64c3814
==== TABLE OF CONTENTS ====
Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//assembly.tex
Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//counts.tex
Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//geometry.tex
Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//handles.tex
Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//history.tex
Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//introduction.tex
Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//involution.tex
Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//main.tex
Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//morse.tex
Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//references.tex
===========================
==== BEGIN Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//assembly.tex ====
\section{Proof of the counterexample}
\label{sec:assembly}
We now combine the geometric Morse comparison, the two manifold constructions, and the perturbation of the involution.
\begin{proof}[Proof of Theorem~\ref{thm:main}]
Fix an integer $N\geq704$ and take the simply connected Kähler manifold $X$ of Section~\ref{sec:geometry}. Let $Y$ be the Kähler surface of Proposition~\ref{prop:surface}, and put
\[
M=X\times Y,\qquad C=X\times\CP^1,
\]
with the product symplectic form on $M$. Both $M$ and $C$ are simply connected. Their real dimensions are $2N+4$ and $2N+2$, respectively, so Theorem~\ref{thm:smale} gives
\[
\SM(M)=\lambda(M),\qquad \Morse(C)=\lambda(C).
\]
Choose a Morse function on $C$ with exactly $\lambda(C)$ critical points. Corollary~\ref{cor:product-realization}, applied to the four fixed components of $\id_X\times g_Y$, provides a smooth one-periodic Hamiltonian $H$ whose time-one map has exactly these $4\lambda(C)$ fixed points, all nondegenerate. Every corresponding periodic orbit is contractible because $M$ is simply connected. Proposition~\ref{prop:gap} now gives
\[
\#\Fix_0(\phi_H^1;H)=4\lambda(C)=\lambda(M)-16=\SM(M)-16.
\]
More explicitly, the construction has
\[
\SM(M)=456N-2056,\qquad
\#\Fix(\phi_H^1)=456N-2072.
\]
Taking $N=704$ gives the dimension and counts stated in the theorem.
\end{proof}
\begin{remark}[Comparison with the integral Floer bound]\label{rem:periodic-bound}
The distinction between integer and cyclic gradings can be seen numerically in this family. The \emph{minimal Chern number} $\nu$ is the positive generator of the subgroup $\langle c_1(TM),\pi_2(M)\rangle\subset\Z$ when that subgroup is nonzero. An exceptional line in $Y$, regarded as a sphere in $M$ with its $X$-coordinate fixed, has Chern number $2-1=1$: its tangent line has degree $2$ and its normal line has degree $-1$. Hence $\nu=1$ for $M$.
The integral bound of Bai--Xu \cite[Theorem~A]{BaiXu2022}, equivalently \cite[Corollary~1.2]{BaiXu2026}, therefore uses the two groups
\[
\overline H_{\bar i}(M;\Z)=\bigoplus_{k\equiv i\ (\mathrm{mod}\ 2)}H_k(M;\Z),
\qquad \bar i\in\Z/2.
\]
In each parity the torsion is
\[
(\Z/2)^{16(N-3)}\oplus(\Z/3)^{32(N-5)},
\]
by summing the profiles of Section~\ref{sec:counts}. Its minimum number of generators is $32(N-5)$. Thus their lower bound evaluates to
\[
b(M)+2\sum_{\bar i\in\Z/2}d\bigl(\Tor\overline H_{\bar i}(M;\Z)\bigr)
=8(41N-183)+128(N-5)=456N-2104,
\]
where $d$ denotes the minimum number of generators. Our Hamiltonian has $456N-2072$ fixed points, thirty-two more than this bound. The counterexample therefore distinguishes the integer-graded geometric Morse count from the established cyclically graded integral count.
\end{remark}
==== END Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//assembly.tex ====
==== BEGIN Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//counts.tex ====
\section{The difference in Morse counts}\label{sec:counts}
The four fixed components of the involution on $X\times Y$ are copies
of $X\times\CP^1$. We now compare their combined Morse count with the
stable Morse number of the ambient manifold. Set
\begin{equation}\label{eq:counts-products}
C=X\times\CP^1,\qquad M=X\times Y.
\end{equation}
The calculation rests on the fact that torsion at distinct primes can
share a generator. For nonnegative integers $a,q$,
\begin{equation}\label{eq:counts-generators}
\text{the minimum number of generators of }
(\Z/2)^a\oplus(\Z/3)^q
\text{ is }\max(a,q).
\end{equation}
Reduction modulo $2$ and modulo $3$ gives the lower bound; pairing
cyclic summands of different orders into copies of $\Z/6$ gives the
upper bound.
\begin{proposition}\label{prop:gap}
For the manifolds in \eqref{eq:counts-products},
\begin{equation}\label{eq:counts-values}
\lambda(C)=114N-518,\qquad
\lambda(M)=456N-2056.
\end{equation}
In particular,
\begin{equation}\label{eq:counts-gap}
\lambda(M)-4\lambda(C)=16.
\end{equation}
\end{proposition}
\begin{proof}
Recall from Section~\ref{sec:geometry} that $L=N-3$ and that, in each of the two lists of degrees
$1+2j$ and $2+2j$, the torsion in $H_*(X;\Z)$ has multiplicities
\[
a_j=2,\qquad
(q_1,\ldots,q_L)=(1,3,4,\ldots,4,3,1),
\qquad 1\leq j\leq L,
\]
at the primes $2$ and $3$, respectively. Extend both sequences by
zero to all $j\in\Z$. Put $c_j=\max(a_j,q_j)$ and
\begin{equation}\label{eq:counts-S}
S=\sum_jc_j=2+3+4(L-4)+3+2=4L-6=4N-18.
\end{equation}
The homology of $\CP^1$ is free with ranks $(1,1)$ in degrees
$(0,2)$, and that of $Y$ is free with ranks $(1,6,1)$ in degrees
$(0,2,4)$. Thus the integral K\"unneth formula has no torsion-product
terms in either product. It replaces the profiles $a,q$ by their
convolutions with
\[
e_C=(1,1),\qquad e_M=(1,6,1),
\]
where these sequences start at index $0$ and
$(e*a)_j=\sum_ke_ka_{j-k}$. By
\eqref{eq:counts-generators}, the sum of torsion generator numbers
in either one of the two degree lists is therefore
\[
\sum_j\max\bigl((e*a)_j,(e*q)_j\bigr).
\]
We compute how much this sum falls short of
$\sum_j(e*c)_j=(\sum_ke_k)S$. Define the nonnegative sequences
\[
u_j=\max(a_j-q_j,0),\qquad
v_j=\max(q_j-a_j,0).
\]
Since $c=a+v=q+u$, the loss at index $j$ is exactly
\begin{equation}\label{eq:counts-loss}
(e*c)_j-\max\bigl((e*a)_j,(e*q)_j\bigr)
=\min\bigl((e*u)_j,(e*v)_j\bigr).
\end{equation}
Here $u_1=u_L=1$ and $u_j=0$ elsewhere, whereas
$v_2=v_{L-1}=1$, $v_j=2$ for $3\leq j\leq L-2$, and $v_j=0$
elsewhere. Consequently the loss can occur only near the two ends
of the profiles. Table~\ref{tab:counts-loss} gives every index at
which $(e*u)_j$ is nonzero. The ends do not overlap, since $L\geq7$.
\begin{table}[htbp]
\centering
\begin{tabular}{@{}ccrrr@{}}
\toprule
Kernel $e$ & Indices $j$ & $(e*u)_j$ & $(e*v)_j$ & Loss \\
\midrule
$(1,1)$ & $1,\ L+1$ & $1$ & $0$ & $0$ \\
& $2,\ L$ & $1$ & $1$ & $1$ \\
\midrule
$(1,6,1)$ & $1,\ L+2$ & $1$ & $0$ & $0$ \\
& $2,\ L+1$ & $6$ & $1$ & $1$ \\
& $3,\ L$ & $1$ & $8$ & $1$ \\
\bottomrule
\end{tabular}
\caption{The loss in the torsion generator count under convolution.
Both indices in each pair have the displayed values; every unlisted
index has zero loss.}
\label{tab:counts-loss}
\end{table}
The total loss is $2$ for $e_C$ and $4$ for $e_M$. All degree shifts
are even, so the two parity lists remain separate and give identical
contributions. Hence
\begin{equation}\label{eq:counts-total-torsion}
\sum_i t_i(C)=2(2S-2),\qquad
\sum_i t_i(M)=2(8S-4).
\end{equation}
The free ranks satisfy $b(C)=2b(X)$ and $b(M)=8b(X)$. It follows
that
\begin{align*}
\lambda(C)&=2b(X)+8S-8,\\
\lambda(M)&=8b(X)+32S-16.
\end{align*}
Subtracting four times the first identity from the second proves
\eqref{eq:counts-gap}. Finally, substituting $b(X)=41N-183$ and
$S=4N-18$ gives \eqref{eq:counts-values}.
\end{proof}
It remains to realize $4\lambda(C)$ as a Hamiltonian fixed-point count on $M$.
