sections/introduction}
\input{sections/morse}
\input{sections/geometry}
\input{sections/dynamics}
\input{sections/counts}
\appendix
\input{appendices/handles}
\input{references}
\end{document}
==== END Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two//main.tex ====

==== BEGIN Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two//morse.tex ====
\section{Integral homology and Morse numbers}\label{morse:section}\label{sec:morse}

The construction requires two facts about Morse numbers: integral torsion
survives stabilization, and its resulting lower bound is attained by an
ordinary Morse function on the simply connected manifolds used here.
We prove the first fact directly for the precise condition at infinity
in the definition of $\SM$.  The second is the classical theorem of
Smale; a handle proof is included in Appendix~\ref{handle:appendix}.
The presentation follows the corresponding arguments in
\cite[Section~2 and Appendix~A]{OpenAICompanion}.

Recall that $t_i(W)$ is the minimum number of generators of
$\Tor H_i(W;\Z)$.  Put $r_i(W)=\operatorname{rank}H_i(W;\Z)$, so that
\[
 \lambda(W)=\sum_i r_i(W)+2\sum_i t_i(W).
\]
All homological indices are integers, and these quantities are zero
outside the homological range of $W$.

\begin{lemma}\label{morse:chain-bound}
Let $C_*$ be a finite chain complex of finitely generated free abelian
groups.  If $\rho_i$ is the rank of $H_i(C_*)$ and $\tau_i$ is the
minimum number of generators of its torsion subgroup, then
\[
 \operatorname{rank}C_i\geq\rho_i+\tau_i+\tau_{i-1},
 \qquad
 \sum_i\operatorname{rank}C_i\geq\sum_i\rho_i+2\sum_i\tau_i.
\]
\end{lemma}

\begin{proof}
Write $Z_i=\ker\partial_i$ and $B_i=\operatorname{im}\partial_{i+1}$.
These groups are free, and the two defining short exact sequences give
\begin{equation}\label{morse:chain-rank}
 \operatorname{rank}C_i
 =\rho_i+\operatorname{rank}B_i+\operatorname{rank}B_{i-1}.
\end{equation}
Smith normal form for $B_i\subset Z_i$ expresses
$Z_i/B_i=H_i(C_*)$ as a direct sum of a free group and cyclic groups,
one for each nonunit invariant factor of this inclusion.
Consequently $\operatorname{rank}B_i\geq\tau_i$.
Substitution in \eqref{morse:chain-rank} proves both inequalities.
\end{proof}

\begin{proposition}\label{morse:stable-lower}
For every connected closed smooth manifold $W$,
\[
 \SM(W)\geq\lambda(W).
\]
\end{proposition}

\begin{proof}
Let $F\colon W\times\R^k\to\R$ be an admissible Morse function.
We will construct a finite free integral chain complex with one
generator per critical point and with homology equal to that of $W$,
up to a degree shift.  For $k=0$, the ordinary Morse complex has these
properties.  Suppose $k>0$.

After a linear change of the auxiliary coordinates, the quadratic form
at infinity is
\[
 Q(u,v)=-|u|^2+|v|^2,
 \qquad (u,v)\in\R^d\times\R^{k-d}.
\]
Choose $R_0>0$ such that $F=Q$ for $|(u,v)|\geq R_0$ and every
critical point lies in $W\times B_{R_0}$.  Choose constants $A,T>0$ with
\[
 A>\max_{W\times\overline B_{R_0}}\{|F|,|Q|\},
 \qquad T>R_0,\qquad T^2>A.
\]
Set
\[
 K=W\times\overline B_T,\qquad
 K_-=K\cap\{F\leq-A\},\qquad
 K_+=K\cap\{F\leq A\}.
\]
The endpoint sublevels of $F$ and $Q$ agree:
\begin{equation}\label{morse:endpoint-pair}
 K_\pm=K\cap\{Q\leq\pm A\}.
\end{equation}
Indeed, outside the core the functions agree, while inside it both
sublevels at $-A$ are empty and both sublevels at $A$ contain the entire
core.

We next check that the truncation by $K$ introduces no extra cells.
On the side boundary $W\times\partial B_T$, the function is $Q$.
Its differential restricted to that boundary can vanish only at $u=0$
or $v=0$, with respective values $T^2$ and $-T^2$ when those loci
exist.  There are therefore no side-boundary critical points in the
level band $[-A,A]$.  A gradient-like vector field for $F$ on that
band can be chosen tangent to the side boundary: choose such a field
on the boundary using the nonvanishing restricted differential, extend
it to a collar, and combine it with an interior gradient-like field
by a partition of unity.  The condition $dF(V)>0$ away from critical
points is preserved by this combination.

