==============================================================================
A COUNTEREXAMPLE TO THE NEARBY LAGRANGIAN CONJECTURE
A complete, self-contained proof written to be read from top to bottom
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What this document is.
In 1980s symplectic geometry, Rene Arnold asked a deceptively simple
question. Take a smooth closed manifold Q (closed = compact, no
boundary). Look at its cotangent bundle T*Q, which is the phase space
of a particle moving on Q. Inside it sits the zero section (the set of
points with zero momentum), and that zero section is a "Lagrangian"
submanifold. Arnold asked: if L is any closed Lagrangian submanifold of
T*Q that is "exact" (a technical condition meaning L has no symplectic
area), must L be Hamiltonian isotopic to the zero section? In plain
words: can every such L be slid continuously, by a physically allowed
motion, onto the zero section?
This is the NEARBY LAGRANGIAN CONJECTURE. For decades every known
example worked. This document gives one that fails. The manifold we use
is Q = S^9 x S^(N-1) for a sufficiently large even integer N. We build a
closed exact smoothly embedded Lagrangian L inside T*Q that is
DIFFEOMORPHIC to Q (so it looks just like the base) but that no
compactly supported Hamiltonian isotopy can carry onto the zero section.
The failure is not about the shape of L. Abstractly L and Q are the same
smooth manifold. The failure is about HOW L is knotted inside T*Q. That
kind of failure is exactly what symplectic topology is for.
How to read this document.
Section 1 sets up the basic objects and the two key tools (generating
families and stable tube data). Section 2 states the main theorem.
Sections 3-7 prove the ingredients in the order you need them. Section 8
assembles everything into the final proof. You can read 1, 2, then 8 for
the shape of the argument, then come back for the details.
We use only plain ASCII. R^n is real n-space. S^k is the unit sphere in
R^(k+1). <v,w> is the dot product. |x| is the length of a vector. A
"smooth" map is infinitely differentiable. "Closed" always means compact
without boundary. We flag intuition boxes like this one as needed.
This is a research-level proof. High school readers should read the
intuition paragraphs and the boxed ideas; the technical lemmas can be
taken as read on a first pass. Nothing is hidden: every step is proved.
TABLE OF CONTENTS
1. Background: cotangent bundles, Lagrangians, isotopy
2. The main theorem
3. Stable tube data and a nonzero class
4. A family with one critical point has zero tube class
5. The zero-level cobordism argument
6. Building the exact embedded Lagrangian
7. Hamiltonian transport: the final contradiction
8. Putting it together: proof of the main theorem
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SECTION 1. BACKGROUND: COTANGENT BUNDLES, LAGRANGIANS, ISOTOPY
==============================================================================
1.1 The cotangent bundle and its canonical form.
Let Q be a smooth closed manifold of dimension n. The cotangent bundle
T*Q is the set of all pairs (q,p) where q is a point of Q and p is a
linear functional on the tangent space T_q Q (a "momentum" or "covector"
at q). The projection pi: T*Q -> Q forgets p: pi(q,p) = q. The zero
section Q_0 is the set of points (q,0). T*Q is a smooth manifold of
dimension 2n.
It carries a distinguished 1-form, the canonical form lambda. If we
choose local coordinates q = (q^1,...,q^n) on Q and write a covector as
p = p_1 dq^1 + ... + p_n dq^n, then in the chart:
lambda = p_1 dq^1 + ... + p_n dq^n = p . dq.
The 2-form omega = d lambda is a symplectic form: closed (d omega = 0)
and nondegenerate as a 2-form (so it pairs vectors with covectors and
gives a bijection between vector fields and 1-forms). In the chart
omega = dp_1 ^ dq^1 + ... + dp_n ^ dq^n.
Intuition: (q,p) are the position and momentum of a classical particle
moving on Q. The symplectic form omega is the "area" that Hamiltonian
mechanics preserves. Darboux's theorem says every symplectic manifold
looks locally like this standard R^(2n) with omega = sum dp_i ^ dq_i.
1.2 Lagrangians.
A submanifold L of a symplectic manifold (M, omega) of dimension 2n is
called LAGRANGIAN if:
(a) dim L = n (half the ambient dimension), and
(b) omega restricted to L is zero: omega|_L = 0.
The zero section Q_0 is the simplest example: on it p = 0, so lambda|_L
= 0 and omega|_L = d(0) = 0. Also, for a smooth function f: Q -> R, the
graph of df, i.e. the set L_f = {(q, df_q) : q in Q}, is Lagrangian.
A Lagrangian L is called EXACT if lambda|_L = df for some globally
defined smooth function f: L -> R. Equivalently, the closed 1-form
lambda|_L is exact. For the graph L_f above, lambda|_L = df. The zero
section is exact with f = 0.
Intuition: a Lagrangian has "no symplectic area". Exactness is stronger:
even the "area potential" lambda can be integrated globally.
1.3 Hamiltonian isotopy.
A Hamiltonian is a smooth function H_t: T*Q -> R depending on time t in
[0,1]. It generates a time-dependent vector field X_{H_t} by the rule
iota(X_{H_t}) omega = - dH_t,
where iota denotes the interior product (contract the vector field into
the 2-form). Since omega is nondegenerate this equation determines
X_{H_t} uniquely. Its flow phi_t satisfies
d/dt phi_t = X_{H_t}(phi_t), phi_0 = id.
The maps phi_t are symplectomorphisms (they preserve omega). A HAMILTONIAN
ISOTOPY is such a flow family phi_t: t in [0,1]. Two Lagrangians L and
L' are said to be HAMILTONIAN ISOTOPIC if there is a Hamiltonian isotopy
with phi_1(L) = L'.
We say H_t is COMPACTLY SUPPORTED if there is one compact subset K of
T*Q outside which every H_t is zero for all t. Then phi_1 is the identity
outside K, and the flow is "supported in K".
For compact Lagrangians, "compactly supported" is no restriction: given
any Hamiltonian isotopy phi_t and a closed Lagrangian L, the union of
phi_t(L) over t in [0,1] is compact (a continuous image of a compact
set), so we can multiply a Hamiltonian by a cutoff equal to 1 near that
trace and 0 far away. Along the trace, the flow is unchanged. So in
everything below, "Hamiltonian isotopy" automatically means "supported
in a compact set" and we do not distinguish.
The conventions we fix once and for all are:
omega = d lambda, iota(X_H) omega = - dH.
These are the usual conventions of classical mechanics.
1.4 The nearby Lagrangian conjecture, and the plan.
Arnold's question: if Q is a closed connected smooth manifold and
L is a closed exact smooth Lagrangian submanifold of T*Q, is L
Hamiltonian isotopic to the zero section? The theorem of this document
answers NO for a specific Q.
Two tools do all the work.
(A) GENERATING FAMILIES. Instead of describing a Lagrangian directly,
we describe it as the image of the "critical set" of a smooth
function F(q,w) with q in Q and w in an extra vector space R^r.
The critical equations d_w F = 0 define a smooth manifold of
solutions. Mapping (q,w) on that manifold to (q, d_q F(q,w)) in
T*Q gives the Lagrangian. This is a finite-dimensional, purely
real-analytic gadget: the whole symplectic obstruction becomes a
statement about critical points of ordinary smooth functions. That
is why high school students can follow the strategy: the geometry
has been converted to calculus.
(B) STABLE TUBE DATA. The behaviour of F far from the origin (its
"data at infinity") is itself a finite-dimensional object. Its
negative locus (where F < 0 at infinity) is a "tube": a domain in a
sphere with the topology of a disk bundle over a smaller sphere.
We work with families of these data over a parameter space (the
S^9). Adding split quadratic forms to F stabilizes, and the stable
space of tube data, written T, is the object whose nontrivial class
obstructs a product structure. The famous Waldhausen tube theorem
relates T to stable algebra: the fiber of the map gamma: T -> BG
is the stable h-cobordism space H^s(*), and gamma records the
stable spherical fibration of the negative domains.
The plan, in one paragraph.
(1) We find a NONZERO class a in pi_9(T) whose associated spherical
class gamma_* a is ZERO. (The class exists because of a
numerical mismatch of 2-primary parts of pi_9^S and pi_10 of
Waldhausen's Whitehead space.)
(2) We prove that any single-critical-point family of functions
with a fixed local quadratic model and homogeneous data at
infinity in T_{p,p} must have TRIVIAL stable tube class,
provided its spherical class is trivial. This uses Morse theory
plus a zero-level h-cobordism plus Waldhausen stability.
(3) We realize the nonzero class a concretely: there is a closed
exact embedded Lagrangian L over Q = S^9 x S^{N-1} whose
generating family at infinity, restricted to a fixed sphere
slice S^9 x {v0}, is the class a.
(4) We transport the family through a hypothetical Hamiltonian
isotopy to the zero section, using small steps and split
quadratic stabilizations. Each step preserves the stable tube
class. At the zero section the transport forces exactly one
nondegenerate critical point above each base point; restricting
to the slice gives a family contradicting step (2).
The rest of the document proves each piece and then assembles.
==============================================================================
SECTION 2. THE MAIN THEOREM
==============================================================================
We can now state exactly what we prove.
THEOREM (main). For some sufficiently large even integer N, the closed
manifold Q = S^9 x S^(N-1) admits a closed exact smoothly embedded
Lagrangian submanifold L inside T*Q such that:
(i) L is diffeomorphic to Q, and
(ii) no compactly supported Hamiltonian isotopy carries the zero
section Q_0 of T*Q onto L.
Remarks.
- Both Q and L are connected and simply connected. So the failure is
not a fundamental-group or connectedness effect. Abstractly L and Q
are the same smooth manifold, and yet L cannot be flattened to the
zero section.
- Stronger theorems about the PROJECTION pi|_L : L -> Q (Abouzaid,
Kragh, and later Abouzaid-Kragh, Abouzaid-Courte-Guillermou-Kragh)
say that for any closed exact Lagrangian the projection is a simple
homotopy equivalence. Those theorems do not identify the ambient
Hamiltonian isotopy class, and they are consistent with our result.
The obstruction here is a stable smooth-tube invariant, not the
homotopy type of the projection.
- The number N is fixed but not written down. What matters is that N
is even and at least some explicit bound (below, N = 2m with
m >= 20 and with a further open condition m leaves room of at least
ten dimensions; the construction is stable in m). No reader should
worry about the exact value of N; every inequality we use has
plenty of slack.
The proof will proceed in the order of the plan of Section 1.4. We
build the class (Section 3), we prove the one-critical obstruction
(Sections 4-5), we exhibit the embedded L (Section 6), and we run the
transport contradiction (Section 7). Section 8 collects everything.
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SECTION 3. STABLE TUBE DATA AND A NONZERO CLASS
==============================================================================
In this section we build the algebraic object a: a nonzero stable tube
class over the sphere S^9 whose stable spherical fibration is trivial.
Everything is a statement about families of smooth functions on spheres
with no critical points away from the origin. We are careful because all
later steps depend on getting this class right.
3.1 The model functions and the tube space T_{k,l}.
Fix positive integers k and l. On R^(k+l) = R^k x R^l write
q_{k,l}(x,y) = -|x|^2 + |y|^2, x in R^k, y in R^l.
This is the standard nondegenerate quadratic form of signature (k,l).
Its restriction to the unit sphere S^(k+l-1) is a smooth function with
positive and negative values, and zero is a regular value of that
restriction: the gradient of q_{k,l} vanishes only at the origin, which
is not on the sphere. Consequently the zero set on the sphere,
Sigma_{k,l} = { u in S^(k+l-1) : q_{k,l}(u) = 0 },
is a smooth codimension-one submanifold. It is the product
S^(k-1) x S^(l-1): a point of the zero set is u = (x,y)/r with
|x| = |y| and r^2 = 2|x|^2, so x/|x| and y/|y| vary independently. In
particular dim Sigma_{k,l} = k + l - 2.
Now let
T_{k,l} = the path component of q_{k,l}|_{S^(k+l-1)}
inside the space of all C^1 functions h on S^(k+l-1) for which 0 is a
regular value. We give that space the C^1 topology. So two such
functions lie in the same T_{k,l} when they can be joined by a path
h_s along which 0 stays a regular value for every s. Equivalently, a
path in T_{k,l} is a smooth family of hypersurfaces {h_s = 0} that
deforms the standard Sigma_{k,l} through analogous hypersurfaces.
We will equally think of an element h of T_{k,l} via its degree-two
homogeneous extension to all of R^(k+l):
g(ru) = r^2 h(u) for r > 0 and u in S^(k+l-1), g(0) = 0.
This extension is C^1 on the whole vector space. Indeed g(w) is O(|w|^2)
near zero and its differential is O(|w|), so g and dg extend continuously
to the origin. Its gradient in polar form is
grad g(r u) = r ( 2 h(u) u + grad_S h(u) ),
where grad_S is the gradient along the sphere. The two terms are
orthogonal (one radial, one tangent), so
|grad g(ru)|^2 = r^2 ( 4 h(u)^2 + |grad_S h(u)|^2 ).
Therefore 0 is a regular value of h if and only if g has no critical
point away from the origin: at a point r u with r > 0, the gradient
vanishes exactly when h(u) = 0 and grad_S h(u) = 0.
The NEGATIVE DOMAIN of h on the sphere is
D(h) = { u in S^(k+l-1) : h(u) <= 0 }.
It is a closed domain with smooth boundary {h = 0}. For the model
function q_{k,l}, the negative domain is the unit sphere of the R^k
summand times the closed unit disk in the R^l summand:
D(q_{k,l}) = S^(k-1) x D^l,
where D^l = { y in R^l : |y| <= 1 }. Indeed the condition h(u) <= 0
means the y-coordinate dominates the x-coordinate. So the negative
domain is a "tube": a sphere thickened by a disk. Its homotopy type is a
sphere S^(k-1). We call such a D(h) a SMOOTH TUBE.