==== END Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//counts.tex ====
==== BEGIN Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//geometry.tex ====
\section{A simply connected K\"ahler manifold with two torsion primes}
\label{sec:geometry}
Fix an integer $N\geq 704$. We construct a simply connected compact
K\"ahler manifold $X$ of complex dimension $N$ whose integral homology has
prescribed $2$-primary and $3$-primary torsion. The construction has two
steps. First, quotients of products of elliptic curves provide surfaces
with the required torsion groups. Blowing up projective space along these
surfaces, with two projective-line factors added to one center, then
places that torsion in the required degrees.
\subsection{Two quotient surfaces}
Write
\[
E=\C/(\Z+i\Z),\qquad
\zeta_p=\exp(2\pi i/p),\qquad
\Lambda_2=\Z+i\Z,\qquad
\Lambda_3=\Z+\zeta_3\Z.
\]
For $p\in\{2,3\}$, let $E_p=\C/\Lambda_p$ and define
\[
\sigma_p:E\times E_p\longrightarrow E\times E_p,
\qquad
\sigma_p(z,w)=(z+1/p,\zeta_p w).
\]
Multiplication by $\zeta_p$ preserves $\Lambda_p$, so this map is
well-defined and holomorphic. It has order $p$, and its nonidentity
powers have no fixed points: translation by $k/p$ on $E$ is nontrivial
for $0<k<p$. Thus
\[
S_p=(E\times E_p)/\langle\sigma_p\rangle
\]
is a smooth compact connected complex surface. These are classical bielliptic surfaces; see \cite{Serrano1991,Blomme2025} for their quotient descriptions and integral topology. We give the specific computations needed here.
\begin{lemma}
\label{lem:torsion-surfaces}
The surfaces $S_2$ and $S_3$ admit holomorphic embeddings into
$\CP^{44}$ and $\CP^{164}$, respectively. Set
\[
T_2=(\Z/2)^2,\qquad T_3=\Z/3.
\]
Their integral homology is
\begin{equation}
\label{eq:surface-homology}
H_i(S_p;\Z)\cong
\begin{cases}
\Z,&i=0,4,\\
\Z^2\oplus T_p,&i=1,2,\\
\Z^2,&i=3,\\
0,&\text{otherwise}.
\end{cases}
\end{equation}
In particular, $b(S_p)=8$.
\end{lemma}
\begin{proof}
We first construct the projective embeddings. On an elliptic curve, a
line bundle of degree three has three independent sections. Riemann--Roch
shows that imposing one or two zeros, with multiplicities allowed,
lowers this dimension by one or two. Its complete linear system is
therefore base-point-free and separates points and tangent directions,
giving an embedding in $\CP^2$. Applying this to both curves and then
using the Segre embedding gives a holomorphic embedding
\[
\iota_p:E\times E_p\longrightarrow\CP^8.
\]
View a point of $\CP^8$ as the projective class of a nonzero linear
polynomial on $(\C^9)^*$. To the orbit of $x\in E\times E_p$ associate
the degree-$p$ polynomial obtained by multiplying representatives of
the points
$\iota_p(x),\iota_p(\sigma_p x),\ldots,
\iota_p(\sigma_p^{p-1}x)$.
Changing representatives changes the product by a nonzero scalar.
Multiplication of linear polynomials is a holomorphic map
\[
(\CP^8)^p\longrightarrow
\mathbb P\bigl(\operatorname{Sym}^p\C^9\bigr),
\]
and the product is invariant under cyclically permuting its factors.
Consequently it defines a holomorphic map from $S_p$ to the projective
space on the right. Unique factorization of polynomials shows that this
map separates orbits.
It is also immersive. Indeed, choose local representatives
$L_0,\ldots,L_{p-1}$ of the orbit points and suppose that a tangent
vector makes their product stationary as a projective point. Its
derivatives then satisfy
\[
\sum_{j=0}^{p-1}\dot L_j\prod_{k\ne j}L_k
=a\prod_{k=0}^{p-1}L_k
\]
for some scalar $a$. The orbit points are distinct, so the linear
polynomials $L_j$ are pairwise nonproportional. Restricting the identity
to the hyperplane $L_j=0$ shows that $\dot L_j$ vanishes on that
hyperplane: the product of the remaining factors is a nonzero
polynomial there. Hence $\dot L_j$ is a scalar multiple of $L_j$.
Every projective orbit point is stationary, and the immersion
$\iota_p$ forces the original tangent vector to vanish. The quotient
map is a local biholomorphism, so the induced map on $S_p$ is immersive.
A compact injective holomorphic immersion is an embedding. Finally,
\[
\dim\operatorname{Sym}^2\C^9=45,
\qquad
\dim\operatorname{Sym}^3\C^9=165,
\]
which gives the stated ambient dimensions.
For the homology computation, the universal covering $\C^2\to S_p$
has deck group generated by
\[
a(z,w)=(z+1/p,\zeta_p w),\qquad
d(z,w)=(z+i,w),\qquad
t_\lambda(z,w)=(z,w+\lambda)\quad(\lambda\in\Lambda_p).
\]
Here $a^p$ is translation by $1$ in the first coordinate. The elements
$a,d$ commute and generate a copy of $\Z^2$; conjugation by $a$ on
$\Lambda_p$ is multiplication by $\zeta_p$, while conjugation by $d$
is trivial. Thus the deck group is $\Lambda_p\rtimes\Z^2$, and
abelianization gives
\begin{equation}
\label{eq:surface-abelianization}
H_1(S_p;\Z)
\cong\Z^2\oplus\Lambda_p/(\zeta_p-1)\Lambda_p.
\end{equation}
For $p=2$, multiplication by $\zeta_p-1$ is $-2I$ on the lattice.
For $p=3$, its matrix in the basis $(1,\zeta_3)$ is
\[
\begin{pmatrix}-1&-1\\1&-2\end{pmatrix}.
\]
The latter has determinant $3$ and an entry equal to $1$, so its Smith
normal form has diagonal $(1,3)$. These computations identify the
torsion groups in \eqref{eq:surface-abelianization} as $T_p$.
The complex orientation makes $S_p$ an oriented closed real
four-manifold. Integral Poincar\'e duality and the universal coefficient
theorem give
\[
\Tor H_2(S_p;\Z)\cong\Tor H^2(S_p;\Z)
\cong\Tor H_1(S_p;\Z)=T_p,
\]
whereas $H_3(S_p;\Z)\cong H^1(S_p;\Z)$ is free of rank two.
The degree-$p$ covering $E\times E_p\to S_p$ gives
$p\chi(S_p)=\chi(E\times E_p)=0$. Since the ranks in degrees
$0,1,3,4$ are $1,2,2,1$, respectively, the rank in degree two is also
two. This proves \eqref{eq:surface-homology}.
\end{proof}
\subsection{Integral homology under blow-up}
We next recall the blow-up formula with integral coefficients. The
coefficient ring matters here: the centers just constructed have
torsion, and we need that torsion to survive as direct summands. The
following standard formula is recorded, in this integral generality,
in \cite[Theorem~4.1]{DuanLi2009}; we include its proof.
\begin{proposition}
\label{prop:blowup}
Let $A$ be a compact connected K\"ahler manifold, and let $Z\subset A$
be a closed connected complex submanifold of complex codimension
$c\geq2$. Its complex blow-up $\pi:\widehat A\to A$ is compact,
connected, and K\"ahler. If $A$ is simply connected, then so is
$\widehat A$. For every integer $i$ there is an isomorphism of abelian
groups
\begin{equation}
\label{eq:blowup-homology}
H_i(\widehat A;\Z)
\cong H_i(A;\Z)\oplus
\bigoplus_{j=1}^{c-1}H_{i-2j}(Z;\Z).
\end{equation}
\end{proposition}
\begin{proof}
Let $\nu$ denote the complex normal bundle of $Z$ in $A$. In local
coordinates transverse to $Z$, the blow-up replaces a vector in
$\C^c$ by a pair consisting of a line and a vector on that line.
These local models give a smooth complex manifold and a proper
holomorphic map $\pi$, which is an isomorphism away from $Z$. The
exceptional divisor is $E=\mathbb P(\nu)$, with projection
$\rho:E\to Z$, and its normal line bundle in $\widehat A$ is the
tautological bundle $\mathcal O_E(-1)$. The model also shows that
$\widehat A$ is connected; properness over compact $A$ gives
compactness.