The Morse attachment theorem now applies on this compact manifold with
corners.  Between critical values the tangent field gives the usual
deformation of sublevels, including at the side boundary; at each
critical point the change occurs in an interior Morse chart.
Thus, up to homotopy relative to $K_-$, the space $K_+$ is obtained
from $K_-$ by attaching one $j$-cell for each index-$j$ critical point
of $F$; see
\cite[Section~3]{Milnor1963} for the local attachment argument.
Coincident critical values can be separated by an arbitrarily small
perturbation supported in their Morse charts, without changing the
endpoint pair or the critical points and their indices.
Both endpoint sublevels have finite CW type.  Taking a finite CW model
for $K_-$ and cellular approximations of the attachments gives a finite
free relative cellular complex $C_*$ satisfying
\begin{equation}\label{morse:critical-complex}
 \sum_j\operatorname{rank}C_j=\#\Crit(F),
 \qquad H_j(C_*)\cong H_j(K_+,K_-;\Z).
\end{equation}

It remains to identify this homology.  The homotopy
$(u,v)\mapsto(u,(1-s)v)$, $0\leq s\leq1$, decreases both $Q$ and
the Euclidean norm.  By \eqref{morse:endpoint-pair}, it is a deformation
retraction of the pair onto
\[
 W\times\bigl(D_T^d,\{u\in D_T^d:|u|\geq\sqrt A\}\bigr).
\]
The disk--annulus pair has relative integral homology $\Z$ in degree
$d$ and zero in every other degree.  When $d=0$ it is the pair
$(\{0\},\varnothing)$, with the same assertion.  The relative
K\"unneth theorem therefore gives
\begin{equation}\label{morse:shifted-homology}
 H_j(C_*)\cong H_{j-d}(W;\Z).
\end{equation}
This calculation includes both definite signatures of $Q$.
Applying Lemma~\ref{morse:chain-bound} to
\eqref{morse:critical-complex}--\eqref{morse:shifted-homology} gives
$\#\Crit(F)\geq\lambda(W)$.  Taking the minimum over $F$ proves
the proposition.
\end{proof}

\begin{remark}\label{morse:infinity-conventions}
Some formulations of the stable Morse number allow
$\|dF-dQ\|$ to be uniformly bounded in the product of a metric on
$W$ and the Euclidean metric on $\R^k$.
This gives the same minimum as exact equality at infinity.
Indeed, write $h=F-Q$ and assume $\|dh\|\leq L$.
Compactness of $W$ gives $|h(x,z)|\leq K_0+L|z|$, while
nondegeneracy gives $\|dQ(z)\|\geq c|z|$ for some $c>0$.
Thus all critical points lie in $|z|\leq L/c$.
Choose a radial cutoff $\chi_R$ equal to one for $|z|\leq R$ and
zero for $|z|\geq2R$, with $\|d\chi_R\|\leq C/R$.
On the transition annulus,
\[
 \|d(\chi_Rh)\|\leq L+C(K_0/R+2L),
 \qquad \|dQ\|\geq cR.
\]
For sufficiently large $R$, the function $Q+\chi_Rh$ therefore
has exactly the original critical points, agrees with $F$ near them,
and equals $Q$ outside a compact set.  The converse inclusion is
immediate.  This also covers functions equal to $f(x)+Q(z)$ near
infinity, since $df$ is bounded on the closed manifold $W$.
\end{remark}

\begin{theorem}[Smale]\label{morse:equality}
Let $W$ be a simply connected closed smooth manifold of dimension
$D\geq9$.  There is an ordinary Morse function on $W$ with exactly
\[
 r_i(W)+t_i(W)+t_{i-1}(W)
\]
critical points of index $i$ for every $i$.  Consequently
\begin{equation}\label{morse:three-equal}
 \SM(W)=\Morse(W)=\lambda(W).
\end{equation}
\end{theorem}

\begin{proof}
The existence statement is the torsion-inclusive form of Smale's
Morse theorem \cite[Theorem~2.4 and the following paragraph,
pp.~134--135]{Smale1963}.  Appendix~\ref{handle:appendix} gives the
handle-cancellation proof in the stated dimension range, including the
case of handles of indices $2$ and $3$.
Summing the prescribed index counts gives $\lambda(W)$ critical points.
Since ordinary Morse functions are admissible in the definition of
$\SM$, Proposition~\ref{morse:stable-lower} yields
\[
 \lambda(W)\leq\SM(W)\leq\Morse(W)\leq\lambda(W).
\]
\end{proof}

The torsion count in these formulas combines different primes in the
same degree.  In particular, for distinct primes $p,q$,
\begin{equation}\label{morse:primary-maximum}
 d\bigl((\Z/p)^a\oplus(\Z/q)^b\bigr)=\max(a,b),
\end{equation}
where $d$ denotes the minimum number of generators.
Projection to each primary subgroup gives the lower bound, and pairing
cyclic summands by the Chinese remainder theorem gives the upper bound.
The degreewise maximum in \eqref{morse:primary-maximum} is the source of
the strict numerical comparison in our construction.
==== END Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two//morse.tex ====

==== BEGIN Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two//references.tex ====
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==== END Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two//references.tex ====