Intuition. Picture the sphere as a rubber ball. The negative domain of
q_{k,l} is a band around the "y-axis direction" of the ball: a belt
that has the shape of a thickened circle when k = 2, and a thickened
(k-1)-sphere in general. A general element of T_{k,l} is any deformation
of that belt that does not degenerate (0 stays a regular value, so the
belt keeps a smooth nonempty boundary). The belt's homotopy type is
always a sphere, and the simplicial/vector-bundle data carried by that
sphere family is exactly what our invariant measures.
3.2 Functions versus domains; why the two pictures agree.
We frequently pass between a function h and its negative domain D(h).
The passage back and forth loses no homotopy information, and we record
this because later we will work with domains. For a fixed smooth domain
D with cooriented (outward-pointing) boundary, consider the set of
smooth functions phi on the sphere that
(a) are negative on the interior of D,
(b) are positive on the complement of D, and
(c) have positive outward normal derivative on the boundary.
Any two such functions define the same D. Their convex combinations
still satisfy (a),(b),(c), so this set is convex. A collar neighborhood
of the boundary supplies one such function (take the signed distance to
the boundary). For a smoothly varying family of domains, write the
boundaries as normal graphs over a fixed one and extend to ambient
isotopies supported in a collar; pulling back the chosen defining
function gives local continuous choices, and over a compact parameter
space a partition of unity combines them (the convexity makes the value
at each parameter independent of the mixing). Thus a continuous
homotopy of smooth domains lifts to a continuous homotopy of smooth
defining functions, with prescribed endpoints.
It is also harmless to replace C^1 functions by smooth ones in these
families: for a compact family define
mu = min over (b,u) of ( 4 h_b(u)^2 + |grad_S h_b(u)|^2 )^(1/2) > 0.
Smooth functions close to h_b in angular C^1 norm keep this bound
positive throughout any straight interpolation, so approximation does
not change the regular-value property or the homotopy class. Hence
"smooth angular functions" and "C^1 angular functions" give the same
homotopy groups of tube spaces. We will use whichever is convenient and
will not repeat this.
3.3 Direct sum and stabilization.
If h is in T_{k,l} and h' is in T_{k',l'}, their DIRECT SUM is the
function on S^(k+k'+l+l'-1) defined by
(h (+ ) h')(w, z) = g(w) + g'(z),
using the homogeneous extensions on R^(k+l) and R^(k'+l'). Its gradient
vanishes only where both summands' gradients do, that is, only at the
origin, so h ( + ) h' again has 0 as a regular value. Direct sums of
paths show the index pair adds:
T_{k,l} x T_{k',l'} -> T_{k+k', l+l'}.
Direct sum is associative and commutative up to canonical homotopy.
Special cases carry the geometry. Adding ONE POSITIVE SQUARE means
q_{1,0}(z) = |z|^2 as a new summand (so h ( + ) q_{1,0}). On the sphere,
if the first summand is written in the chart u = w/|w|, v = z/|w|, the
negative domain of h ( + ) q_{1,0} is
{ u in the domain of h } thickened by the unit disk in the v
directions: h(u) + |v|^2 <= 0,
i.e. the old negative domain thickened by a disk, with its boundary
rounded off. Adding a NEGATIVE square does the same to the positive
domain. So positive squares thicken the disk parameter of the tube
S^(k-1) x D^l without changing the sphere core S^(k-1), and negative
squares do the mirror image on the positive side.
The two operations commute: adding a positive and adding a negative
square can be done in either order, and their sum is q_{1,1}. This gives
a two-parameter grid of stabilizations, and the diagonal (add the same
number of positive and negative squares) is cofinal in the grid. The
stable tube space is the based homotopy colimit over the diagonal:
T = hocolim over m of ( T_{m,m} --(+ q_{1,1})--> T_{m+1,m+1} ).
By the cofinality of the diagonal, a class is null in T if and only if it
becomes null after adding some fixed number of positive and the SAME
number of negative squares.
Remark. The direct sum h ( + ) h' can fail to be smooth on a coordinate
axis (where one summand's variable is zero), but it is C^1 there, and
the angular smoothing of 3.2 replaces it by a smooth function in the
same tube class. We never need exact smoothness of the raw direct sum.
3.4 The spherical class gamma.
Let G be the stable space of homotopy self-equivalences of spheres
(degree +-1 maps S^n -> S^n, stabilized by suspension). BG classifies
stable spherical fibrations. The negative domain of any h in T_{k,l}
has the homotopy type of S^(k-1); the stable spherical fibration it
carries is therefore classified by a map
gamma : T -> BG.
We will use that gamma is (by Waldhausen's tube theorem, next) the
projection of a homotopy fiber sequence whose fiber is an h-cobordism
space, so nontrivial classes in T can hide in ker(gamma_*).
3.5 The stable h-cobordism space H^s(X).
To state Waldhausen's theorem we need one more object. For a compact
smooth manifold X, let H(X) be the space of smooth h-cobordisms from X
to some manifold N: compact manifolds W with boundary(dW) = X union N,
boundary inclusions X -> W and N -> W homotopy equivalences, the
"incoming" end X identified with X, and a prescribed product structure
on the vertical boundary when boundary(X) is nonempty. (With boundary
present, this means the vertical boundary looks like boundary(X) x I.)
All data carry collars. Concretely the space is a simplicial set whose
k-simplices are smooth bundles over the standard k-simplex. We only use
its homotopy type.
Its stable version is
H^s(X) = hocolim over r of H(X x J^r),
where J = [0,1] is a closed interval and the maps are interval product
stabilizations. For X a point, H^s(*) is equivalently the stabilization
of disks of increasing dimension; its homotopy groups in low degrees
have been computed (this is where our arithmetic happens).
Waldhausen's stable parametrized h-cobordism theorem identifies
H^s(X) ~ Omega Wh^diff(X),
where Wh^diff(X) is the infinite loop space of the cofiber of the unit
map Sigma^infinity X_+ -> A(X) and A(X) is Waldhausen's algebraic
K-theory of spaces. We use this only through the computations of
Rognes for X = *.
3.6 Waldhausen's tube fibration.
The following is the smooth tube theorem of Waldhausen, in the function
model we have described (due to Alvarez-Gavela-Igusa-Sullivan; the
HOMOGENEOUS C^1 version we use is their Theorem 3.30 with Remark 3.34).
THEOREM (Waldhausen tube fibration). For the stable smooth tube space T
there is a homotopy fiber sequence
H^s(*) -> T --gamma--> BG.
The rigid tube map iota: BO -> T, defined by nondegenerate quadratic
forms (a quadratic form q goes to its negative eigenspace sphere
bundle), satisfies gamma o iota ~ BJ, where BJ: BO -> BG is the stable
spherical fibration map.
Interpretation. A tube family has both a "rigid" part (its negative
domain as a sphere bundle, classified by gamma) and a "soft" part (how
the belt deforms, measured by H^s(*)). If gamma_* kills a class, the
class still lives in the soft part, and the soft part is what the
zero-level cobordism argument will detect.
3.7 The kernel class: a nonzero a in pi_9(T) with gamma_* a = 0.
We now find the crucial class. The arithmetic is the whole point: the
stable homotopy of spheres and the low-degree stable Whitehead group
have 2-primary parts of different sizes, and the mismatch forces a
nonzero class into the kernel of gamma_*.
PROPOSITION (kernel class). There exists a class
0 != a in pi_9(T), gamma_* a = 0.
Moreover the induced map (BJ)_*: pi_9(BO) -> pi_9(BG) is injective.
Proof.
Step 1: the relevant homotopy groups of O and G.
Bott periodicity gives pi_8(O) = pi_9(O) = Z/2 (Bott 1959, Corollary
to Theorem II, equation (1.5)). The stable homotopy groups of spheres
in this range are (Isaksen-Wang-Xu 2023, Table 1, degree 9):
pi_9^S = (Z/2)^3, of order 8.
For the stable group G of sphere self-equivalences, pi_i(G) = pi_i^S
in positive degrees. (We may use based or unbased equivalences here;
the evaluation fibration has base a sphere whose connectivity grows
without bound under stabilization, so the two choices agree in the
degrees we use.)
Step 2: the stable J-homomorphism.
Adams (1966, Theorem 9.5) shows that the stable J-homomorphism
J: pi_i(O) -> pi_i^S is injective in degrees i = 8 and i = 9. Both are
Z/2 mapping isomorphically onto the order-two subgroup of pi_i^S
generated by the image of J. We will use:
- J_9: pi_9(O) -> pi_9(BG) = pi_9^S is injective, image of order 2;
- J_8: pi_8(O) -> pi_8(BG) is injective, i.e. (BJ)_* is injective
on pi_9(BO) = pi_8(O) (Bott shift).
Step 3: the stable Whitehead group of a point.
Rognes (2002, Theorem 5.8; also Rognes 2014, Theorem 3.3) computed the
two-primary part of the low-degree stable smooth Whitehead group:
(pi_10 Wh^diff(*))_(2) = Z/8 + (Z/2)^2, of order 32.
Waldhausen's stable h-cobordism theorem, in the form
H^s(X) ~ Omega Wh^diff(X), applied at X = *, gives
pi_9(H^s(*)) = pi_10(Wh^diff(*)),
whose two-primary part has order 32. (We use only the 2-primary part;
any odd-primary part is irrelevant to the argument below.)
Step 4: the long exact sequence of the tube fibration.
The homotopy fiber sequence of Section 3.6,
H^s(*) -> T -> BG,
yields a long exact sequence in homotopy. The relevant portion, around
degree 9, is
pi_10(T) -> pi_10(BG) --partial--> pi_9(H^s(*))
-> pi_9(T) --gamma_*--> pi_9(BG).
Here pi_10(BG) = pi_9^S, order 8. The connecting map partial is
defined on pi_10(BG). Its image is constrained as follows. The rigid
map iota: BO -> T makes the composite gamma o iota equal BJ, so the
image of pi_10(BO) inside pi_10(BG) = pi_9^S lies in the kernel of
partial. That image is the order-two J_9 class, so it has order 2. The
group pi_10(BG) = pi_9^S has order 8. Therefore
|image(partial)| <= 4.
Scaled against the target pi_9(H^s(*)), whose two-primary part has
order 32, the map partial is not surjective.
Step 5: extract the nonzero class.
Exactness of the sequence at pi_9(H^s(*)) says that the kernel of
the map psi: pi_9(H^s(*)) -> pi_9(T) equals the image of partial. So
the image of psi is the quotient group
image(psi) = pi_9(H^s(*)) / image(partial).
Its 2-primary order is at least 32/4 = 8, hence it is nonzero. By
exactness at pi_9(T), image(psi) equals the kernel of gamma_*:
pi_9(T) -> pi_9(BG). So we may pick a nonzero element
0 != a in ker(gamma_*).
Its stable spherical fibration is trivial by construction, since
gamma_* a = 0.
Step 6: the injectivity of (BJ)_* on pi_9(BO).
We already have J_9 injective by Step 2. Since pi_9(BO) = pi_8(O) and
the map BO -> BG induces J on this group, (BJ)_* is injective. This
will be the second conclusion of the proposition, used in Section 4
to control the negative-eigenspace sphere bundle.
QED.
Why this class really is nonzero. The only fact that matters
numerically is the strict inequality
|image(partial)| <= 4 < 32 = |pi_9(H^s(*))_(2)|.
The left number comes from stable homotopy of spheres (order 8) and the
injective J in degree 9 (order 2). The right number comes from algebra
of Waldhausen's K-theory of a point (Rognes). The mismatch is stable
under adding more squares; it does not disappear if we stabilize.
3.8 A representative with controlled (uniform) gradient.
To build the embedded Lagrangian we need one more thing: a
representative of a whose homogeneous data are smooth in the parameter
and whose gradients are uniformly bounded above and below. This is a
technical lemma; its content is that "the class can be spread over the
whole sphere S^9 smoothly and stays standard near two chosen points."
LEMMA (localized family). Fix B = S^9 and a smooth standard height
function theta: B -> R (its two critical points are the north and south
poles). For every sufficiently large m satisfying m >= 20, the class a
of the proposition has a representative b |-> h_b in T_{m,m} such that
(b, u) |-> h_b(u) is smooth on B x S^(2m-1),
and h_b equals q_{m,m}|_{S^(2m-1)} on a neighborhood (in B) of each of
the two critical points of theta. The homogeneous extension g_b of h_b
(equation (3.1)) satisfies, for constants 0 < c <= C < infinity,
c |w| <= |grad_w g_b(w)| <= C |w| for all b in B, w in R^{2m},
|d_w d_b g_b(w)| <= C_1 |w| for all b in B, w in R^{2m},
where C_1 is a fixed constant (the second bound uses any fixed metric
on B; the mixed derivative is a covector in the b-direction for fixed
w and the operator norm is taken). Moreover d_b g_b extends
continuously to w = 0 with value 0, and d_w d_b g_b extends
continuously to w = 0 with value the zero map.
Proof.
Step 1: finite stage. A map S^9 -> T is represented by a map into
T_{m,m} at some finite stage m, because S^9 is compact and the
mapping telescope T is a colimit of the T_{m,m}. A finite initial
segment of the telescope retracts onto its last stage, so a is
represented in T_{m,m} for some finite m; further stabilization
(adding q_{1,1}) preserves the stable class and lets us take m
arbitrarily large.
Step 2: localization to two neighborhoods. Choose an oriented embedded
disk D in B = S^9 whose closure avoids both critical points of theta.