To obtain a K\"ahler form, let $\omega_A$ be one on $A$. Choose a
Hermitian metric on $\nu$ and its induced metric on
$\mathcal O_E(1)$, whose curvature is positive on each projective
fiber. Since
$\mathcal O(-E)|_E\cong\mathcal O_E(1)$, this metric extends to a
Hermitian metric on $\mathcal O(-E)$. Let $\eta$ be its real Chern
curvature form. The form $\pi^*\omega_A$ is semipositive; along $E$
its kernel is precisely the tangent space to the projective fiber,
and it is positive away from $E$. On those kernel directions $\eta$
is positive. Compactness of the unit tangent bundle now shows that
\[
\pi^*\omega_A+\varepsilon\eta
\]
is positive for every sufficiently small $\varepsilon>0$. More
explicitly, $\eta$ is positive in a neighborhood of the kernel
directions, while on the complement $\pi^*\omega_A$ has a positive
lower bound and $\eta$ is bounded. The displayed form is closed and
of type $(1,1)$, hence K\"ahler.
Suppose that $A$ is simply connected. Since $Z$ has real codimension
at least four, general position allows paths and contracting disks
to avoid $Z$; consequently $A\setminus Z$ is simply connected.
Every loop in $\widehat A$ can be perturbed off the real codimension
two submanifold $E$. It then lies in
$\widehat A\setminus E\cong A\setminus Z$ and contracts there.
This proves simple connectivity of $\widehat A$.
It remains to establish the integral homology formula. We first work
in cohomology, always with coefficients in $\Z$. Write
$\iota:E\hookrightarrow\widehat A$ for the inclusion and put
$h=c_1(\mathcal O_E(1))$. By Leray--Hirsch, the map
\[
\bigoplus_{j=0}^{c-1}H^{k-2j}(Z;\Z)
\longrightarrow H^k(E;\Z),\qquad
(u_0,\ldots,u_{c-1})\longmapsto
\sum_{j=0}^{c-1}h^j\rho^*u_j
\]
is an isomorphism. It uses an integral basis on every fiber and
requires no restriction on torsion in $H^*(Z;\Z)$.
The Gysin map for the oriented normal line of $E$ satisfies
\[
\iota^*\iota_*(v)=c_1(\mathcal O_E(-1))v=-hv.
\]
For $1\leq j\leq c-1$ define
\[
s_j:H^{k-2j}(Z;\Z)\longrightarrow H^k(\widehat A;\Z),
\qquad s_j(u)=-\iota_*\bigl(h^{j-1}\rho^*u\bigr).
\]
Then $\iota^*s_j(u)=h^j\rho^*u$, so these maps lift all the
positive powers of $h$ in the preceding decomposition.
We claim that every class $v\in H^k(\widehat A;\Z)$ with
$\iota^*v\in\rho^*H^k(Z;\Z)$ lies in the image of $\pi^*$.
The map of pairs
\[
(\widehat A,E)\longrightarrow(A,Z)
\]
induces an isomorphism on relative cohomology: after collapsing the
subspaces, it induces a homeomorphism
$\widehat A/E\cong A/Z$. Write $\iota^*v=\rho^*z$.
Naturality of the connecting homomorphisms of the two pairs implies
that the connecting image of $z$ vanishes, because its pullback is
the connecting image of $\iota^*v$, which is zero. Hence $z$ extends
to a class $u\in H^k(A;\Z)$. The difference $v-\pi^*u$ restricts
to zero on $E$ and therefore comes from $H^k(\widehat A,E;\Z)$.
Pulling that relative class back from $H^k(A,Z;\Z)$ proves the
claim.
Subtracting suitable classes $s_j(u_j)$ from an arbitrary $v$ leaves
a class covered by this claim. Thus
\[
\pi^*+\sum_{j=1}^{c-1}s_j:
H^k(A;\Z)\oplus\bigoplus_{j=1}^{c-1}H^{k-2j}(Z;\Z)
\longrightarrow H^k(\widehat A;\Z)
\]
is surjective. It is injective as well. First, $\pi^*$ is injective
over $\Z$: the map $\pi$ has degree one, so the pushforward defined
using integral Poincar\'e duality is a left inverse to $\pi^*$.
If a sum in the displayed map is zero, restriction to $E$ and
Leray--Hirsch force all its $u_j$ for $j\geq1$ to vanish; injectivity
of $\pi^*$ then forces the remaining class to vanish. We have proved
\begin{equation}
\label{eq:blowup-cohomology}
H^k(\widehat A;\Z)
\cong H^k(A;\Z)\oplus
\bigoplus_{j=1}^{c-1}H^{k-2j}(Z;\Z).
\end{equation}
If $\dim_{\C}A=m$, integral Poincar\'e duality applied to
\eqref{eq:blowup-cohomology} in degree $2m-i$ gives center summands
$H_{i-2c+2j}(Z;\Z)$. Replacing $j$ by $c-j$ yields
\eqref{eq:blowup-homology}.
\end{proof}
\subsection{The manifold \texorpdfstring{$X$}{X}}
The blow-up formula now allows us to place both torsion primes in one
simply connected manifold. The different dimensions of the two centers
will give their torsion contributions different behavior near the ends
of the degree range.
\begin{proposition}
\label{prop:torsion-space}
For the fixed integer $N\geq704$, there is a simply connected compact
K\"ahler manifold $X$ of complex dimension $N$ obtained by blowing up
$\CP^N$ along disjoint copies of
\[
Z_2=S_2,\qquad Z_3=S_3\times\CP^1\times\CP^1.
\]
Set $L=N-3$. Its only nonzero integral homology torsion groups occur
in degrees $1+2j$ and $2+2j$, where $1\leq j\leq L$, and are
\begin{equation}
\label{eq:X-torsion}
\Tor H_{1+2j}(X;\Z)\cong\Tor H_{2+2j}(X;\Z)
\cong(\Z/2)^{a_j}\oplus(\Z/3)^{q_j},
\end{equation}
where
\begin{equation}
\label{eq:torsion-profiles}
a_j=2,\qquad
q_j=
\begin{cases}
1,&j=1,L,\\
3,&j=2,L-1,\\
4,&3\leq j\leq L-2.
\end{cases}
\end{equation}
Moreover,
\begin{equation}
\label{eq:rank-X}
b(X)=41N-183.
\end{equation}
\end{proposition}
\begin{proof}
By Lemma~\ref{lem:torsion-surfaces}, $Z_2$ embeds in $\CP^{44}$.
The embedding $S_3\hookrightarrow\CP^{164}$ and the Segre embedding
give
\[
Z_3\hookrightarrow
\CP^{164}\times\CP^1\times\CP^1
\hookrightarrow\CP^{659},
\]
since $165\cdot2\cdot2=660$. The direct-sum decomposition
$\C^{705}=\C^{45}\oplus\C^{660}$ places $\CP^{44}$ and
$\CP^{659}$ in disjoint linear subspaces of $\CP^{704}$.
Including that space linearly in $\CP^N$ gives the required disjoint
centers. Define $X$ by blowing up $Z_2$ and then $Z_3$; the latter
center is unchanged by the first blow-up because they are disjoint.
Both centers have complex codimension at least two, so
Proposition~\ref{prop:blowup} shows that $X$ is compact, connected,
simply connected, and K\"ahler.
The complex codimensions of $Z_2$ and $Z_3$ are $N-2$ and $N-4$,
respectively. The integral blow-up formula therefore gives
\begin{equation}
\label{eq:X-homology}
\begin{split}
H_i(X;\Z)\cong{}&H_i(\CP^N;\Z)
\oplus\bigoplus_{r=1}^{N-3}H_{i-2r}(S_2;\Z)\\
&\oplus\bigoplus_{r=1}^{N-5}
H_{i-2r}(S_3\times\CP^1\times\CP^1;\Z).
\end{split}
\end{equation}
The first center contributes $(\Z/2)^2$ in each of degrees
$1+2j$ and $2+2j$, for $1\leq j\leq L$. For the second center,
the homology of $\CP^1\times\CP^1$ is free, with ranks $(1,2,1)$
in degrees $(0,2,4)$. K\"unneth therefore has no additional
$\operatorname{Tor}$ terms. In either of the two degree lists, its
$3$-primary multiplicity is
\[
q_j=
\sum_{r=1}^{L-2}
\bigl(\mathbf 1_{j=r}+2\mathbf 1_{j=r+1}
+\mathbf 1_{j=r+2}\bigr),
\]
where $\mathbf 1$ denotes the indicator of the stated equality.
At the endpoints this gives $q_1=q_L=1$, next to them it gives
$q_2=q_{L-1}=3$, and at every remaining index it gives $q_j=4$.
There is no torsion outside these lists by
\eqref{eq:surface-homology} and \eqref{eq:X-homology}. Thus the
profiles are
\[
(a_1,\ldots,a_L)=(2,\ldots,2),\qquad
(q_1,\ldots,q_L)=(1,3,4,\ldots,4,3,1).
\]
The projective-line factors in the second center produce the multiplicities
$(1,2,1)$: the resulting $3$-primary multiplicity is smaller than the
$2$-primary multiplicity at the two endpoints and larger at every interior
index. This reversal is what the product comparison in Section~\ref{sec:counts}
will exploit.
Finally, $b(S_2)=b(S_3)=8$ and
$b(S_3\times\CP^1\times\CP^1)=32$. Summing ranks in
\eqref{eq:X-homology} yields
\[
b(X)=(N+1)+8(N-3)+32(N-5)=41N-183.
\]
\end{proof}
==== END Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//geometry.tex ====
==== BEGIN Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//handles.tex ====
\section{Morse attainment by handle cancellation}\label{app:handles}
We prove the existence assertion of Theorem~\ref{thm:smale} for a simply
connected closed smooth $D$-manifold $W$, with $D\geq9$.
The argument is the classical handle proof of Smale's theorem
\cite[Theorem~2.4]{Smale1963}. Its purpose here is to keep the geometric
attainment of the integral torsion bound explicit.
We use the standard correspondence between Morse functions and ordered
handle decompositions, together with handle slides, insertion of
cancelling pairs, and handle cancellation. A slide over a handle of the
same index changes the corresponding integral handle basis by an
elementary addition. Two consecutive handles of indices $i$ and $i+1$
cancel when the attaching sphere of the latter meets the belt sphere of
the former transversely in one point. Handle isotopies, slides, and
reorderings carry the subsequent attaching maps along. Thus they preserve
the represented smooth manifold. The Morse attachment correspondence is
reviewed in \cite{Milnor1963}. We use the cancellation and basis-change
theorems in \cite[Theorems~5.4 and~7.6]{Milnor1965}.
\subsection{Removing handles of index one and coindex one}
Choose an ordered handle decomposition of $W$. Connectedness allows us
to combine the $0$-handles along a tree of $1$-handles, leaving one
$0$-handle. The dual operation leaves one $D$-handle. We next remove
all $1$-handles, allowing the insertion of cancelling $2/3$ pairs.
This is the handle exchange underlying
\cite[Theorem~8.1]{Milnor1965}; we describe it to specify the role of
simple connectivity.
Let $P$ be the union of the $0$- and $1$-handles. It retracts onto a
graph, and $\pi_1(P)$ is the free group generated by the $1$-handles.
The boundary $\partial P$ is connected, and its inclusion into $P$
induces an isomorphism on fundamental groups: viewed relative to
$\partial P$, the reversed handles have indices $D-1$ and $D$.
The attaching circles of the $2$-handles normally generate
$\pi_1(\partial P)$, because the higher-index handles do not affect
fundamental groups and $\pi_1(W)=0$.
Fix one free generator. Insert a cancelling $2/3$ pair whose
$2$-handle initially attaches along a trivial circle in a small ball in
$\partial P$. The new $2$-handle can be placed with the other
$2$-handles in the ordered decomposition. Slide it over those handles
until its attaching circle represents the chosen generator. To justify
this step, express the generator as a finite product of conjugates of
the old attaching words and their inverses. A slide is a band sum
with an attaching circle, and choosing the band path and its orientation
realizes multiplication by the corresponding conjugate or inverse.
The needed paths can be chosen away from the other attaching circles by
general position; their tubular attaching regions can be made thinner.
The new circle is homotopic to a standard circle running once over the
chosen $1$-handle and avoiding the other $1$-handles. Choose this standard
circle disjoint from the other $2$-handle attaching circles. In the
$(D-1)$-dimensional boundary, general position turns the homotopy into an
isotopy through embedded circles avoiding those other circles. Indeed,
both self-intersections in a one-parameter family and intersections with
the other circles have expected dimension at most $3-(D-1)<0$.
Transport the attaching framing along the isotopy. The new attaching
circle now meets the chosen belt sphere once, so the $1$-handle and the
new $2$-handle cancel. Repeating removes all $1$-handles.
Apply the same construction to the reversed decomposition to eliminate
the $(D-1)$-handles. Its inserted $2/3$ pairs have original indices
$D-2,D-3$, so this operation does not reintroduce $1$-handles. We
obtain an ordered decomposition with one handle in each of indices $0$
and $D$, none in indices $1$ and $D-1$, and all other handles in indices
$2,\ldots,D-2$.
\subsection{An algebraic unit whenever the count is excessive}
The simply connected manifold $W$ is orientable. Fix an orientation,
orient the handles, and let $C_*$ be their integral chain complex. The
coefficient of an $i$-handle in the boundary of an $(i+1)$-handle is the
algebraic intersection number of its belt sphere and the latter's
attaching sphere in the level between these indices. Write
$Z_i=\ker\partial_i$ and $B_i=\operatorname{im}\partial_{i+1}$ as in
\eqref{eq:chain-rank}. That identity gives
\begin{equation}\label{eq:handle-excess}
\sum_i\rank C_i-\lambda(W)
=2\sum_i\bigl(\rank B_i-t_i(W)\bigr).
\end{equation}
Each summand on the right is nonnegative. If the total handle count
exceeds $\lambda(W)$, some $B_i$ therefore has rank greater than
$t_i(W)$. Since the chain groups in degrees $1$ and $D-1$ vanish,
such an index satisfies $2\leq i\leq D-3$.
The subgroup $Z_i$ is a direct summand of $C_i$, because
$C_i/Z_i\cong B_{i-1}$ is free. The nonzero Smith invariant factors of
$\partial_{i+1}$ are consequently those for the inclusion
$B_i\subset Z_i$. Exactly $t_i(W)$ of them are nonunits: these are the
cyclic invariant factors of $\Tor H_i(W;\Z)$. The strict rank inequality
thus gives at least one unit invariant factor. Row and column operations
can place a $1$ in the boundary matrix. They are realized geometrically
by handle slides, relabelings, and choices of orientation in indices $i$
and $i+1$.
The relevant levels are connected, so the slide bands can be chosen.
More precisely, the union $P_i$ of all handles of index at most $i$ is
simply connected, since it has no $1$-handles. Its reversed handles
relative to the boundary $L=\partial P_i$ have indices at least
$D-i\geq3$. They induce isomorphisms on components and fundamental
groups; hence $L$ is connected and simply connected. The same reasoning
applies to the level before the $i$-handles. Attaching spheres of one
index have codimension at least two in their attaching level, which
allows slide paths to avoid their disjoint attaching regions after those
regions have been made sufficiently thin.
We have produced an attaching sphere $A\cong S^i$ and a belt sphere
$B\cong S^{D-1-i}$ in $L$, with algebraic intersection number $1$.
To cancel their handles, we must make their geometric intersection a
single point. This is the remaining geometric step.
\subsection{Converting the unit to a single intersection}
First suppose $i\geq3$ and $D-1-i\geq3$. Put all attaching and belt
spheres at this level in transverse position. A pair of opposite-sign
intersections of $A$ and $B$ determines a Whitney circle, using one arc
in each sphere and choosing the arcs to avoid the other intersection
points. Since $L$ is simply connected, the circle bounds a Whitney
disk. The high-dimensional Whitney trick
\cite[Theorem~6.6]{Milnor1965} removes the pair. Its
dimensional hypotheses are precisely available here: both sphere
dimensions are at least three, and the interior of the two-dimensional
Whitney disk can avoid every attaching and belt sphere. The Whitney
move can thus be performed without disturbing their other intersections.
Repeating leaves one intersection of $A$ and $B$.
The cases in which one sphere has dimension two require control of the
complement of the other sphere, as in the low-index refinement of the
Whitney theorem in \cite[Theorem~6.6]{Milnor1965}. Here this control
can be seen directly. By reversing the decomposition if necessary, it is enough to
treat $i=2$. Here $L$ is the boundary after the $2$-handles have been
attached to the unique $0$-handle. Let
\[
B_1,\ldots,B_h\subset L
\]
be all their belt spheres, and put
$V=L\setminus\bigcup_{j=1}^h B_j$.
The plan is to move a disk in the chosen attaching sphere into $V$ by a
relative homotopy, then use general position to turn the resulting sphere
homotopy into an isotopy. We first determine the fundamental group of this
complement.
The manifold $L$ is obtained from $S^{D-1}$ by surgery on the attaching
circles. Each surgery replaces
$S^1\times D^{D-2}$ by $D^2\times S^{D-3}$, whose belt sphere is
$\{0\}\times S^{D-3}$. After removing this belt sphere, the replacement
piece is
\[
(D^2\setminus\{0\})\times S^{D-3}.
\]
It retracts onto its attaching boundary
$S^1\times S^{D-3}$. Performing these retractions simultaneously shows
that $V$ has the homotopy type of the exterior of the original attaching
circles in $S^{D-1}$. That exterior is simply connected: a loop can be
filled in the sphere, and a filling disk can be put in general position
away from the circles. Moving away from their cores and away from their
small tubular neighborhoods gives the same fundamental group. Thus
\begin{equation}\label{eq:belt-complement-pi1}
\pi_1(V)=\pi_1(L)=0.
\end{equation}
Let $A\cong S^2$ be the chosen attaching sphere of a $3$-handle, and
choose a pair of opposite-sign intersections with the chosen belt
sphere. Join them by an arc in $A$ avoiding all other intersections
with all the $B_j$. A small disk $\Delta\subset A$ around this arc
contains exactly that pair; its boundary and a collar outside it lie in
$V$. The oriented normal bundles of the belt spheres have rank two.