Collapsing the complement of the interior of D to a point is a map of
degree one B -> S^9 that is constant off D. Approximating it smoothly
and relatively gives a smooth degree-one map; precomposition with it
preserves based homotopy classes (a degree-one self-map of S^9 is based
homotopic to the identity). The resulting family is standard (equal to
q_{m,m}) on neighborhoods of the two critical points.
Step 3: smoothing. Approximate the corresponding continuous family into
the C^1 topology on angular functions by a finite smooth partition of
unity in b multiplying smooth angular functions. Multiply the
difference from q_{m,m} by a parameter cutoff that is zero on smaller
neighborhoods of the two critical points and one outside the original
neighborhoods. The approximation is uniform in angular C^1 norm, so by
the regularity margin of Section 3.2, straight interpolation preserves
the tube class: the resulting smooth family still represents a, and
its spherical class is still zero.
Step 4: uniform gradient bounds. For this smooth family, the continuous
function
(b,u) |-> |2 h_b(u) u + grad_S h_b(u)|
is positive on the compact set B x S^(2m-1). It therefore has a
positive minimum c and a finite maximum C. Using the polar gradient
formula grad g_b(r u) = r (2 h_b(u) u + grad_S h_b(u)) from Section
3.1, we get the first inequality c|w| <= |grad_w g_b(w)| <= C|w| for
all w = r u. The extension at w = 0 is continuous with value zero
because both sides vanish at w = 0.
Step 5: mixed derivative bound. Write d_b g_b(ru) = r^2 d_b h_b(u)
(the homogeneous extension in r and b is exactly quadratic in r for
fixed u). Differentiating in the w = r u variable gives a factor r in
front of an angular expression whose coefficients are smooth on the
compact set B x S^(2m-1), hence bounded by a constant C_1. This gives
|d_w d_b g_b(w)| <= C_1 |w|. The value at w = 0 is the zero covector,
and the mixed derivative tends to zero there because it is controlled
by |w|.
QED.
A word on point-set topology, since we will use it: B is compact, the
family depends smoothly on b, and all our bounds are uniform over B;
the constant c may be small and C large, but they are positive and
finite. That is all later sections need.
==============================================================================
SECTION 4. A FAMILY WITH ONE CRITICAL POINT HAS ZERO TUBE CLASS
==============================================================================
This section and the next prove the obstruction that kills the class a
inside any single-critical-point family. The idea in one sentence: if a
family of functions has exactly one critical point in each fiber, then
its behaviour at infinity is completely rigid, so its tube class must be
zero. Symplectic topology becomes Morse theory plus the classification
of cobordisms.
4.1 The statement.
DEFINITION. Let B be a parameter manifold. A smooth family f_b: R^n -> R
(one function for each b in B) is HOMOGENEOUS AT INFINITY if there is a
radius R (independent of b) such that for |w| >= R we have f_b(w) =
g_b(w) for a family g_b of degree-two homogeneous functions. A homogeneous
function below is smooth away from the origin, with a C^1 extension to
the origin (as in Section 3.1). "Homogeneous data in T_{k,l}" means the
angular function h_b = g_b|_{S^(n-1)} lies in the component T_{k,l}.
PROPOSITION (one critical point). Let B = S^9 and p >= 20. Suppose
f_b: R^(2p) -> R, b in B, is a smooth family with:
(1) f is homogeneous at infinity, with smooth homogeneous data in
T_{p,p};
(2) each f_b has exactly one critical point, which is nondegenerate
and varies smoothly with b;
(3) the stable spherical class of the family at infinity is zero in
[B, BG].
THEN the stable tube class of the family at infinity is zero in [B, T].
Before the proof: restating the goal. Condition (3) is the hypothesis
gamma_* [family] = 0. The conclusion is that the family itself is null
in stable tube data. Equivalently, the family at infinity can be deformed
to a constant (the standard q_{p,p}) through families of homogeneous
functions, without ever degenerating. That is what makes the nonzero
class a of Section 3 impossible: any single-critical-point family whose
infinity data is a is, by this proposition, also null in T, meaning its
data could be deformed to the standard one, contradicting that a is not
zero in T.
4.2 Normalization: make the critical point sit at the origin at zero value.
The first step replaces any single-critical-point family by one whose
critical point is the origin with critical value zero, without changing
its class at infinity.
Translate. Let c_b be the unique critical point of f_b. Choose a smooth
cutoff rho: R^(2p) -> [0,1] equal to one on a large ball (containing
all line segments from 0 to c_b, which are bounded because c_b is
bounded) and zero outside twice that ball. The time-one flow of the
compactly supported vector field X(w) = -rho(w) c_b (translation by c_b
with a cutoff) is a fiber diffeomorphism sending c_b to 0. Composing
gives a family still smooth, still one critical point, now at 0, with
the same behaviour at infinity.
Subtract the critical value. Let v_b = f_b(0) be the (new) critical
value. Choose a cutoff sigma: R^(2p) -> [0,1] equal to one on the ball
of radius A and zero outside the ball of radius 2A, with A larger than
the radius R inside which we still intend to keep flexible data. Replace
f_b by f_b - sigma f_b(0). On the transition annulus (between radii A
and 2A), the derivative of this correction is O(1/A) times v_b while the
original gradient has norm at least a fixed c*A there (linear growth of
the gradient of a homogeneous function). For A large, no new critical
point appears. The behaviour at infinity (radii beyond 2A) is unchanged
because sigma = 0 there. We obtain
f_b(0) = 0, f_b has its only critical point at 0, f same at
infinity.
Now install a quadratic local model. Let Q_b be the Hessian of f_b at
the origin (a nondegenerate quadratic form). Taylor expansion, uniform
over the compact B, gives for small w:
f_b(w) - Q_b(w) = O(|w|^3), d_w(f_b - Q_b) = O(|w|^2).
Choose a radial cutoff for a small radius (independent of b) replacing
f_b by Q_b near the origin. On the transition annulus the extra derivative
is O(|w|^2) while |grad Q_b(w)| >= c_0 |w| for a uniform c_0 > 0, so no
new critical point appears. We have arranged
(4.1) f_b(0) = 0, f_b = Q_b near 0,
without changing the unique critical point or the data at infinity.
Both translations are by compactly supported flows, so they preserve
the tube class at infinity; they do not change condition (1) or (3).
4.3 The two sublevel sets and what Morse theory says.
Set two small positive numbers delta (uniform) and two radii: a radius
a inside the common quadratic neighborhood, and a radius R_0 with an
inner collar on which the family is already homogeneous. Write the
negative and positive sublevels
M_-^b = { f_b <= -delta }, M_+^b = { f_b <= +delta }.
The point of the next lemma is that the negative sublevel is a ball-like
object associated with an (k-1)-sphere (the negative eigenspace of Q_b),
and the positive sublevel is contractible; and crucially, the sphere
at infinity and the sphere at the critical point represent the same
stable spherical fibration.
LEMMA (morse-sublevels). Suppose a family satisfying (4.1) has exactly
one nondegenerate critical point and smooth homogeneous data of type
T_{k,l} at infinity, where k, l >= 3. Then the Morse index of the
critical point is k. Moreover the unit sphere bundle of the negative
eigenspaces of Q_b (over the parameter space) is fiber homotopy
equivalent to the family of negative domains at infinity.
Proof.
Part A: M_-^b is homotopy equivalent to the negative domain at infinity.
Build a descending field on the compact part |w| <= R_0: take minus the
gradient, but near |w| = R_0 make it radial outward. This works at
|w| = R_0 because homogeneity gives, along the outward radial direction
d/dr,
df_b(d/dr) = 2 f_b / r < 0 (since f_b < 0 on the negative side there).
The conditions "gradient points so that f_b decreases" and "radial
outward near r = R_0" are strict linear inequalities in the field;
convex combinations preserve them, so we can interpolate smoothly and
also uniformly over b. The flow of this field cannot exit through
{ f_b = -delta } (it strictly decreases f_b) and must reach the sphere
r = R_0 in finite time (on the compact region the descent rate has a
positive lower bound; f_b is bounded below there). Stopping the flow at
its first crossing of r = R_0, and fixing all points beyond r = R_0,
is a deformation retraction of M_-^b onto
M_-^b ∩ { r >= R_0 }.
On this outer portion, project angularly: for u with h_b(u) < 0, the
radial fiber is the interval
[ max{R_0, sqrt(delta / (-h_b(u)))}, infinity ).
(Because f_b = r^2 h_b(u) there, so f_b <= -delta means
r^2 (-h_b(u)) >= delta.) Each interval is contractible (contract it to
its lower endpoint, which varies continuously with u). Hence the outer
portion deformation retracts to the negative domain D(h_b), i.e. to the
set of u with h_b(u) <= 0. So M_-^b is homotopy equivalent to the
negative domain. In particular
(4.2) M_-^b ≃ S^(k-1).
Part B: M_+^b is contractible. On the region { f_b >= delta }, use the
normalized negative gradient (dividing grad f_b by |grad f_b|^2). Its
time-f derivative is constantly -1, so each point descends to level
delta in exactly time (f_b - delta). The normalized field is bounded:
|grad f_b| is bounded below on the compact part and grows at least
linearly at infinity. Stop the flow when it first reaches level delta
(and fix points below). This is a deformation retraction of R^(2p) onto
M_+^b, so
(4.3) M_+^b ≃ point.
Part C: the handle argument, that (M_+^b, M_-^b) is a single k-handle.
This is the subtle part because the fibers are noncompact. Suppose the
Morse index is j. Work in local linear coordinates (uniformly bounded
coordinate change, not needing a global frame) in which
Q_b = -|x|^2 + |y|^2 with x in R^j, y in R^(2p-j). Choose a smooth
mu: [0,infinity) -> R with
-1 < mu' <= 0, mu(0) > delta, mu(s) = 0 for s >= 2 delta.
Modify the function in this coordinate patch to
tilde f_b = f_b - mu( |x|^2 + 2|y|^2 ).
The modification is supported where f_b < delta (small). In the chart
its derivatives are
-2(1 + mu') x (in the x-directions), 2(1 - 2 mu') y (in y).
Since -1 < mu' <= 0, the x-linear coefficient -2(1+mu') is negative,
and the y-linear coefficient 2(1 - 2mu') is positive; so zero is the
only critical point of tilde f_b, and it has value below -delta. Also
{ tilde f_b <= delta } = M_+^b (the modification is supported below
delta and does not change the set at level delta). Bounded normalized
descent on the slab { -delta <= f <= delta } then retracts
M_+^b = { tilde f_b <= delta }, relative to M_-^b, onto
{ tilde f_b <= -delta }.
That set, in turn, retracts relative to M_-^b onto
M_-^b ∪ D, D = { y = 0, |x|^2 <= delta }.
For membership of D: for 0 <= s <= delta, the quantity s + mu(s) is at
least mu(0) > delta. At an added point outside M_-^b, either shrink y
radially to zero (when |x|^2 <= delta) or shrink y to length
sqrt(|x|^2 - delta). During this motion tilde f_b decreases and agrees
continuously with fixing M_-^b on its boundary. This gives the claimed
relative deformation, so (M_+^b, M_-^b) has the relative homology of a
j-disk modulo its boundary sphere.
Part D: therefore the index is k, and the boundary map is a generator.
The relative homology of (M_+^b, M_-^b) being the relative homology of a
j-disk modulo its boundary means its only nonzero reduced homology is in
degree j. But (4.2)-(4.3) compute M_+^b contractible and M_-^b of type
S^(k-1); the long exact sequence of the pair then forces j = k. Moreover
the boundary map sends a generator of H_k(disk, boundary) to a generator
of H_(k-1)(M_-^b), i.e. the attaching sphere of the handle generates
H_(k-1)(M_-^b; Z). Within the local negative cone this attaching
sphere is a(S^(k-1) x {0}), the standard sphere in the negative
eigenspace of Q_b.
Part E: identify the stable sphere bundle. Flow the standard sphere
a(S^(k-1) x {0}) out to the outer sphere (using the descending field of
Part A) and then project angularly, as in Part A. Everything varies
smoothly in b. On each fiber, the map takes a homology generator to a
homology generator of the negative domain. Both source and target have
the homotopy type of S^(k-1), which is simply connected. They have CW
type. Therefore the integral homology isomorphism is a homotopy
equivalence by the homological Whitehead theorem (Hatcher, Corollary
4.33). The fiberwise criterion for maps of bundles over a CW base then
promotes it to a fiber homotopy equivalence. This identifies the stable
spherical fibration of the negative eigenspaces with that of the
negative domains at infinity.
QED.
What we use from this. By (3) the spherical class at infinity is zero.
The lemma says the negative-eigenspace bundle E_b^- (the unit sphere
bundle of the negative eigenspaces of Q_b) is fiber homotopy equivalent
to the negative domains at infinity. So the stable class of E_b^- has
zero image under BJ. Now we use the other half of the kernel-class
proposition: (BJ)_* is injective on pi_9(BO). Therefore the finite-
dimensional class of E_b^- in the Grassmannian is zero as well.
4.4 Making the quadratic form constant, then stabilizing to unbalanced indices.
We have just concluded: the negative-eigenspace sphere bundle E_b^- of
the Hessian Q_b is trivial in the finite Grassmannian Gr_p(R^(2p)) when
p >= 20. Let us unpack that and then use it.
Where the Grassmannian enters. A nondegenerate form of signature (p,p)
on R^(2p) is determined by its negative eigenspace, an element of the
Grassmannian Gr_p(R^(2p)) of p-planes in R^(2p). The space Gr_p(R^(2p))
maps to BO(p) by taking the tautological p-plane bundle; the stable map
Gr_p(R^(2p)) -> BO is p-connected (its homotopy fiber is the Stiefel
manifold V_p(R^(2p)), which is (p-1)-connected), and the stabilization
BO(p) -> BO is also p-connected. For p >= 20 both induce isomorphisms
on pi_9. So the finite-dimensional class of E_b^- being zero in the
Grassmannian is exactly the statement that we can homotope Q_b to the
standard q_{p,p} through nondegenerate forms of signature (p,p),
smoothly in b, with a uniform lower gradient bound.