Excision and the Thom isomorphism identify
\begin{equation}\label{eq:relative-intersection-group}
H_2(L,V;\Z)\cong\bigoplus_{j=1}^h\Z,
\end{equation}
where the coordinates of $[\Delta]$ are its intersection numbers with
the $B_j$. They all vanish, so $[\Delta]=0$.
The same conclusion holds in relative homotopy. To see explicitly why
no assumption on $\pi_2(L)$ is needed, \eqref{eq:belt-complement-pi1}
and the long exact sequences give
\begin{align*}
\pi_2(L,V)&\cong
\operatorname{coker}\bigl(\pi_2(V)\longrightarrow\pi_2(L)\bigr),\\
H_2(L,V;\Z)&\cong
\operatorname{coker}\bigl(H_2(V;\Z)\longrightarrow H_2(L;\Z)\bigr).
\end{align*}
Absolute Hurewicz in degree two is an isomorphism for both simply
connected spaces. It therefore induces an isomorphism between these
cokernels. The disk $\Delta$, regarded as a map of pairs into $(L,V)$,
can consequently be homotoped into $V$, with its boundary moving in
$V$. The homotopy extension property extends this boundary motion
through maps of the collar into $V$, fixing its outer edge. Together
with the disk homotopy this gives a homotopy of $A$ stationary outside
$\Delta$ and the collar. At its endpoint the
chosen pair has disappeared and every other intersection with the belt
spheres is unchanged.
We can realize this homotopy by an isotopy of the attaching sphere.
First smooth it relative to the initial sphere and the stationary
region. General position for a one-parameter family of maps of a
$2$-sphere eliminates self-intersections and intersections with the
other attaching $2$-spheres: their expected dimensions are at most
$2+2+1-(D-1)<0$. The immersion condition holds by the same parametric
general-position argument. The level has dimension at least eight, so
these perturbations can be made relative to the prescribed region.
At the endpoint the altered disk and collar form a compact subset of
the open set $V$; a sufficiently small approximation preserves their
avoidance of the belt spheres. Keep neighborhoods of the remaining
transverse intersections stationary. Intermediate crossings with belt
spheres impose no restriction on this isotopy. Isotopy extension
transports the attaching framing, and the other attaching tubes can be
shrunk to remain disjoint. Repeating the operation leaves exactly one
intersection with the chosen belt sphere.
\subsection{Completing the cancellation}
In every case, the unit coefficient has now been represented by one
geometric intersection. Order the chosen $i$-handle last among the
$i$-handles and the chosen $(i+1)$-handle first among the
$(i+1)$-handles. They are consecutive and cancel. Intersections with
other handles do not obstruct this cancellation; their attaching maps
are transported by the cancellation diffeomorphism. The resulting
decomposition still has no handles of indices $1$ or $D-1$.
Every cancellation decreases the total number of handles by two.
Whenever that count exceeds $\lambda(W)$, Equation
\eqref{eq:handle-excess} produces another cancellable pair by the
argument above. The process therefore terminates with exactly
$\lambda(W)$ handles. At this stage every nonnegative summand in
\eqref{eq:handle-excess} vanishes, so $\rank B_i=t_i(W)$ for every $i$.
Equation~\eqref{eq:chain-rank} then gives precisely
\[
\rank C_i=r_i(W)+t_i(W)+t_{i-1}(W).
\]
The Morse function associated with the final handle decomposition has
these critical-point counts. This proves the existence assertion in
Theorem~\ref{thm:smale}.
==== END Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//handles.tex ====
==== BEGIN Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//history.tex ====
\subsection*{Context and prior work}
Arnold's fixed-point questions ask how much of the critical-point theory of a small autonomous Hamiltonian persists for a general Hamiltonian path \cite{Arnold1986}. Conley and Zehnder established the torus case by variational methods \cite{ConleyZehnder1983}. Floer's use of pseudoholomorphic curves turned the problem into a comparison between a Hamiltonian chain complex and the topology of the ambient manifold \cite{Floer1989}. Hofer--Salamon and Ono extended the theory in the weakly monotone setting \cite{HoferSalamon1995,Ono1995}; virtual-moduli methods subsequently yielded the general rational homological bound through work of Fukaya--Ono, Liu--Tian, and Ruan \cite{FukayaOno1999,LiuTian1998,Ruan1999}. The general rational bound counts Betti numbers. The Morse and stable Morse formulations ask for a geometric critical-point minimum, which can be larger.
The stable formulation is recorded explicitly in \cite[Conjecture~1.2]{Golovko2020}. Its distinction from the ordinary Morse formulation is essential: Damian exhibited closed manifolds with different ordinary and stable Morse numbers \cite{Damian2002}. In the positive direction, Dimitroglou Rizell and Golovko proved the stable Morse lower bound for generic Hamiltonians when both the symplectic area class and the first Chern class vanish on $\pi_2(M)$ \cite[Corollary~1.4]{DRG2017}. Their argument controls the simple homotopy type of the Floer complex. An independent proof was obtained by P\"oder Balkest\aa hl \cite[Corollary~1.2]{PoderBalkestahl2021}. The almost-quadratic convention for stable Morse functions used by Dimitroglou Rizell and Golovko agrees with ours by Lemma~\ref{lem:infinity-conventions}. Theorem~\ref{thm:main} disproves the unrestricted stable formulation, including its version that counts all fixed points.
Recent integral refinements explain why torsion alone does not force the bound in \eqref{eq:proposed-bound}. Abouzaid and Blumberg proved homological lower bounds over fields of arbitrary positive characteristic \cite{AbouzaidBlumberg2021}. Bai and Xu obtained an integral bound incorporating torsion at all primes \cite[Theorem~A]{BaiXu2022}, and their subsequent foundations give the same bound in \cite[Corollary~1.2]{BaiXu2026}. Its homology is graded modulo twice the minimal Chern number, so classes in different integer degrees may enter the same group. The geometric stable Morse number retains the integer degrees. For our examples the two bounds differ: Remark~\ref{rem:periodic-bound} computes the cyclically graded integral bound and shows that our fixed-point count exceeds it.
A separate construction gives nondegenerate Hamiltonian counterexamples to the ordinary Morse bound in real dimension twelve, with arbitrarily large deficit \cite[Theorem~1.1 and Corollary~1.2]{OpenAI2026Morse}. Those examples use a nontrivial fundamental group and the distinction between ordinary and stable Morse theory. In the construction below, the ambient manifold and the four fixed components of an auxiliary involution are simply connected, and Smale's theorem identifies both Morse numbers with the minimum allowed by integral homology \cite[Theorem~2.4]{Smale1963}. The separation instead comes from distributing torsion at two primes across different integer degrees and realizing the smaller Morse counts on the fixed components of an involution. The proof below is independent of that companion construction.
==== END Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//history.tex ====
==== BEGIN Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//introduction.tex ====
\section{Introduction}
\label{sec:introduction}
The Arnold fixed-point problem compares periodic orbits of Hamiltonian systems with critical points of smooth functions. We consider the proposed lower bound given by the stable Morse number, with stabilization understood geometrically.
\begin{definition}
Let $W$ be a connected closed smooth manifold. Its \emph{stable Morse number} $\SM(W)$ is the minimum of $\#\Crit(F)$ over all integers $k\geq0$ and smooth Morse functions
\[
F:W\times\R^k\longrightarrow\R
\]
such that, when $k>0$, there are a nondegenerate quadratic form $Q$ on $\R^k$ and $R>0$ with $F(x,z)=Q(z)$ for every $x\in W$ and $|z|\geq R$. The signature of $Q$ is unrestricted. For $k=0$, there is no condition at infinity. The ordinary Morse number $\Morse(W)$ is the same minimum restricted to $k=0$.
\end{definition}
These functions have finitely many critical points, and $\SM(W)\leq\Morse(W)$. The definition is a minimum over actual smooth functions; an algebraic complex of small rank alone does not provide an upper bound.
Let $(M,\omega)$ be a connected closed symplectic manifold, and let $H\in C^\infty((\R/\Z)\times M,\R)$. We use the convention $\omega(X_{H_t},\cdot)=dH_t$ and denote its flow by $\phi_H^t$. A fixed point $x$ of $\phi_H^1$ is \emph{nondegenerate} if $1$ is not an eigenvalue of $d_x\phi_H^1$. Write
\[
\Fix_0(\phi_H^1;H)=\{x\in\Fix(\phi_H^1):t\mapsto\phi_H^t(x)\text{ is contractible in }M\}.
\]
This counts distinct initial points, with no multiplicity from choices of capping disks. The stable Morse version of the fixed-point bound asks whether
\begin{equation}\label{eq:proposed-bound}
\#\Fix_0(\phi_H^1;H)\geq\SM(M)
\end{equation}
holds whenever every fixed point is nondegenerate. The following theorem disproves this unrestricted assertion.
\begin{theorem}\label{thm:main}
There exist a simply connected closed Kähler manifold $(M,\omega)$ and a smooth one-periodic Hamiltonian $H$ such that every fixed point of $\phi_H^1$ is nondegenerate and
\[
\#\Fix_0(\phi_H^1;H)=\#\Fix(\phi_H^1)=\SM(M)-16.
\]
One may take $\dim_{\R}M=1412$, with $\SM(M)=318968$ and $\#\Fix(\phi_H^1)=318952$.
\end{theorem}
\input{sections/history}
\subsection*{The construction and its mechanism}
For a closed manifold $W$, let $b(W)$ be its total integral homology rank, and let $t_i(W)$ be the minimum number of generators of $\Tor H_i(W;\Z)$. The quantity
\[
\lambda(W)=b(W)+2\sum_i t_i(W)
\]
is a lower bound for $\SM(W)$. For simply connected manifolds of the dimensions used here, Smale's theorem gives an ordinary Morse function attaining this bound. We prove the geometric lower bound in Section~\ref{sec:morse} and recall a handle proof of attainment in Appendix~\ref{app:handles}.
The dynamical ingredient is a Kähler surface $Y$, obtained by blowing up the four points $\{0,\infty\}\times\{0,\infty\}$ of $\CP^1\times\CP^1$. A half-turn of the diagonal circle action lifts to a Hamiltonian involution whose fixed set consists of four exceptional projective lines. For the high-dimensional simply connected Kähler manifold $X$ constructed below, the product involution $g$ on $M=X\times Y$ has four fixed components, each diffeomorphic to $C=X\times\CP^1$. Smale's theorem supplies a Morse function on $C$ with $\lambda(C)$ critical points.
Extend copies of this function to a $g$-invariant Hamiltonian on $M$, and let $\psi^s$ denote its flow. The flow commutes with $g$, so $(\psi^\epsilon\circ g)^2=\psi^{2\epsilon}$. For small $\epsilon$, the absence of short nonconstant periodic trajectories on the compact manifold $M$ forces every fixed point of $\psi^\epsilon\circ g$ to be an equilibrium on a fixed component of $g$. Section~\ref{sec:involution} proves that these are precisely the $4\lambda(C)$ chosen critical points and that they are all nondegenerate.
It remains to arrange $4\lambda(C)<\lambda(M)$. Section~\ref{sec:geometry} constructs $X$ by blowing up projective space along two disjoint centers with $2$-primary and $3$-primary torsion. The relevant elementary fact is
\[
d\bigl((\Z/2)^a\oplus(\Z/3)^q\bigr)=\max(a,q),
\]
where $d$ denotes the minimum number of generators. Multiplication by $\CP^1$ and by $Y$ combines the torsion groups in nearby degrees with different multiplicities. When the two primes are combined into cyclic summands, these products permit different savings in the number of generators. Section~\ref{sec:counts} shows that only the ends of the degree range contribute to the resulting difference
\[
\lambda(X\times Y)-4\lambda(X\times\CP^1)=16.
\]
The final assembly appears in Section~\ref{sec:assembly}. This use of mixed-prime torsion, together with the explicit involution, is the mechanism that puts the Hamiltonian count below the geometric stable Morse number. The proof does not require Floer theory.
==== END Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//introduction.tex ====
==== BEGIN Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//involution.tex ====
\section{Perturbing a Hamiltonian involution}\label{sec:involution}
We construct Hamiltonian maps whose fixed points are prescribed by Morse
functions on the fixed components of an involution. The involution used in
our example acts on a K\"ahler surface and fixes four projective lines. Its
product with the identity on another manifold will therefore replace one
Morse problem on the product by four Morse problems on smaller manifolds.
We use the basic Hamiltonian constructions recalled in \cite{McDuffSalamon2017}. Throughout this section the Hamiltonian vector field of a function $f$ on a
symplectic manifold $(P,\omega)$ is defined by
\[
\omega(X_f,\,\cdot\,)=df.
\]
We first record the elementary period estimate that makes the perturbation
count exact. This is a short-period estimate of the same kind as
Yorke's Lipschitz bound~\cite{Yorke1969}; the weaker estimate below suffices.
\begin{lemma}\label{lem:short-period}
Let $V$ be a smooth vector field on a closed smooth manifold $P$, and let
$\psi^s$ denote its flow. There is a number $\tau>0$ such that, for every
$0<T<\tau$,
\[
\psi^T(x)=x \quad\Longrightarrow\quad V(x)=0.
\]
\end{lemma}
\begin{proof}
Choose finitely many coordinate balls with compact closures inside their
charts, together with smaller balls whose closures lie in the corresponding
coordinate balls and whose union covers $P$. Bounds on the coordinate
speeds give a time $\delta>0$ such that a
trajectory starting in any smaller ball stays in its larger ball for time
$\delta$. In these finitely many balls choose a common positive Lipschitz
bound $L$ for the coordinate representatives of $V$.
Suppose a trajectory of period $T<\delta$ starts in one of the smaller balls.
Write its coordinate representation as $y:[0,T]\to\R^d$ and the coordinate
vector field as $v$. Set
\[
m=\max_{0\le t\le T}|v(y(t))|,
\]
and choose $t_0$ at which this maximum is attained. Since $y(T)=y(0)$,
\[
\int_0^T v(y(t))\,dt=0.
\]
Moreover, $|y(t)-y(t_0)|\le Tm$ for every $t$. It follows that
\[
\begin{split}
m
&=\left|\frac1T\int_0^T
\bigl(v(y(t_0))-v(y(t))\bigr)\,dt\right|\\
&\le \frac LT\int_0^T|y(t_0)-y(t)|\,dt
\le LTm.
\end{split}
\]
Taking $\tau=\min\{\delta,1/L\}$ forces $m=0$ whenever $0<T<\tau$.
\end{proof}
\begin{proposition}\label{prop:involution}
Let $(P,\omega)$ be a closed symplectic manifold, and let $g:P\to P$ be a
Hamiltonian diffeomorphism satisfying $g^2=\mathrm{id}_P$. Write
\[
\Fix(g)=F_1\sqcup\cdots\sqcup F_r
\]
for its connected fixed components. For each $j$, choose a smooth Morse
function $h_j:F_j\to\R$. There exists a $g$-invariant smooth function
$f:P\to\R$ with $f|_{F_j}=h_j$ such that, for all sufficiently small
$\epsilon>0$, the map
\[
\Phi_\epsilon=\psi^\epsilon\circ g,
\]
where $\psi^s$ is the Hamiltonian flow of $f$, satisfies
\begin{equation}\label{eq:involution-fixed-set}
\Fix(\Phi_\epsilon)
=\bigsqcup_{j=1}^r\operatorname{Crit}(h_j).
\end{equation}
Every fixed point of $\Phi_\epsilon$ is nondegenerate. Moreover,
$\Phi_\epsilon$ is the time-one map of a smooth one-periodic Hamiltonian.
If $P$ is simply connected, all its fixed-point loops for this Hamiltonian
are contractible.
\end{proposition}
\begin{proof}
We first describe the geometry of the fixed locus and extend the component
functions. We then identify every fixed point of the perturbation and
check its derivative.
Average a Riemannian metric over the group $\{\mathrm{id}_P,g\}$. At a fixed
point $x$, its exponential chart intertwines $g$ with $d_xg$. Thus the fixed
locus is a smooth closed submanifold, with tangent space
\[
E_x^+=\ker(d_xg-I).
\]
It has finitely many connected components by compactness and this local
description. Since $g$ is symplectic, the eigenspace decomposition
\[
T_xP=E_x^+\oplus E_x^-,
\qquad E_x^-=\ker(d_xg+I),
\]
is symplectically orthogonal: for $u\in E_x^+$ and $v\in E_x^-$,
\[
\omega_x(u,v)
=\omega_x(d_xg(u),d_xg(v))=-\omega_x(u,v).
\]
Nondegeneracy of $\omega_x$ then implies that its restrictions to both
summands are nondegenerate. Each $F_j$ is therefore symplectic, and
$E_x^-$ is its symplectic normal space.
Extend the functions $h_j$ to a smooth function $\widetilde f$ on $P$, using
disjoint tubular neighborhoods of the closed components. Define
\[
f=\tfrac12(\widetilde f+\widetilde f\circ g).
\]
Then $f\circ g=f$ and $f|_{F_j}=h_j$. Invariance and the symplectic property
of $g$ imply $g_*X_f=X_f$, so $g$ commutes with $\psi^s$. At a point
$x\in F_j$, invariance also gives $d_xf(v)=0$ for $v\in E_x^-$. Consequently
$X_f$ is tangent to $F_j$, and its restriction is the Hamiltonian vector
field of $h_j$ for $\omega|_{F_j}$. In particular,
\begin{equation}\label{eq:ambient-equilibria}
x\in F_j,\quad X_f(x)=0
\quad\Longleftrightarrow\quad d_xh_j=0.