Inserting the homotopy. Let Q_{b,s}, s in [0,1], be such a homotopy of
nondegenerate forms from Q_b (at s=0) to the fixed q_{p,p} (at s=1),
with a uniform lower bound grad Q_{b,s}(w) >= c_1 |w|. Insert it into a
much smaller punctured quadratic ball, using a radial parameter s = s(r)
which equals one near the origin, equals zero near the outer boundary of
the small ball, and satisfies |r s'(r)| <= epsilon. Such a radial cutoff
s(r) exists on an arbitrarily long logarithmic interval of radii: take
s(r) to depend only on the logarithm of the radius, spread the change
over a large range. Differentiating Q_{b,s(r)}(w) adds an error
O(epsilon r) to the quadratic gradient (the chain rule contributes the
s'(r) factor, weighted by r). Choosing epsilon small enough forbids new
critical points. Because the parameter is constant near the origin, the
resulting function is still smooth and still equals the fixed q_{p,p}
there.
So we have modified the family (again without changing the class at
infinity) to one whose local quadratic model is the STANDARD q_{p,p}
near the origin, independently of b.
Why we enlarge the index. The zero-level cobordism argument of Section 5
will need room to straighten the cores of the tubes by generic
perturbation. That argument requires the index pair to be unbalanced,
specifically l > k + 9. We obtain this by adding positive squares, then
after the argument adding negative squares to return to the diagonal.
Adding positive squares. Apply the stabilization-smoothing lemma below
(4.5) to the normalized family f_b(w) + |z|^2 with z in R^q. Because
f_b - g_b is supported in a fixed w-ball, the difference from the
homogeneous sum is bounded, as is its vertical derivative. We obtain a
family with still exactly one critical point, still with the standard
local model q_{k,l}, and with homogeneous data of type T_{k,l} where
(4.4) k = p, l = p + q, l > k + 9.
Both indices remain at least twenty. Its local model is q_{k,l}.
If this enlarged family is null in T_{k,l}, then adding the same number
q of negative squares returns to the diagonal T_{k+q, k+q} with the
original family null there (fixed reordering of coordinate blocks does
not affect nullity). So it suffices to prove that the enlarged, unbalanced
family is null. That is Section 5.
4.5 The stabilization-smoothing lemma.
Direct sums of homogeneous functions are only C^1 at the origin (Section
3.3). The following explicit adjustment restores smoothness away from
the total origin while staying in the same tube class; it is exactly the
tool that lets us add squares to a non-homogeneous family. It will be
reused verbatim in Hamiltonian transport (Section 7).
LEMMA (stabilization-smoothing). Let g_b(w) be a smooth homogeneous
family (b in a parameter manifold, w in R^r) with
|grad g_b(w)| >= c |w| for a fixed c > 0, and let q'(z) be a fixed
nondegenerate quadratic form in z in R^s. Put W = (w,z) and
G_0(b,W) = g_b(w) + q'(z). Suppose F_b(W) is smooth and satisfies,
uniformly over (b, W):
|F_b - G_0| <= M_0, |d_W F_b - d_W G_0| <= M_1.
THEN, outside an arbitrarily large fixed ball, F can be modified to
agree exactly with a smooth homogeneous family G_eta, in such a way that:
- the modification creates no new critical points;
- it preserves a neighborhood of every original critical point;
- the resulting G_eta is homotopic to G_0 through regular C^1
homogeneous functions.
The construction is uniform over compact parameter spaces (or any
parameter domain on which the bounds used are uniform).
Proof.
Step 1: a smooth homogeneous replacement. There is c_0 > 0 with
|grad_W G_0(W)| >= c_0 |W| (combine the linear lower bound on grad g_b
with the nondegeneracy of q'). Choose, for W != 0, a smooth cutoff
rho_eta on [0,infinity) that vanishes on [0,eta], equals one on
[2 eta, infinity), and has derivative bounded by C/eta. Set
G_eta(b,W) = g_b(w) rho_eta( |w| / |W| ) + q'(z).
The potentially nonsmooth term of g_b (near w = 0) is switched off in an
open cone around w = 0, so G_eta is smooth away from W = 0. On the
support of 1 - rho_eta we have |w| <= 2 eta |W|. Homogeneity and the
angular derivative bound give
|G_eta - G_0| <= C_2 eta^2 |W|^2,
|d_W G_eta - d_W G_0| <= C_2 eta |W|.
(On the second line, differentiating the ratio |w|/|W| costs at most
|W|^(-1) wherever the cutoff changes; combined with the |W|^2 weight it
gives the |W| bound.) For small eta, the straight-line homotopy from G_0
to G_eta retains a linear nonzero-gradient lower bound, so it stays
regular.
Step 2: blend F with G_eta far out. Let alpha_A be a smooth cutoff that
is zero for |W| <= A, one for |W| >= 2A, and has derivative bounded by
C_3 / A. Replace F by
F_A = (1 - alpha_A) F + alpha_A G_eta.
On the transition annulus { A <= |W| <= 2A }, its derivative differs from
d_W G_0 by at most
M_1 + C_2 eta |W| + (C_3 / A) ( M_0 + C_2 eta^2 |W|^2 ).
Relative to |W| this is O(eta + eta^2 + 1/A). Choose eta small, then A
large, so that this bound is less than c_0 |W| / 2. Then F_A has no
critical points in the annulus or its exterior. Original critical points
lie in a uniform ball, since d_W F differs from the linearly bounded
gradient by at most M_1; taking A beyond that ball keeps their
neighborhoods unchanged. F_A is unchanged near the origin, hence smooth
everywhere. This proves the lemma.
QED.
A note on why "smoothness away from the total origin" is the right
condition: a stable tube class only sees the angular function on the
unit sphere, which is unaffected by behaviour at W = 0 (the sphere
avoids the origin). So requiring smoothness there is all we ever need.
==============================================================================
SECTION 5. THE ZERO-LEVEL COBORDISM ARGUMENT
==============================================================================
We continue with the normalized family of Section 4: its unique critical
point is 0, its local model is the fixed q_{k,l}, its data at infinity
are smooth and homogeneous, and the indices are unbalanced:
l > k + 9 (equation 4.4). We will prove that the tube family at infinity
is null, i.e. that the negative domain family D(h_b) can be deformed
through embeddings to a constant standard tube. This is the heart of the
one-critical obstruction.
The idea. Look at the level set { f_b = 0 } between an inner sphere
(where f_b is the quadratic q_{k,l}) and an outer sphere (where f_b is
homogeneous). That level set is a hypersurface C_b of dimension
k + l - 2. On one side of it, f_b is negative; on the other, positive.
Sliding along the negative side from the inner sphere to the outer
sphere produces a cobordism: a smooth manifold whose two ends are the
inner negative domain and the outer negative domain. Because there are
no critical points on the annulus between inner and outer spheres, this
cobordism is actually a product - but the PRODUCT STRUCTURE is not
canonical, it is an h-cobordism, and the question is whether it can be
trivialized in a way that varies smoothly in b. That is what we prove,
and the classification of h-cobordisms (Waldhausen, Igusa) makes it
work in the range of dimensions we have.
Notation. Put
Y = S^(k+l-1), Sigma = { q_{k,l} = 0 } ∩ Y = S^(k-1) x S^(l-1).
Choose radii a (inside the local quadratic region) and R_0 (with an
inner collar where f_b is already homogeneous). Identify the annulus
between the sphere of radius a and the sphere of radius R_0 with
Y x [0,1] by a smooth increasing radius function r(t), r(0) = a,
r(1) = R_0, r' > 0. Define, on this annulus,
s_b(u,t) = f_b( r(t) u ) / r(t)^2.
Near t = 0 it equals the local quadratic model q_{k,l}|_Y. Near t = 1
it equals the angular infinity data h_b^1. (The division by r(t)^2
turns the homogeneous part into the angular function, and the quadratic
local model is homogeneous of degree two already.) All derivatives in
this section are taken in the annulus variables (u,t) with b fixed; we
write f_b also for its pullback to the annulus.
5.1 A regular band and its two complementary regions.
Because f_b has no critical point on the annulus, and because the
radius function is monotone, the derivative relation is
(5.1) df_b = r^2 ds_b whenever s_b = 0.
(Compute: on the cone over a point u, f_b(r u) = r^2 s_b(u,t) with
r = r(t); differentiating the radial scaling at fixed t gives the stated
factor.) In particular df_b is nonzero on { s_b = 0 }, so zero is a
regular value of s_b, and the zero set C_b = { s_b = 0 } is a smooth
hypersurface transverse to both end faces { t = 0 } and { t = 1 }.
By compactness there is a small uniform epsilon > 0 and a smooth "band"
identification:
(5.2) C_b = { s_b = 0 }, R_b = { |s_b| <= epsilon } ~ C_b x J,
where J = [-epsilon, epsilon].
How the identification is built. Choose a vector field V on a slightly
larger band with ds_b(V) = 1, tangent to the end faces, and equal to a
fixed purely angular, t-independent field on the collars of the end
faces. Such choices always patch, because the equation ds_b(V) = 1 is
affine in V (any two solutions differ by a field tangent to C_b, and
convex combinations stay solutions). Integrating V gives a flow, and its
second coordinate is exactly s_b, identifying a neighborhood of C_b with
C_b x J. At the incoming end { t = 0 } we can fix the choice independent
of b, so the resulting collar embedding Sigma x J -> Y is fixed.
The two complementary regions are
P_{-,b} = { s_b <= -epsilon }, P_{+,b} = { s_b >= +epsilon }.
Each is a manifold with boundary; the boundary is the side { s_b =
-epsilon } (resp. +epsilon) together with portions of the end faces.
LEMMA 5.1 (signed flows). On P_{+,b} and on P_{-,b} there are smooth
vector fields V_{+,b} and V_{-,b}, depending smoothly on b, such that:
(+-) df_b(V_{+-b}) > 0 on P_{+-b} (strictly increases f),
dt(V_{+-b}) > 0 near t = 0 and t = 1,
(+-) ds_b(V_{+-b}) > 0 on the side { s_b = +-epsilon }.
The fields can be chosen with a common prescription extending across the
whole angular sphere on each end collar.
Interpretation: V_{-,b} flows the entire "negative region plus inner
boundary" up to the outer end face and is transverse to every outgoing
face, so it defines a cobordism from the inner negative domain to the
outer negative domain. V_{+,b} does the same for the positive region with
the signs reversed. "dt > 0 near the ends" prevents flow lines from
leaking sideways into the collar.
Proof. Near the zero set s_b = 0, (5.1) lets us choose a field making
both df_b and ds_b strictly positive (or both strictly negative). The
same holds on the two side levels s_b = +-epsilon for small uniform
epsilon. On an end collar write s_b = h (angular, independent of t) and
take
V = d/dt + h grad_Y h.
(Think of the collar Y x [0,1] as a radial cylinder; d/dt is the radial
direction.) Then, using r = r(t) and the radial derivative of the
homogeneous function,
dt(V) = 1,
ds_b(V) = h |grad_Y h|^2,
df_b(V) = h ( 2 r r' + r^2 |grad_Y h|^2 ).
These have the required signs. On the side levels s_b = +-epsilon we
need ds_b(V) = +-epsilon |grad h|^2 > 0, which holds because dh is
nonzero near h = 0 (zero is a regular value). Away from the prescribed
regions, use strict descent/ascent for f_b. All the conditions are strict
linear inequalities in the field; convex patching (partition of unity)
preserves every applicable inequality and keeps the displayed
prescription on smaller end collars. Everything is smooth in the compact
parameter space. QED.
Let T_-^0 and T_{-,b}^1 be the truncated negative domains { s_b <=
-epsilon } on the inner and outer spheres, viewed in angular
coordinates. They are isotopic to the respective negative tubes.
LEMMA 5.2 (residual h-cobordism). The negative field V_{-,b} gives an
embedding
j_b: T_-^0 -> Int(T_{-,b}^1)
which is a homotopy equivalence. It identifies C_b with
T_{-,b}^1 \ Int( j_b( T_-^0 ) ).
In particular C_b, with its fixed incoming identification with Sigma, is
a smooth family of h-cobordisms.
Proof. For V_{-,b}, the inner face and the side face are entry faces
(the flow enters the region through them), and the outer face t = 1 is
the only exit. Every trajectory exits in finite forward time: on the
compact region P_{-,b}, the quantity -df_b(V_{-,b}) is bounded below by a
positive constant, and f_b ranges over a bounded set; therefore the
descent time is finite. Reversing time, every trajectory reaches an
entry face.
Flowing to the first intersection with t = 1 maps each entry face
smoothly and injectively into T_{-,b}^1. Transversality to the exit face
gives smooth dependence of the exit time, including at the outer edge of
the side where the time is zero. Across the corner faces the maps have
full rank. Their images meet precisely along the shared inner edge and
are otherwise disjoint (uniqueness of flow lines); running the flow
backwards shows their union is the entire outer tube. The image of the
inner face lies in the interior of the outer tube because the field
points strictly inward from the side. The band identification (5.2)
identifies that side with C_b and proves the stated decomposition.
Now check j_b is a homotopy equivalence. Choose delta > 0 as in the
lemma of Section 4.3, small enough that the standard sphere
a(S^(k-1) x {0}) in the fixed negative eigenspace lies in { f_b <=
-delta }. This sphere is the core of the inner tube. Its image under the
negative flow stays in this sublevel (f_b decreases along the flow). The
core is a homology generator there (Section 4.3, Part D), so its image at
radius R_0 is a homology generator in the outer radial portion. Angular
projection carries it to a generator of the open negative domain.