\end{equation}
Indeed, at a critical point of $h_j$, the differential of $f$ vanishes on
both summands $E_x^+$ and $E_x^-$.
Let $\tau$ be supplied by Lemma~\ref{lem:short-period} for $X_f$. Commutation
and $g^2=\mathrm{id}_P$ give
\[
\Phi_\epsilon^2=\psi^{2\epsilon}.
\]
If $2\epsilon<\tau$ and $\Phi_\epsilon(x)=x$, then
$\psi^{2\epsilon}(x)=x$, so the lemma implies $X_f(x)=0$. Hence
$\psi^\epsilon(x)=x$, and the original fixed-point equation gives $g(x)=x$.
Equation~\eqref{eq:ambient-equilibria} now shows that $x$ is one of the
prescribed component critical points. Every such point is conversely
fixed by both maps. This proves~\eqref{eq:involution-fixed-set}.
It remains to check full nondegeneracy. At a point in this finite fixed
set, let $A_x:T_xP\to T_xP$ be the linearization of $X_f$. The point is an
equilibrium, so
\[
d_x\psi^\epsilon=\exp(\epsilon A_x).
\]
Equivariance implies that $A_x$ commutes with $d_xg$ and preserves $E_x^+$
and $E_x^-$. Write its restrictions as $A_x^+$ and $A_x^-$. On
$E_x^+=T_xF_j$, the map $A_x^+$ is the linearization of the Hamiltonian
vector field of $h_j$. The nonsingular Hessian of $h_j$ and the
nondegenerate form $\omega|_{F_j}$ make $A_x^+$ invertible. Therefore
\[
d_x\Phi_\epsilon|_{E_x^+}=\exp(\epsilon A_x^+)
\]
has no eigenvalue $1$ for all sufficiently small positive $\epsilon$:
its eigenvalues are $e^{\epsilon\lambda}$ with $\lambda\ne0$. On the normal
space,
\[
d_x\Phi_\epsilon|_{E_x^-}=-\exp(\epsilon A_x^-)
\longrightarrow -I
\qquad\text{as }\epsilon\longrightarrow0,
\]
so this block also has no eigenvalue $1$ for small $\epsilon$. One choice
of $\epsilon$ works for the finite set of component critical points.
Finally, choose a smooth Hamiltonian $K_s$, $0\le s\le1$, whose time-one
map is $g$. Choose nonnegative smooth functions $a,b$ supported in
successive disjoint subintervals of $(0,1)$, each with integral $1$, and
put
\[
A(t)=\int_0^t a(u)\,du,
\qquad
H_\epsilon(t,x)=a(t)K_{A(t)}(x)+\epsilon b(t)f(x).
\]
During the first interval, the flow traverses the path of $K_s$; during
the second, it traverses $\psi^s$ from $s=0$ to $s=\epsilon$. Its time-one
map is thus $\psi^\epsilon\circ g$. The function $H_\epsilon$ vanishes near
$t=0$ and $t=1$, so its periodic extension is smooth. If $P$ is simply
connected, every fixed-point loop along this specified path is
contractible.
\end{proof}
We now construct a surface to which the proposition applies. The crucial
feature is that its half-turn fixes entire exceptional curves, including
those on which the full circle action is nontrivial.
\begin{proposition}\label{prop:surface}
Let $Y$ be the complex blow-up of $\CP^1\times\CP^1$ at the four points
\[
\{0,\infty\}\times\{0,\infty\}.
\]
There is a circle-invariant K\"ahler form $\omega_Y$ for which the lifted
diagonal circle action is Hamiltonian. Its half-turn $g_Y$ has precisely
four fixed components, the exceptional curves $E_1,\ldots,E_4$, each
isomorphic to $\CP^1$. Along each $E_j$, the derivative of $g_Y$ is the
identity on the tangent bundle and minus the identity on the symplectic
normal bundle. The surface $Y$ is simply connected, and
\begin{equation}\label{eq:Y-homology}
H_i(Y;\Z)\cong
\begin{cases}
\Z,&i=0,4,\\
\Z^6,&i=2,\\
0,&\text{otherwise}.
\end{cases}
\end{equation}
\end{proposition}
\begin{proof}
On the standard affine chart $\C^2\subset\CP^1\times\CP^1$, let the circle act by
\[
t\cdot(z,w)=(e^{2\pi it}z,e^{2\pi it}w),
\qquad t\in\R/\Z.
\]
The four centers are invariant, so the action lifts holomorphically to
the blow-up. The surface $Y$ is K\"ahler and simply connected by
Proposition~\ref{prop:blowup}. Averaging a K\"ahler form over the
holomorphic circle action gives an invariant K\"ahler form $\omega_Y$.
If $V$ denotes the infinitesimal generator of the circle action, invariance
gives
\[
d(\omega_Y(V,\,\cdot\,))=\mathcal L_V\omega_Y=0.
\]
Since $H^1(Y;\R)=0$, this closed one-form is $dk$ for a smooth function
$k:Y\to\R$. Hence the circle action is Hamiltonian for our sign
convention, and the autonomous Hamiltonian $k/2$ has time-one map $g_Y$.
The downstairs half-turn fixes exactly the four centers. At each center
there are complex coordinates in which the circle has weights
$(\alpha,\beta)$ with $\alpha,\beta\in\{1,-1\}$. Regardless of these
signs, the half-turn is
\[
(z_1,z_2)\longmapsto(-z_1,-z_2).
\]
The local blow-up consists of pairs $(\ell,v)$, where $\ell\subset\C^2$ is
a complex line and $v\in\ell$. The lifted half-turn is
\[
(\ell,v)\longmapsto(\ell,-v).
\]
It fixes exactly the zero section, acts identically on its tangent, and
acts by $-1$ on its normal line. Off the exceptional curves the blow-down
is an equivariant isomorphism, so no additional fixed points occur. The
symplectic normal assertion follows also from the eigenspace decomposition
in Proposition~\ref{prop:involution}.
The homology formula~\eqref{eq:blowup-homology} adds one free generator in
degree two for each point blown up. Since the homology of
$\CP^1\times\CP^1$ is free with ranks $1,2,1$ in degrees $0,2,4$,
respectively, it gives~\eqref{eq:Y-homology}.
\end{proof}
For later use, we state explicitly the product construction. Its input
is an ordinary Morse function; its critical points will become genuine
Hamiltonian fixed points.
\begin{corollary}\label{cor:product-realization}
Let $(X,\omega_X)$ be a connected closed simply connected K\"ahler manifold,
let $(Y,\omega_Y)$ be the surface in Proposition~\ref{prop:surface}, and give
$M=X\times Y$ the product symplectic form
$\omega_X\oplus\omega_Y$. Put $C=X\times\CP^1$.
For every smooth Morse function $h:C\to\R$, there is a smooth one-periodic
Hamiltonian $H$ on $M$ such that every fixed point of its time-one map is
nondegenerate and contractible for its Hamiltonian path, and
\[
\#\Fix(\phi_H^1)=\#\Fix_0(\phi_H^1;H)
=4\,\#\operatorname{Crit}(h).
\]
\end{corollary}
\begin{proof}
The Hamiltonian involution $g=\mathrm{id}_X\times g_Y$ fixes exactly the four
symplectic submanifolds $X\times E_j$, each diffeomorphic to $C$. Transport
$h$ to each of them and apply Proposition~\ref{prop:involution}. The product
$M$ is simply connected, so all the fixed-point loops are contractible.
More concretely, use the path generated by the pullback of $k/2$ in the
proof of Proposition~\ref{prop:surface}. For a resulting fixed point
$(x,y)\in X\times E_j$, the circle part of the path lies in $\{x\}\times E_j$ and the
perturbation part is stationary. The whole loop therefore contracts in
that projective line.
\end{proof}
==== END Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//involution.tex ====
==== BEGIN Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//main.tex ====
\documentclass[11pt]{amsart}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{amsmath,amssymb,mathtools}
\usepackage{microtype}
\usepackage[hmargin=1.12in,vmargin=1in]{geometry}
\usepackage{booktabs}
\usepackage{xcolor}
\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
\hypersetup{pdftitle={Hamiltonian Fixed Points Below the Stable Morse Number},pdfauthor={OpenAI},pdfsubject={A simply connected Kähler counterexample to a stable Morse lower bound}}
\newcommand{\Z}{\mathbb Z}
\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\CP}{\mathbb{CP}}
\DeclareMathOperator{\SM}{SM}
\DeclareMathOperator{\Morse}{Morse}
\DeclareMathOperator{\Fix}{Fix}
\DeclareMathOperator{\Crit}{Crit}
\DeclareMathOperator{\Tor}{Tor}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\id}{id}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}
\title{Hamiltonian Fixed Points Below the Stable Morse Number}
\author{OpenAI}
\date{October 5, 2026}
\begin{document}
\begin{abstract}
We disprove the stable Morse lower bound for nondegenerate Hamiltonian fixed points by constructing a simply connected closed Kähler manifold with sixteen fewer fixed points than its stable Morse number. All the corresponding periodic orbits are contractible. The construction combines a Hamiltonian involution with integral homology torsion at two different primes; its proof uses finite-dimensional Morse theory and complex blowups.