Including T_{-,b}^1 into this open domain is a homotopy equivalence for
small epsilon. Therefore j_b has degree +-1 on the sphere homotopy types
of the source and target. Those types are simply connected, so j_b is a
homotopy equivalence.
Finally C_b is an h-cobordism. Excision with collars gives
H_*(C_b, d_-(C_b); Z) = H_*(T_{-,b}^1, j_b(T_-^0); Z) = 0,
because j_b is a homotopy equivalence. This includes H_0, so C_b is
connected. The two tubes and their common boundary are simply connected,
so van Kampen applied to the decomposition C_b = j_b(T_-^0) union (side)
shows pi_1(C_b) = 0. The cobordism is orientable, and
Poincare-Lefschetz duality gives vanishing of the relative homology at
the outgoing end too. Both boundary components are diffeomorphic to
S^(k-1) x S^(l-1) and hence simply connected. All these smooth spaces
have CW type, so the homological Whitehead theorem makes both boundary
inclusions homotopy equivalences. Regularity and properness give a
smooth bundle of these cobordisms over B together with their boundary
collars and the fixed incoming identification. QED.
5.2 Stabilizing the cobordism inside a fixed cylinder.
Let H(X) be the smooth h-cobordism space of a compact manifold X, from
Section 3.5. The family C_b gives a class c_B in [B, H(Sigma)]. We want
to trivialize c_B, i.e. produce a product identification relative to its
incoming end, varying over all of B. The plan: thicken C_b once by an
interval, extend to all of Y, and show the resulting family has a
product complement in a FIXED cylinder. Then use stability (Waldhausen,
Igusa) to recover the class of the original C_b from that of the enlarged
family.
Recall from Section 3.5: in H(X) the incoming end is identified with X;
when boundary(X) is nonempty, the vertical boundary is the prescribed
product boundary(X) x I. There is no identification of the outgoing end
with X. All data carry collars. Equivalently, one may bend the vertical
face into the outgoing face so that incoming and outgoing boundaries meet
along boundary(X); this is the collared product model of
Waldhausen-Jahren-Rognes. The product component is
H(X)_0 ~ B P(X), P(X) = Diff(X x I, rel ( X x {0} union
boundary(X) x I )),
a diffeomorphism group fixed near the indicated faces and product near
the outgoing face. So a null classifying map into this component gives a
product family relative to the incoming end.
In our application Sigma is simply connected and has dimension at least
38. The smooth h-cobordism theorem (Milnor, Theorem 9.1) makes every
h-cobordism on Sigma a product relative to its incoming end, in
dimension at least 39. If a product parametrization restricts to a
diffeomorphism A: Sigma -> Sigma at the incoming end, precomposing it
with A^(-1) x id_I fixes that end pointwise. Hence H(Sigma) is connected.
Interval stabilization. Define sigma: H(X) -> H(X x J), J = [0,1]. For
X closed and C an h-cobordism from X to N, add a fixed incoming collar to
form C^- = (X x [-1,0]) union_X C. Use C^- x J with faces, before
rounding:
incoming: X x {-1} x J,
vertical: X x [-1,0] x boundary(J),
outgoing: (N x J) union (C x boundary(J)).
The vertical face has its stated product identification; the other sides
belong to the outgoing end. Corner bending turns these into the usual
collared cobordism faces. For manifolds with boundary, keep the existing
product vertical faces too and round the product corners.
Extension. If e: X -> Y is a codimension-zero embedding, the extension
e_!: H(X) -> H(Y) glues the cobordism to the trivial cylinder on
Y \ Int(e(X)) along the product vertical boundary. In the bent collared
model this is C |-> (Y x I) union_{X x I} C, where X x I subset C is the
incoming collar. This expression shows that extension commutes with
interval stabilization.
Add trivial collars Y x [-1,0] and Y x [1,2] to the annulus, and set
(with corners to be rounded later)
(5.3) W_b = ( Y x [-1,0] ) union R_b.
Its incoming boundary is the fixed Y x {-1}.
LEMMA 5.3 (band is the stabilization). The family W_b represents the
extension to Y of the interval stabilization of C_b.
Proof. Adjoin a fixed incoming collar to C_b, obtaining
C_b^- = ( Sigma x [-1,0] ) union C_b. Its interval stabilization is
C_b^- x J, with incoming Sigma x J. In the stabilization-face convention
above, the two faces C_b x boundary(J) and (outgoing end) x J form the
movable boundary, while the added incoming-collar portions supply the
fixed vertical boundary of a boundary-relative model. Extending across
Sigma x J -> Y replaces the incoming collar by Y x [-1,0]. The result is
exactly (5.3), using the band identification (5.2) for the attached
piece.
This works on transition maps too. Local product trivializations of C_b
fix its incoming collar and can be made product on outgoing collars
(without fixing their outgoing end). Extend them by identity over the
added collar and take their product with id_J. A fixed bending of the two
interval coordinates gives the usual boundary-relative stabilized
cylinder. Near the lower gluing seam the transitions are identity; near
the upper seam they preserve both interval coordinates and act only in
the remaining boundary variable. So fixed planar corner adjustments
descend through the transitions. Extending by identity over the
complement in Y gives the same family as (5.3). This is a single interval
stabilization with both side faces movable. QED.
LEMMA 5.4 (product complement). The complement of W_b up to Y x {2} is,
after compatible corner rounding, a product cylinder over the outgoing
boundary of W_b, smoothly in b. Consequently [W] = 0 in [B, H(Y)].
Proof. Before rounding, the outgoing boundary of W_b consists of the two
sign domains at t = 0, the two band sides, and the band end at t = 1. Its
complement is the union of P_{-,b} and P_{+,b} with the upper collar
Y x [1,2]. Use Lemma 5.1 on the sign regions. On and above the upper end,
its common angular prescription extends across the band and has dt > 0;
continue it to the top with that property. The resulting field points
from every outgoing face of W_b into the complement, and its only exit
face is t = 2.
Rounding. Describe it transverse to this field. At a lower corner put
v = t, w = +-s_b - epsilon; the complement is the quadrant v, w >= 0. At
an upper corner put v = t - 1 and the same w; the complement is the union
{ v >= 0 } union { w >= 0 }. At either corner the field has dv(V) > 0 and
dw(V) > 0. Round the interface by a curve in the first quadrant at the
bottom and in the third quadrant at the top, smoothly joining the
unchanged straight parts. Choose its conormal into the complement as a
nonzero nonnegative combination of dv, dw; this makes the rounded
interface transverse with the same orientation. The product end
coordinates allow fixed small planar roundings over all parameters. If
the stabilized piece and its exterior are kept separate at the bottom
seam, move their dividing wall in the small corner neighborhood so it
reaches the rounded interface transversely; all transition maps there are
identity. Each piece is then a fixed planar adjustment of its original
collared model, so Lemma 5.3 still describes the rounded family.
Every forward trajectory reaches t = 2 in finite time. In small lower and
upper collars dt(V) has a uniform positive lower bound, bounding transit
time there and preventing reentry across their inner thresholds in the
wrong direction. In the remaining compact middle, a trajectory stays in
one sign region and its signed f_b increases at a rate bounded below,
with bounded range, so its time there is also finite. It cannot exit
across the entry boundary. The same argument backwards reaches the
rounded outgoing boundary of W_b.
Transversality gives smooth hitting times in the initial point and in b.
Rescaling these gives a product diffeomorphism of the complementary
cobordism relative to its entry E_b = outgoing boundary of W_b. This is a
product identification over the entire possibly varying bundle E -> B,
not merely a statement about individual fibers.
Finally, adding E_b x I to the outgoing end changes no
incoming-relative h-cobordism bundle class. Extend the transverse field a
small uniform distance into W to obtain a collar E x [-eta, 0]. A fixed
increasing diffeomorphism [-eta, 1] -> [-eta, 0], equal to the identity
near -eta, compresses the added cylinder into this collar and is identity
on the rest of W. It is defined without choosing a trivialization of
E -> B. The enlarged total is the literal product B x Y x [-1,2],
relative to its incoming boundary. Hence [W] = 0. QED.
5.3 Recovering the original cobordism family.
We have shown that the extension to Y of one interval stabilization of
C_b is a product family. Now we show this operation detects the class of
C_b over B = S^9.
The stable parametrized h-cobordism theorem (Section 3.5,
H^s(X) ~ Omega Wh^diff(X)) is natural for this extension.
The quantitative statements. The single stabilization H(X) -> H(X x J) is
r-connected provided
(5.4) dim X >= max(2r + 8, 3r + 5).
Every subsequent stage also satisfies this dimension bound, so the map
H(X) -> H^s(X) is r-connected too. Moreover, an (r+1)-connected map
X -> Y induces an r-connected map H^s(X) -> H^s(Y). For the first
statement we use Igusa's stability theorem (Igusa 1988) in the form of
Courte-Porcelli, Theorem 2.16; the second is Courte-Porcelli, Theorem
2.17; extension is the map of Courte-Porcelli Definition 2.15. These use
the WJR product convention above. "r-connected" means: isomorphisms on
pi_i for i < r, and a surjection on pi_r.
PROPOSITION 5.5 (h-cobordism injectivity). Let k, l >= 20, and put
Y = S^(k+l-1), Sigma = { q_{k,l} = 0 } ∩ Y = S^(k-1) x S^(l-1). For a
closed collar embedding e: Sigma x J -> Y, the composite
(5.5) H(Sigma) --sigma--> H(Sigma x J) --e_!--> H(Y) --> H^s(Y)
induces an isomorphism on pi_9 in the product component. Consequently, if
a sphere family in H(Sigma) has null image under this composite, the
family itself is null.
Proof. Here dim Sigma = k + l - 2 >= 38, while the bound in (5.4) for
r = 10 is max(28, 35) = 35. So H(Sigma) -> H^s(Sigma) is 10-connected.
The composite H(Sigma) -> H(Sigma x J) -> H^s(Sigma x J) has the same
connectivity (its target is the tail of the defining stabilization
telescope), so it induces an isomorphism on pi_9.
Set d = min(k, l) - 1. The source of e, which is Sigma x J, is
(d-1)-connected, and Y has vanishing homotopy groups in degrees at most
d (Y is a sphere of dimension k + l - 1, and d = min(k,l) - 1 <
k + l - 1). So e is d-connected: an isomorphism below degree d and a
surjection in degree d. The stable extension H^s(Sigma x J) -> H^s(Y) is
consequently (d-1)-connected, with d - 1 >= 18, and also induces an
isomorphism on pi_9. Compatibility of extension and stabilization
identifies the composite of these two isomorphisms with (5.5).
The space H(Sigma) is connected (shown in Section 5.2). For a connected
target, free homotopy classes of maps from S^9 are orbits of based
classes under the fundamental-group action. The orbit of zero is just
zero. So the same detection of nullity applies to sphere families
without a specified basepoint. QED.
By Proposition 5.5, Lemmas 5.3 and 5.4 imply
(5.6) c_B = 0 in [B, H(Sigma)].
The passage to free sphere classes is harmless: the fundamental-group
action fixes zero. The classification of smooth families by this
h-cobordism space now supplies incoming-relative product trivializations
of the whole family C_b, not just of its fibers (Courte-Porcelli,
Proposition 2.11).
5.4 Removing the cobordism and straightening the cores.
We now finish the proof of the one-critical proposition (Section 4.1).
By (5.6) and Lemma 5.2, each outer negative domain T_{-,b}^1 is
j_b(T_-^0) with a smoothly varying product collar attached. Shrink that
collar: if its product coordinate is [0,1], retain only [0,s] and let s
decrease to zero. The outer boundaries vary through embeddings, including
at s = 0 where they equal the boundary of j_b(T_-^0). So the family is
homotopic as tube domains to j_b(T_-^0).
Fix T_-^0 = S^(k-1) x D^l and let
e_b: S^(k-1) -> Y, e_b(u) = j_b(u, 0),
be its cores. We may replace j_b(T_-^0) by small round normal
neighborhoods of e_b. Here is a precise version that avoids choosing a
framing. First shrink the disk factors to a uniformly small radius. In
metric normal-bundle charts of the cores write
j_b(u,z) = ( psi_b(u,z), xi_b(u,z) ), psi_b(u,0) = u, xi_b(u,0) = 0.
On this smaller domain use, for 0 < t <= 1, the embeddings
(5.7) (u,z) |-> ( psi_b(u, t z), t^(-1) xi_b(u, t z) ).
These are the original embedding composed with dilation in the disk and
normal directions. The estimate |xi_b(u, t z)| <= C t |z| keeps them in
the chosen chart. At t = 0 the formula extends smoothly to
(u, D_z xi_b(u,0) z). The normal derivative is an isomorphism, so the
limit is also an embedding. Change its fiber ellipsoids to round balls by
interpolating positive definite metrics, with a common disk radius small
enough. At the level of domains this needs no fixed normal framing.
The core family is homotopic through embeddings to the fixed standard
core. Indeed, as maps into Y = S^(k+l-1), both it and the constant
standard family are nullhomotopic because
dim( B x S^(k-1) ) = (k-1) + 9 = k + 8 < k + l - 1.
Choose a smooth nullhomotopy, constant on collars of its two time
endpoints. Its parameter dimension, including time, is ten. Two-point
(multijet) transversality avoids double points since
10 + 2(k-1) < k + l - 1,
which is exactly where the unbalanced condition l > k + 9 enters (both
sides are 2k + 8 and k + l - 1 = k + (something > k + 9) - 1 > 2k + 8).
Vertical rank-defect strata have codimension at least l + 1 >
10 + (k-1) and are avoided as well. We use the parametric multijet
theorem in the form Borodzik-Powell-Teichner, Theorem 2.3.4 and
Corollary 2.3.5(iii), applied as follows.
Cover the compact parameter manifold B x [0,1], away from smaller
endpoint collars, by finitely many coordinate charts. In each chart
multiply the local perturbations by a parameter cutoff supported away
from the endpoint collars. Local source cutoffs allow independent
variations at distinct source points, as required for the two-point jets.
Apply transversality successively on compact subsets covering these
charts; sufficiently small later perturbations preserve avoidance already
obtained on compact subsets. The rank estimate first gives fiberwise
immersions. Compactness and C^1 openness then give a uniform neighborhood
of the diagonal on which distinct sufficiently close source points remain
distinct. Apply the two-point argument on the compact complement of this
neighborhood. The strict dimension bounds make each transverse
bad-stratum preimage empty. The endpoint families remain fixed, and their
embedding property persists on smaller endpoint collars by C^1 openness.
So every member of the resulting homotopy is an embedding. Uniformly
small round normal neighborhoods follow this family to a fixed tube.
All the resulting homotopies of domains give homotopies of tube data.
Use boundary collars for local defining-function choices; the fibers of
these choices are convex, so partitions of unity in parameters give
global choices (Section 3.2). Continuous families in the smooth topology
suffice for the homotopy class; when smooth parameter dependence is
needed, relative approximation preserves embeddings and regularity on
compact parameter sets. We have proved nullity of the enlarged family in
T_{k,l}. Adding the missing negative squares returns to the diagonal
system and proves the proposition.
This completes the proof of the one-critical proposition (Section 4.1):
a smooth family over S^9 with exactly one nondegenerate critical point,
homogeneous data in T_{p,p} at infinity, and trivial stable spherical
class, has stable tube class zero.
==============================================================================
SECTION 6. BUILDING THE EXACT EMBEDDED LAGRANGIAN
==============================================================================
We now turn the nonzero tube family of Section 3 into a concrete closed
exact embedded Lagrangian. The idea: add one sphere factor to the base;
its momentum records the DIRECTION of the vertical gradient of g_b, and
a scale R(b) that varies over S^9 records its MAGNITUDE. This makes the
critical-locus projection injective, hence an embedding. The infinity
data on a fixed sphere slice is the class a.
6.1 The generating-family criterion for an exact Lagrangian immersion.
Before the specific construction, we recall the elementary criterion. If
M is a smooth manifold, a smooth function F: M x R^r -> R is a MORSE
FAMILY when zero is a regular value of the full vertical-gradient map
d_wF: M x R^r -> (R^r)^*. "Full" means regular with respect to BOTH
variables (m and w), not that each fiber Hessian is invertible. Write
Crit(F) = { (q,w) : d_wF(q,w) = 0 },
Gamma_F(q,w) = (q, d_qF(q,w)) in T*M.
LEMMA 6.1 (generating map). For a Morse family F, the map
Gamma_F: Crit(F) -> T*M is an exact Lagrangian immersion, with primitive
F restricted to Crit(F).
Proof. Step 1: Crit(F) is a smooth manifold of dimension dim M. This is
the implicit function theorem for the equation d_wF = 0, whose
differential on the (q,w) variables is the surjective matrix A = (F_{wq},
F_{ww}) (surjective on the horizontal m-variable by the Morse-family
hypothesis).
Step 2: Gamma_F is an immersion. Suppose a tangent vector (delta q,
delta w) is killed by dGamma_F. Then delta q = 0 and the w-component
satisfies F_{ww} delta w = 0 and F_{qw} delta w = 0; equivalently
A^T delta w = 0 (using that F_{ww} is symmetric). Surjectivity of A
forces delta w = 0. So the differential is injective.
Step 3: exactness. On Crit(F), where d_wF = 0, the derivative of
F restricted to Crit equals its derivative in the q-direction only,
which is exactly the pullback of the canonical one-form lambda:
Gamma_F^* lambda = d( F | Crit(F) ).
Hence the pullback of omega = d lambda is d(df) = 0: the immersion is
isotropic. Dimension dim M = half of dim T*M, so it is Lagrangian.
QED.
6.2 The specific family and its critical locus.
Let B = S^9, N = 2m, and Q = B x S^(N-1). Take the localized family g_b
of Section 3.8 (b in B, w in R^N), together with a standard height
function theta: B -> R. Fix a metric on B, and choose a union U_0 of
neighborhoods of the two critical points of theta on which g_b is the
fixed quadratic form q_{m,m}. Set
delta_0 = min over B \ U_0 of |d theta| > 0.
Section 3.8 gives the mixed derivative bound |d_w d_b g_b(w)| <= C_1 |w|,
and the two-sided gradient bound c |w| <= |grad_w g_b(w)| <= C |w|.
Construct the family. Choose first a constant K with
(6.1) K delta_0 > C_1 / c.
Set
R(b) = Lambda exp( K theta(b) ).
So R varies from a positive minimum to a positive maximum; its logarithmic
gradient is K d theta, which is large away from the critical points.
Let G_b be a smooth function on all of R^N agreeing with g_b for
|w| >= 1; multiply by a radial cutoff that vanishes near the origin to
get such a G_b. Let beta: [0,infinity) -> [0,1] be smooth, equal to one
on [0,1] and zero on [2,infinity). Define
(6.2) F(b, v, w) = G_b(w) - R(b) beta( |w| / T ) <v, w>,
where (b,v) run over Q = B x S^(N-1) and w over R^N.
Choose Lambda so large that
R_min > max( C, sup over b, |w| <= 1 of |grad G_b(w)| ).
With
D_beta = sup over s >= 0 of ( |beta(s)| + s |beta'(s)| ),
choose
(6.3) T > max( 1, R_max / c, D_beta R_max / c ).
The order of choices is: K first (6.1), then Lambda, then T (6.3).
Step 1: no critical points in |w| <= 1. In that ball, F = G_b(w) - R(b)
<v,w> (no cutoff), and the vertical derivative is grad G_b(w) - R(b) v.
On the sphere base v with |v| = 1, |R(b) v| = R(b) > sup |grad G_b|,
so this never vanishes. So all critical points have |w| >= 1.
Step 2: no critical points in |w| >= T. There the cutoff term's vertical
derivative has norm at most D_beta R(b). So
|d_w F| >= c |w| - D_beta R(b) >= c T - D_beta R(b) > 0
by (6.3). So no critical points in |w| >= T either.
Step 3: therefore all critical points satisfy exactly the simpler
equation, since in 1 <= |w| <= T we have G_b = g_b and (for beta's
relevant argument range) the beta-cutoff is not yet zero but does not
change the leading term:
(6.4) grad g_b(w) = R(b) v.
Conversely, every solution of (6.4) has, from the two-sided bound,
1 < R(b) / C <= |w| <= R(b) / c < T (the last by 6.3),
so it IS a critical point of (6.2). By homogeneity, the critical locus is
smoothly parametrized by
(6.5) (b,u) |-> ( b, v = grad g_b(u) / |grad g_b(u)|,
w = R(b) u / |grad g_b(u)| ),
with u in S^(N-1). Its inverse recovers u = w / |w|.
Step 4: regularity of the critical equation. This shows F is a Morse
family, so Lemma 6.1 applies. Vary v in v^perp: this changes d_wF by
-R(b) delta v, spanning the directions perpendicular to v. Vary w
radially, delta w = w: this contributes grad g_b(w) - R(b) times the
derivative of the cutoff-linear term in the radial direction. At the
critical point the leading contribution is grad g_b(w) = R(b) v, because
grad g_b is homogeneous of degree one. Together, the v^perp variations
and the radial variation span all of R^N (v^perp has dimension N-1, plus
one radial direction). So the full vertical-gradient differential is
surjective. Hence F is a Morse family, and Lemma 6.1 gives an exact
Lagrangian immersion.
6.3 Injectivity via two momentum tests.
It remains to prove Gamma_F is injective, so the immersion is a
(self-)embedding. Suppose two critical points map to the same point of
T*Q.
Step 1: same base coordinates. Equal images means equal base coordinates
(b, v), so the two points are (b, v, w) and (b, v, w'). At critical
points the cutoffs are constant and G_b = g_b.
Step 2: the sphere momentum test. The covector in the S^(N-1) direction
at a critical point is
d_v F(xi) = -R(b) < w, xi >, for every xi perpendicular to v.
Equality of these covectors for w and w' implies
< w - w', xi > = 0 for every xi perpendicular to v,
hence w - w' = t v for some real t. This is the "sphere momentum"
(the momentum conjugate to the sphere coordinate v) confining the
possible collision to one line.
Step 3: bound on the base-momentum difference. Both w and w' (and the
whole straight segment between them, which is w' + s t v) have norm at
most R(b)/c. The mixed derivative bound (Section 3.8) gives, applying the
fundamental theorem of calculus along the segment,
(6.6) |d_b g_b(w) - d_b g_b(w')| <= (C_1 / c) R(b) |t|.
(This remains valid even if the segment passes through zero, because
d_b g_b is C^1 there.)
Step 4: the base momentum test. The difference in the B-momenta of the
two critical points is
d_b g_b(w) - d_b g_b(w') - K R(b) t d theta_b,
where the last term comes from the derivative of R(b) = Lambda exp(K
theta(b)) in the linear term -R(b) <v,w> = -R(b) <v, w' + t v> = -R(b)
<v,w'> - R(b) t. For b not in U_0, the norm of this difference is at
least
R(b) |t| ( K delta_0 - C_1 / c ).
By (6.1), this is positive unless t = 0. So equal B-momenta force t = 0,
which (with w - w' = t v) forces w = w'.
Step 5: the quadratic region. For b in U_0, the function g_b is the fixed
nondegenerate quadratic form q_{m,m}. Then (6.4) has a unique solution
for each v: w = R(b) v / 2 up to the metric, and the two solutions with
equal covectors coincide. So there is no collision in U_0 either.
Therefore the immersion is injective everywhere.
6.4 The result.
PROPOSITION 6.2 (embedded realization). On Q = B x S^(N-1) the smooth
Morse family F of (6.2) equals g_b(w) outside a uniform vertical ball,
and its generating map is a diffeomorphism from its critical locus onto a
closed exact embedded Lagrangian L inside T*Q. Its critical locus, and
hence L, is diffeomorphic to Q. The infinity class on each slice
B x {v_0} is the chosen class a.
Proof. Smoothness of F: clear from the construction (composition and
products of smooth functions). All critical points lie in the shell
1 < |w| < T (Steps 1-3 above). The full vertical gradient is regular
(Step 4), so F is a Morse family and Lemma 6.1 gives an exact Lagrangian
immersion, injective (Step 5 and the base/ sphere momentum tests). The
parameter space in (6.5) is compact and has no boundary; a compact
injective immersion is an embedding. Its image L is thus a closed
smoothly embedded exact Lagrangian, diffeomorphic to the parameter
space S^9 x S^(N-1) = Q.
Exactness: Lemma 6.1 gives the primitive F restricted to Crit(F). In
fact Euler's identity gives the global primitive F|Crit(F) = -g_b(w),
although the exactness argument did not need this formula.
Infinity class: for |w| >= 2T, the family equals g_b(w) independently of
v. So its restriction at infinity to any fixed sphere slice B x {v_0}
represents the chosen class a (the localized family of Section 3.8).
QED.
This is the closed exact embedded L whose existence is part (i) of the
main theorem. Part (ii), that no compactly supported Hamiltonian isotopy
carries the zero section onto L, is Section 7.
==============================================================================
SECTION 7. HAMILTONIAN TRANSPORT AND THE CONTRADICTION
==============================================================================
We now suppose there IS a compactly supported Hamiltonian isotopy of
T*Q carrying the zero section onto the Lagrangian L of Section 6, and
derive a contradiction. Along the isotopy we follow the generating family,
carrying its homogeneous data at infinity with it. Each small step of the
isotopy is realized by composing the generating family with a generating
function for the step, and each step adds a fixed split quadratic form.
At the zero section the transported family has exactly one nondegenerate
critical point above each base point, so restricting to a fixed S^9 slice
gives a family contradicting the one-critical obstruction (Section 4).
7.1 A Euclidean cylinder and a compactly supported extension.
Retain N = 2m and Q = S^9 x S^(N-1) from Section 6, and put d = 10 + N.
View Q as the product of the unit spheres in R^10 x R^N = R^d. For
0 < nu < 1, the map
iota: Q x (-nu, nu)^2 -> R^d, iota((b,v), alpha) = ((1+alpha_1) b,
(1+alpha_2) v)
is a diffeomorphism onto a bounded open neighborhood U of Q. (Here
alpha = (alpha_1, alpha_2) is the tuple of the two radial scaling
parameters.) Its inverse, the projection r_Q: U -> Q, and the two radial
coordinates extend smoothly to a neighborhood of the closure of U. The
cotangent lift gives the canonical identification
T*U = T*Q x T*((-nu,nu)^2), lambda_U = lambda_Q + p_alpha . d alpha.
If F is the family of Proposition 6.2, then F^(0)(x,w) = F(r_Q(x), w) is
a Morse family whose generated image is the cylinder
L_0 = { (q,alpha; p_q, 0) : (q,p_q) in L, alpha in (-nu,nu)^2 }.
Its critical parametrization is a diffeomorphism (F has that property),
all its vertical critical points lie in the uniform ball of Section 6,
and it has the stated smoothness and homogeneous estimates uniformly up
to the closure of U.
LEMMA 7.1 (extension of the reverse isotopy). Suppose a smooth compactly
supported Hamiltonian isotopy of T*Q sends the zero section Q_0 to L.
Then there is a smooth Hamiltonian on T*R^d, supported for all times in
one compact subset of T*U, whose flow phi~_t has the following properties.
Every trajectory starting on L_0 has constant radial position alpha. For
some nu_c > 0, the final image over U_c = iota(Q x (-nu_c,nu_c)^2) is
precisely the zero section:
(7.1) phi~_1(L_0) ∩ T*U_c = (U_c)_0.
Proof. Let phi_t be the assumed original flow, generated by H_t. The path
phi_(1-t) o phi_1^(-1) starts at the identity, ends at phi_1^(-1), and is
generated by
(7.2) H_t^Q = - H_(1-t).
Indeed, differentiating this path gives -X_(H_(1-t)) at its current
point; the fixed composition on the right introduces no conjugation of
the Hamiltonian.
Choose a cutoff 0 <= zeta <= 1 supported in (-nu,nu)^2 and identically 1
on (-nu_c,nu_c)^2. Choose a cutoff 0 <= xi <= 1 of compact support in R^2
which will equal 1 on a ball to be specified. On T*U set
(7.3) H~_t(q,alpha;p_q,p_alpha) = H_t^Q(q,p_q) zeta(alpha) xi(p_alpha),
and extend by zero to T*R^d. This is smooth with uniform compact support:
Q is compact, H_t^Q has uniformly compact momentum support, and both
additional coordinate pairs are cut off.
The convention iota(X_H) omega = -dH gives the radial equations
(7.4) d/dt alpha = H_t^Q zeta grad xi(p_alpha),
d/dt p_alpha = - H_t^Q (grad zeta) xi(p_alpha).
Set M = (sup over t,q,p_q of |H_t^Q|) (sup over alpha of |grad zeta(alpha)|).
This is independent of the plateau of xi. Take xi = 1 on
{ |p_alpha| <= M+1 }. A trajectory starting on L_0 starts with
p_alpha = 0; the second equation of (7.4) then gives |p_alpha(t)| <= M t
for 0 <= t <= 1, so it stays in this plateau. Hence the first equation of
(7.4) vanishes, so alpha is constant. When alpha is in (-nu_c,nu_c)^2, also
grad zeta = 0, so p_alpha = 0 and the T*Q motion is exactly the reverse
isotopy; it sends L to Q_0. Since all radial positions on the whole
cylinder are fixed, no other part of the cylinder enters the central slab.
This proves (7.1). QED.
7.2 A compact mixed generating function for each small step.
Fix a compact subset K contained in the interior of T*U that contains the
support of H~_t for every t. Points outside K are stationary, and
uniqueness of flow lines prevents a trajectory from crossing into the
stationary exterior in finite time. In particular all flow maps and their
successive time increments are supported in this common compact region.
Smoothness in time and compactness allow a finite subdivision
0 = t_0 < t_1 < ... < t_s = 1 such that every map
psi_j = phi~_(t_j) o phi~_(t_(j-1))^(-1) is as close to the identity in
C^1 as required below.
LEMMA 7.2 (mixed coordinates and the action primitive). For a sufficiently
small increment, write psi(x,p) = (X,P) in Euclidean cotangent coordinates.
The pair (X,p) is a global coordinate system on the graph of psi. There is
a smooth compactly supported function sigma(X,p) such that
(7.5) x = X + sigma_p(X,p), P = p + sigma_X(X,p).
One can choose a single compact subset D contained in U containing the
spatial support projection of all these functions, independently of a
later refinement of the subdivision. Also the matrix I + sigma_(pX) is
invertible, and refinement makes |sigma_p|_infinity arbitrarily small.
Proof. Require |D_x X - I| < epsilon < 1 uniformly. For each fixed p, the
map x |-> X(x,p) is a local diffeomorphism and satisfies
|X(x_1,p) - X(x_2,p)| >= (1-epsilon)|x_1 - x_2|. It is injective and
proper (identity outside a compact set). Its image is both open and closed
in R^d, hence all of R^d. The inverse x = x(X,p) is smooth jointly in
(X,p).
For a step from time a to time b, let phi_(s,a) denote its partial flow.
Cartan's formula and the sign convention give
L_(X_(H~_s)) lambda = d( lambda(X_(H~_s)) - H~_s ).
Consequently the action integral
(7.6) A(x,p) = integral from a to b of
( lambda(X_(H~_s)) - H~_s )( phi_(s,a)(x,p) ) ds
satisfies psi^* lambda - lambda = dA. It is compactly supported: outside K
the trajectory is stationary and its integrand is zero.
On the graph, expressed in mixed coordinates, put
S(X,p) = A(x(X,p),p) + x(X,p) . p. Then dS = P.dX + x.dp. Writing
S = X.p + sigma gives (7.5). Explicitly,
(7.7) sigma(X,p) = A(x(X,p),p) + ( x(X,p) - X ) . p.
This expression vanishes outside the compact mixed-coordinate image of K.
Its spatial projection lies in a fixed compact subset of U, because both
the input and output trajectories remain in K. Enlarge that projection
slightly to obtain D contained in U. So for every X in U \ D, sigma and all
its derivatives vanish on a neighborhood of X, for all momenta. Finally
I + sigma_(pX) = D_X x is invertible, and sigma_p = x - X tends uniformly
to zero as the time increment tends to zero. QED.
7.3 Transport of one Morse family.
The following proposition supplies the analytic induction step. The
variables called w include all stabilizing variables introduced at earlier
steps.
PROPOSITION 7.3 (one transport step with controlled infinity data).
Suppose F_o: U x R^r -> R is a smooth Morse family, with Gamma_(F_o) a
diffeomorphism onto its generated Lagrangian L_o in T*U. Suppose it is
smooth up to the closure of U, with uniform bounds on compact vertical
sets, and equals a homogeneous function g_o(x,w) for |w| >= R_o, where g_o
is smooth for w != 0 and
(7.8) |grad_w g_o(x,w)| >= c_o |w|,
|d_x g_o(x,w)| <= C_o |w|^2, c_o > 0.
For a small step psi as in Lemma 7.2, there is a smooth Morse family on
U x R^(r+2d) generating psi(L_o) with the same properties. Its critical
locus corresponds diffeomorphically to Crit(F_o). On every fixed compact
parameter slice in U, its homogeneous data represent the split quadratic
stabilization of the preceding tube class.
Proof. Introduce z, p in R^d. Before choosing cutoffs, consider the local
expression
F_o(X+z, w) + sigma(X,p) - z . p, X+z in U.
Its p- and z-equations are z = sigma_p(X,p) and p = grad_x F_o(X+z,w).
Together with the old w-equation, these are precisely the mixed graph
equations for composition with psi: z is the displacement from the new
base point X to the old one, and p is the old momentum. The cutoffs keep
the shifted base point in U and switch off the shift for large |w|. We
then compare the result with g_o(X,w) - z.p to restore homogeneity at
infinity.
Choice of cutoffs. Choose zeta_1: U -> [0,1] of compact support,
identically 1 on an open set V containing D. Fix delta_0 > 0 so that
(7.9) 2 delta_0 < dist(supp zeta_1, boundary(U)),
|sigma_p|_infinity < delta_0.
The first choice is made before refining the subdivision to obtain the
second. Let beta_1: R^d -> [0,1] be 1 for |z| <= delta_0 and 0 for
|z| >= 2 delta_0. Choose a smooth radial chi: R^r -> [0,1], equal to 1 on
a ball strictly larger than { |w| <= R_o }, of compact support, and such
that
(7.10) |grad chi(w)| <= delta_1 / |w| (w != 0),
where delta_1 > 0 may be arbitrarily small. (Construction: after fixing an
inner radius R_1 > R_o, take chi(w) = theta(log(|w|/R_1)/T) outside that
radius, with theta a fixed smooth cutoff equal to 1 near (-inf,0] and 0
near [1,inf); choosing T large gives (7.10). Extend by 1 at smaller radii.)
Define
(7.11) kappa(X,z,w) = zeta_1(X) beta_1(z) chi(w),
x = X + kappa(X,z,w) z,
F^(X,w,z,p) = F_o(X + kappa(X,z,w) z, w) + sigma(X,p) - z.p.
The shifted argument always lies in U. When the shift is nonzero,
X is in supp zeta_1 and its length is less than 2 delta_0; the buffer in
(7.9) places that entire ball in U. When the shift vanishes the assertion
is immediate.
The full critical equations. Write a = grad_x F_o(x,w), identifying
Euclidean covectors and vectors. Direct differentiation, before
discarding any cutoff terms, gives
(7.12) F^_p = sigma_p(X,p) - z,
F^_z = kappa a + (a.z) grad_z kappa - p,
F^_w = grad_w F_o(x,w) + (a.z) grad_w kappa,
F^_X = a + (a.z) grad_X kappa + sigma_X(X,p).
At any vertical critical point, F^_p = 0 forces z = sigma_p(X,p), hence
|z| < delta_0 and beta_1 = 1 on a neighborhood of that point. When
|w| >= R_o, the old function is homogeneous at every allowed shifted base
point. Equations (7.8) and (7.10) therefore imply
(7.13) |F^_w| >= (c_o - C_o delta_0 delta_1) |w|.
Choose delta_1 so that C_o delta_0 delta_1 < c_o/2. Then no vertical
critical point has |w| >= R_o; at every one of them chi = 1 on a
neighborhood as well.
There are two open regions, namely V and U \ D; they cover U.
Region X in V: the three cutoff factors are 1 near a solution, and the
critical equations become
(7.14) grad_w F_o(x,w) = 0, p = grad_x F_o(x,w),
x = X + sigma_p(X,p), x = X + z.
They express exactly composition with the graph (7.5), and the generated
covector is
(7.15) F^_X = p + sigma_X(X,p) = P.
The critical equation is regular here. In local total coordinates
(x,w,p,X), first impose the regular old equation grad_w F_o = 0. With
(x,w) fixed, the equation p - grad_x F_o = 0 has identity coefficient in
p. With (x,w,p) fixed, the last graph equation has invertible coefficient
I + sigma_(pX) in X. This triangular order proves surjectivity of the
differential of the full new vertical-gradient map.
Region X in U \ D: the function sigma vanishes on a spatial neighborhood
for all p. The equations instead reduce to
(7.16) z = 0, x = X, grad_w F_o(X,w) = 0,
p = zeta_1(X) grad_x F_o(X,w).
The auxiliary variable p may differ from the old momentum. Nevertheless,
(7.12) gives exactly
(7.17) F^_X = grad_x F_o(X,w),
because all the spatial cutoff terms are multiplied by z = 0. This is the
identity correspondence, the true graph of psi over this region. For
regularity, first use -z = 0 to eliminate z, then zeta_1 grad_x F_o - p = 0
to eliminate p; the remaining equation is the regular old vertical-gradient
equation. This also checks the entire transition region of zeta_1. The two
descriptions agree on the overlap, where zeta_1 = 1.
The correspondence of critical loci. For an old critical point (x,w) put
p_o = grad_x F_o(x,w) and (X,P) = psi(x,p_o). Define
(7.18) (x,w) |-> ( X, w, z = x - X, p = zeta_1(X) p_o ).
If X is in V these are the mixed graph equations, with p = p_o. If X is
not in D, the true graph is the identity, so x = X, z = 0, and (7.16)
applies. Thus (7.18) is a smooth map into Crit(F^) on all of Crit(F_o).
Its inverse is
(7.19) (X,w,z,p) |-> ( X + kappa(X,z,w) z, w ).
The preceding equations show these maps are inverse and that the generated
image is exactly psi(L_o). The step preserves T*U, since its support is
compactly contained there. So no old or new critical point is lost at the
base boundary, and Gamma_(F^) is a diffeomorphism onto the transported
image. All new solutions obey a uniform bound: |w| < R_o, |z| < delta_0,
and |p| <= sup{ |grad_x F_o(x,w)| : x in closure(U), |w| <= R_o } < infinity.
Restoring exact homogeneity at infinity. Set W = (w,z,p) and consider the
homogeneous reference
(7.20) G_0(X,W) = g_o(X,w) - z.p.
It is C^1 everywhere, homogeneous of degree two, and satisfies
(7.21) |grad_W G_0|^2 = |grad_w g_o|^2 + |z|^2 + |p|^2
>= min(c_o^2, 1) |W|^2.
Both the value error and its FULL vertical first derivative are uniformly
bounded. To verify this, write E(X,W) = F^(X,W) - G_0(X,W) as
E = [ F_o(X + kappa z, w) - F_o(X,w) ]
+ [ F_o(X,w) - g_o(X,w) ] + sigma(X,p). (7.22)
The first bracket and every term in its vertical derivative can be nonzero
only with w, z in fixed bounded sets, where the required derivatives of
F_o are uniformly bounded. The second bracket and its vertical derivative
are bounded because it vanishes for |w| >= R_o. The final term has compact
support. The expression -z.p, although unbounded, has cancelled exactly.
Thus for some finite M_0, M_1,
(7.23) |E(X,W)| <= M_0, |grad_W E(X,W)| <= M_1
(X in closure(U), W in R^(r+2d)).
No compact-support assertion for E in all its variables is needed.
Apply the stabilization-smoothing lemma (Section 4.5) with q'(z,p) =
-z.p, uniformly over closure(U). Its homogeneous smoothing is
(7.24) G_eta(X,W) = rho_eta( |w| / |W| ) g_o(X,w) - z.p (W != 0),
where rho_eta vanishes for arguments at most eta and is 1 for arguments at
least 2 eta. Choose eta small and then A sufficiently large as in that
lemma, with A beyond the uniform critical-locus bound. Let tau_A be the
radial cutoff from that construction and set
(7.25) F_n = (1 - tau_A) F^ + tau_A G_eta.
The lemma shows this function is smooth, equals G_eta for |W| >= 2A, and
has no new critical points. It leaves a neighborhood of the entire critical
locus unchanged, so F_n has the same regular critical locus and generated
parametrization as F^. It also supplies a straight homotopy from G_0 to
G_eta through regular C^1 homogeneous functions.
The new homogeneous function G_eta is smooth off zero, smoothly dependent
on X up to closure(U), and has a uniform positive gradient margin on the
unit sphere. Homogeneity and compactness there give the next constants of
the form (7.8); the remaining uniform smooth bounds follow directly from
the construction. The quadratic form -z.p is nondegenerate with signature
(d,d): in fixed linear coordinates z = a + b, p = a - b it equals
-|a|^2 + |b|^2. The space of nondegenerate forms of signature (d,d) is
connected. Choose a fixed path in this space from -z.p to the standard
split form in the chosen added coordinates. Its direct sum with g_o(X,w)
has nonzero total gradient off the origin, since both block gradients can
vanish only there. This path is independent of X, so direct sum defines a
homotopy over the whole base. Together with the straight homotopy to
G_eta, it preserves the free stable tube class on every fixed compact
parameter slice. This proves all the assertions. QED.
7.4 The final critical point and the contradiction.
We can now prove part (ii) of the main theorem. Suppose, for contradiction,
that a compactly supported Hamiltonian isotopy carries Q_0 onto L.
Take the class a != 0 with gamma_* a = 0 (Section 3.7) and the closed exact
embedded Lagrangian L in T*(S^9 x S^(N-1)) (Section 6). Lemma 7.1 gives a
compactly supported Euclidean Hamiltonian whose final cylinder is the zero
graph over U_c. Start with F^(0) (equation for the initial family) and
apply Proposition 7.3 to each of its s increments. Only finitely many steps
are required. The subdivision is fixed by the mixed-coordinate and
spatial-buffer conditions, which depend on the Hamiltonian and its support.
At each step the new family constants change only the logarithmic cutoff,
smoothing parameter, and outer radius; they impose no further restriction
on the step length. So the estimates and cutoff radii may be chosen anew at
each step. We obtain a smooth family
F^(s): U x R^(2p) -> R, p = m + d s,
with exactly homogeneous smooth-off-zero data at infinity, whose regular
critical locus maps diffeomorphically onto phi~_1(L_0).
Over U_c the full critical-locus projection
(7.26) Crit(F^(s)) restricted to U_c -> U_c
is a diffeomorphism: it is the generating parametrization followed by
projection of the zero graph. In particular there is precisely one
critical point over every X in U_c, and these points form a smooth section.
Their vertical Hessians are invertible. Indeed, if v in R^(2p) were in the
kernel of such a Hessian, then
D( D_W F^(s) )_(X,W) (0,v) = D_W^2 F^(s)(X,W) v = 0.
Since the full critical equation is regular, its zero set has this kernel as
its tangent space. So (0,v) would be tangent to Crit(F^(s)) and in the
kernel of the differential of (7.26). That projection is a diffeomorphism,
so v = 0; the square Hessian is nonsingular.
Fix v_0 in S^(N-1) and restrict the base to
b |-> iota( (b, v_0), 0 ), b in S^9.
The resulting smooth family f_b: R^(2p) -> R therefore has exactly one
nondegenerate fiber critical point, varying smoothly with b. Its
homogeneous data belong to T_(p,p), since each transport step added
signature (d,d). On this fixed slice the initial homogeneous family was
g_b, and every subsequent step preserved its class after split
stabilization. Hence, writing g_infinity: S^9 -> T for the final stable
tube family,
(7.27) [g_infinity]_free = [a]_free != [*], gamma o g_infinity ~ *.
This formulation does not require the smoothing homotopies to fix a chosen
basepoint; equivalently, the basepoint paths supplied by these homotopies
identify the resulting based class with a.
Now all hypotheses of the one-critical proposition (Section 4.1) hold: a
smooth family over B = S^9, exactly one nondegenerate critical point in
each fiber, homogeneous data in T_(p,p) at infinity, and trivial stable
spherical class (7.27). That proposition concludes the family is freely
null in the stable tube space, contradicting the non-null class a of
(7.27). Therefore the assumed compactly supported Hamiltonian isotopy does
not exist. This proves part (ii) of the main theorem.
==============================================================================
SECTION 8. PUTTING IT TOGETHER: PROOF OF THE MAIN THEOREM
==============================================================================
We now assemble the pieces. The proof is short because all the work is
in the lemmas of Sections 3-7.
THEOREM (main, restated). For some sufficiently large even integer N,
Q = S^9 x S^(N-1) admits a closed exact smoothly embedded Lagrangian L in
T*Q such that L is diffeomorphic to Q and no compactly supported
Hamiltonian isotopy carries the zero section Q_0 of T*Q onto L.
Proof.
Step 1: choose the parameter size. Take N = 2m with m large and even,
m >= 20, and m satisfying the room condition that emerges in Step 4
(m > k + 9 = p + 9 for the appropriate p; concretely take m = 2p with
p >= 20). The construction is stable in m, so any sufficiently large even
N works. Fix such an N and set B = S^9, Q = B x S^(N-1).
Step 2: the nonzero tube class. By Proposition (kernel class) of Section
3.7, there exists a class
a in pi_9(T), a != 0, gamma_* a = 0,
where T is the stable smooth tube space and gamma: T -> BG is the
spherical-fibration map. Moreover (BJ)_*: pi_9(BO) -> pi_9(BG) is
injective. The class comes from the mismatch of 2-primary orders
|pi_10(BG)_(2)| = 8 and |pi_9(H^s(*))_(2)| = 32 (Rognes).
Step 3: a concrete representative. By the localized-family lemma of
Section 3.8, the class a is represented, for our m, by a smooth family
b |-> h_b in T_{m,m} over B = S^9, standard (equal to q_{m,m}) on
neighborhoods of the two critical points of a height function theta, with
uniform gradient bounds c|w| <= |grad_w g_b(w)| <= C|w| and mixed bound
|d_w d_b g_b(w)| <= C_1 |w|.
Step 4: the embedded Lagrangian. Apply Proposition 6.2 (embedded
realization) to this family. It produces a smooth Morse family F on
Q x R^N = (B x S^(N-1)) x R^N of the form
F(b,v,w) = G_b(w) - R(b) beta(|w|/T) <v,w>,
with R(b) = Lambda exp(K theta(b)) and K delta_0 > C_1/c. The family
equals g_b(w) outside a uniform vertical ball. Its critical locus is
parametrized by S^9 x S^(N-1) (equation 6.5), the generating map is
injective by the sphere- and base-momentum tests (Section 6.3), so by
Proposition 6.2 and Lemma 6.1 it is a closed exact smoothly embedded
Lagrangian L in T*Q.
Its critical locus, hence L itself, is diffeomorphic to Q. This proves
part (i) of the theorem. On each fixed sphere slice B x {v_0}, the
restriction at infinity of F represents the class a (Section 6.4).
Step 5: the isotopy is impossible. Suppose, for contradiction, that a
compactly supported Hamiltonian isotopy carries Q_0 onto L. By Lemma 7.1
the reverse isotopy extends to a compactly supported Hamiltonian on
T*R^d whose final image of the cylinder over L is the zero section over a
smaller cylinder U_c. Subdivide the isotopy into finitely many small steps
(Section 7.2), and apply Proposition 7.3 (one transport step) to each. We
obtain a smooth family F^(s) on U x R^(2p) (p = m + d s) with exactly
homogeneous smooth-off-zero data at infinity, whose regular critical locus
maps diffeomorphically onto the transported image.
Step 6: one critical point per base point. Over U_c the full critical-locus
projection Crit(F^(s)) restricted to U_c -> U_c is a diffeomorphism (it is
the generating parametrization followed by projection of the zero graph).
Hence there is exactly one critical point over each X in U_c, and its
vertical Hessian is invertible (Section 7.4: a kernel vector would be
tangent to the critical locus and killed by the projection differential, so
it is zero).
Step 7: restrict to the slice. Fix v_0 in S^(N-1) and restrict the base to
b |-> iota((b,v_0), 0), b in S^9. The resulting family f_b: R^(2p) -> R
has exactly one nondegenerate fiber critical point, varying smoothly with
b. Its homogeneous data lie in T_{p,p} (each transport step added
signature (d,d), returning the diagonal). The final stable tube class is
[g_infinity]_free = [a]_free != [*], gamma o g_infinity ~ *,
because each step preserved the class after split stabilization, and
gamma_* a = 0. (= equation 7.27.)
Step 8: the contradiction. The family f_b of Step 7 satisfies all three
hypotheses of the one-critical proposition (Section 4.1): smooth family
over S^9, exactly one nondegenerate critical point per fiber, homogeneous
data in T_{p,p} at infinity, and trivial stable spherical class. That
proposition (proved in Sections 4-5) concludes that the stable
tube class of f_b at infinity is ZERO in [B, T]. But Step 7 says that
stable class is [a] != 0. Contradiction.
Therefore the assumed compactly supported Hamiltonian isotopy does not
exist. This proves part (ii) of the theorem. QED.
That is the whole proof. Everything hinged on two facts: (1) the existence
of a nonzero stable tube class a whose spherical fibration is trivial
(algebra, Section 3), and (2) the rigidity of single-critical-point
families (geometry, Sections 4-5). The construction of Section 6 realizes
a, and the transport of Section 7 forces any hypothetical isotopy to
produce a single-critical family realizing a, contradicting (2).
==============================================================================
APPENDIX A. A GLOSSARY OF THE OBJECTS, FOR REFERENCE
==============================================================================
Q_0 the zero section of T*Q (zero momentum everywhere).
lambda the canonical 1-form p.dq on T*Q.
omega = d lambda the canonical symplectic form on T*Q.
exact L a Lagrangian L with lambda|_L = df for a global f: L -> R.
H_t, phi_t a time-dependent Hamiltonian and its flow (a symplectic
isotopy); "compactly supported" means supported in a fixed
compact set.
Morse family F a smooth F(q,w) with zero a regular value of d_w F.
Gamma_F the generating map (q,w) |-> (q, d_qF) on Crit(F).
q_{k,l} the standard form -|x|^2 + |y|^2 of signature (k,l) on R^(k+l).
T_{k,l} the path component of q_{k,l}|_sphere in regular-valued
angular functions on S^(k+l-1).
D(h) the negative domain { h <= 0 }, a smooth tube.
T the stable smooth tube space, hocolim of T_{m,m}.
G, BG stable sphere self-equivalences and their classifying space.
gamma the map T -> BG sending a tube to its spherical fibration.
H(X), H^s(X) smooth h-cobordism space of X and its stable version.
Wh^diff(X) smooth Whitehead space; H^s(X) ~ Omega Wh^diff(X).
a the nonzero class in pi_9(T) with gamma_* a = 0.
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APPENDIX B. SOURCES FOR THE CLASSICAL INPUTS
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We list the standard facts cited. None of these is proved here; they are
the load-bearing theorems of the subject.
Bott periodicity: pi_8(O) = pi_9(O) = Z/2. (Bott, The stable homotopy of
the classical groups, Annals of Math 70 (1959).)
The stable J-homomorphism is injective in these degrees. (J. F. Adams,
On the groups J(X) IV, Topology 5 (1966), Theorem 9.5.)
Stable homotopy of spheres: pi_9^S = (Z/2)^3. (Isaksen-Wang-Xu, Stable
homotopy groups of spheres: from dimension 0 to 90, Publ. Math. IHES 137
(2023), Table 1.)
Two-primary stable Whitehead group of a point: pi_10 Wh^diff(*) has
2-primary part Z/8 + (Z/2)^2 (order 32). (Rognes, Two-primary algebraic
K-theory of pointed spaces, Topology 41 (2002), Theorem 5.8; also Rognes,
Algebraic K-theory of strict ring spectra, ICM 2014, Theorem 3.3.)
Waldhausen's stable parametrized h-cobordism theorem: H^s(X) ~
Omega Wh^diff(X). (Waldhausen, Jahren, Rognes, Spaces of PL Manifolds and
Categories of Simple Maps, Annals of Math Studies 186, 2013, Theorems 0.1
and 0.3.)
Waldhausen's smooth tube theorem: the tube fibration H^s(*) -> T -> BG,
with gamma o iota ~ BJ on quadratic forms. (Waldhausen 1982, and the
homogeneous C^1 function model in Alvarez-Gavela-Igusa-Sullivan,
Legendrian and Lagrangian higher torsion, arXiv:2603.28007, Theorem 3.30,
Remark 3.34.)
Igusa stability for one-time stabilization: H(X) -> H(X x J) is
r-connected when dim X >= max(2r+8, 3r+5); and the induced map on stable
spaces for an (r+1)-connected inclusion. (Igusa, The stability theorem for
smooth pseudoisotopies, K-Theory 2 (1988); used in the form of
Courte-Porcelli, arXiv:2506.06110, Theorems 2.16 and 2.17.)
The smooth h-cobordism theorem in high dimension: an h-cobordism on a
simply connected closed manifold of dimension at least 5 is a product.
(Milnor, Lectures on the h-Cobordism Theorem, 1965, Theorem 9.1.)
Homological Whitehead theorem. (Hatcher, Algebraic Topology, 2002,
Corollary 4.33.)
Parametric multijet transversality for the core straightening. (Borodzik-
Powell-Teichner, arXiv:2503.04348, Theorem 2.3.4 and Corollary 2.3.5(iii).)
Background on nearby Lagrangians and generating families: Abouzaid
(Invent. Math. 189, 2012), Kragh (Geom. Topol. 17, 2013), Abouzaid-Kragh
(Acta Math. 220, 2018), Abouzaid-Courte-Guillermou-Kragh (Duke 174, 2025),
and the elementary generating-family sources Chaperon (1984), Sikorav
(Comment. Math. Helv. 62, 1987), Eliashberg-Gromov (AMS Transl. 186, 1998).
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