\end{abstract}
\maketitle
\input{sections/introduction}
\input{sections/morse}
\input{sections/geometry}
\input{sections/involution}
\input{sections/counts}
\input{sections/assembly}
\appendix
\input{appendices/handles}
\clearpage
\input{references}
\end{document}
==== END Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//main.tex ====
==== BEGIN Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//morse.tex ====
\section{Integral homology and geometric Morse counts}\label{sec:morse}
We first establish the topological comparison used in the construction.
Integral homology gives a lower bound for every Morse function that is exactly
quadratic outside a compact set. On the simply connected manifolds of high
dimension used below, this bound is attained by an ordinary Morse function.
Both statements concern actual smooth functions.
For a closed smooth manifold $W$, write
\[
r_i(W)=\rank H_i(W;\Z),\qquad
t_i(W)=\text{the minimum number of generators of }\Tor H_i(W;\Z),
\]
and define
\begin{equation}\label{eq:lambda-definition}
b(W)=\sum_i r_i(W),\qquad
\lambda(W)=b(W)+2\sum_i t_i(W).
\end{equation}
All homological degrees in this section are integers. We take $r_i(W)$ and
$t_i(W)$ to be zero outside the homological range of $W$.
Another convention allows a uniformly bounded differential perturbation
of a quadratic form at infinity. It gives the same geometric minimum.
\begin{lemma}\label{lem:infinity-conventions}
Fix a Riemannian metric on a closed smooth manifold $W$ and the Euclidean
metric on $\R^k$, where $k>0$. Suppose $Q$ is a nondegenerate quadratic
form on $\R^k$ and $F\colon W\times\R^k\to\R$ is a smooth function
with nondegenerate critical points such that
\[
\sup_{W\times\R^k}\|dF-dQ\|<\infty
\]
in the product metric. Then there is a smooth function $\widetilde F$
equal to $F$ near all its critical points, with no additional critical
points, and equal to $Q$ outside a compact set. Consequently allowing
these bounded differential perturbations in the definition of the stable
Morse number does not change its value.
\end{lemma}
\begin{proof}
Write $h=F-Q$ and choose $K$ with $\|dh\|\leq K$.
Compactness of $W$ and integration along radial segments give
\[
|h(x,z)|\leq K_0+K|z|,
\qquad K_0=\max_{x\in W}|h(x,0)|.
\]
Nondegeneracy of $Q$ gives a constant $c>0$ with
$\|dQ(z)\|\geq c|z|$, so $F$ has no critical point for $|z|>K/c$.
Choose a smooth cutoff $\chi_R$ equal to one for $|z|\leq R$ and zero
for $|z|\geq2R$, with $0\leq\chi_R\leq1$ and
$\|d\chi_R\|\leq C/R$, where $C$ is independent of $R$.
For $R\geq1$, the differential of $\chi_Rh$ on the intervening annulus
has norm at most
\[
K+C(K_0+2K).
\]
This bound is independent of $R$, whereas $\|dQ\|\geq cR$ there.
For sufficiently large $R$, the function
$\widetilde F=Q+\chi_Rh$ therefore has no critical point in the annulus.
It agrees with $F$ on a ball containing all the critical points and
with $Q$ outside the larger ball. Conversely, equality with $Q$ outside
a compact set implies the displayed boundedness condition. These two
constructions preserve the number of critical points; for $k=0$ both
conventions allow the same ordinary Morse functions.
\end{proof}
\begin{proposition}\label{prop:stable-lower}
For every connected closed smooth manifold $W$,
\[
\SM(W)\geq \lambda(W).
\]
\end{proposition}
\begin{proof}
Let $F\colon W\times\R^k\to\R$ be an admissible function in the definition
of $\SM(W)$. We construct a finite free integral chain complex with one
generator for each critical point of $F$ and with homology equal to that of
$W$, shifted by a fixed integer. When $k=0$, the ordinary Morse complex
has these properties. Suppose $k>0$.
After a linear change of the auxiliary coordinates, the quadratic form is
\[
Q(u,v)=-|u|^2+|v|^2,
\qquad (u,v)\in\R^d\times\R^{k-d}.
\]
The equality $F=Q$ still holds outside a sufficiently large Euclidean ball.
Choose a radius $R_0$ such that this equality holds for $|(u,v)|\geq R_0$
and all critical points lie in $W\times B_{R_0}$. Choose $A>0$ satisfying
\[
A>\max_{W\times\overline B_{R_0}}\{|F|,|Q|\},
\]
and then choose $T>R_0$ with $T^2>A$. At the two levels $-A$ and $A$,
the sublevel sets of $F$ and $Q$ agree, including inside the core
$W\times\overline B_{R_0}$: at $-A$ the core is excluded, and at $A$ it
is included. Thus, with
\[
K=W\times\overline B_T,\qquad
K_-=K\cap\{F\leq-A\},\qquad
K_+=K\cap\{F\leq A\},
\]
we have $K_\pm=K\cap\{Q\leq\pm A\}$.
This truncation introduces no boundary critical point in the level band
$[-A,A]$. Indeed, on the side boundary $W\times\partial B_T$ the function
is $Q$, and its restricted differential can vanish only when $u=0$ or
$v=0$. Its possible critical values there are therefore $T^2$ and $-T^2$.
In the band, a gradient-like vector field can consequently be chosen
tangent to the side boundary. The usual Morse attachment argument on
this compact manifold with corners gives $K_+$ the relative homotopy type,
over $K_-$, of a space obtained by attaching one cell of the appropriate
Morse index for each critical point of $F$;
see \cite[Section~3]{Milnor1963} for the attachment theorem.
The tangent field gives the same deformation argument at the side
boundary as in the interior. If necessary, separate the critical values
by a small perturbation supported near the critical points. The endpoint
sublevels are compact manifolds with corners, or empty, and have finite
CW type. Taking a finite CW model for $K_-$ and using cellular approximation for the
attachments yields a finite free relative cellular complex $C_*$ with
\begin{equation}\label{eq:critical-chain-rank}
\sum_i\rank C_i=\#\operatorname{Crit}(F),\qquad
H_*(C_*)\cong H_*(K_+,K_-;\Z).
\end{equation}
The endpoint pair has a simple geometric description. The homotopy
$(u,v)\mapsto(u,(1-s)v)$, $0\leq s\leq1$, decreases both $Q$ and the
Euclidean norm. It therefore preserves both members of the pair and
deformation retracts $(K_+,K_-)$ onto
\[
W\times\bigl(D_T^d,
\{u\in D_T^d:|u|\geq\sqrt A\}\bigr).
\]
For $d=0$ the second factor is the pair $(\{0\},\varnothing)$.
The relative integral homology of the displayed disk--annulus pair is
$\Z$ in degree $d$ and zero otherwise. The relative K\"unneth formula
therefore gives
\begin{equation}\label{eq:quadratic-pair-homology}
H_i(C_*)\cong H_{i-d}(W;\Z).
\end{equation}
This also covers positive or negative definite quadratic forms.
It remains to count the ranks in $C_*$. For any finite free integral
chain complex, put $Z_i=\ker\partial_i$ and
$B_i=\operatorname{im}\partial_{i+1}$. Since subgroups of free abelian
groups are free,
\begin{equation}\label{eq:chain-rank}
\rank C_i=\rank H_i(C_*)+\rank B_i+\rank B_{i-1}.
\end{equation}
Smith normal form for $B_i\subset Z_i$ shows that $\rank B_i$ is at
least the minimum number of generators of $\Tor H_i(C_*)$.
Summing \eqref{eq:chain-rank} and applying
\eqref{eq:critical-chain-rank}--\eqref{eq:quadratic-pair-homology} gives
$\#\operatorname{Crit}(F)\geq\lambda(W)$. Taking the minimum over
admissible $F$ proves the proposition.
\end{proof}
For the upper bound we use the following form of Smale's Morse theorem.
We state the dimension range for which the handle proof in
Appendix~\ref{app:handles} is given.
\begin{theorem}[Smale]\label{thm:smale}
Let $W$ be a simply connected closed smooth manifold of dimension $D\geq9$.
There is an ordinary Morse function on $W$ with
\[
r_i(W)+t_i(W)+t_{i-1}(W)
\]
critical points of index $i$ for every $i$. In particular,
\begin{equation}\label{eq:simply-connected-morse}
\SM(W)=\Morse(W)=\lambda(W).
\end{equation}
\end{theorem}
\begin{proof}
The existence statement is the torsion-inclusive form of Smale's theorem;
see \cite[Theorem~2.4 and the following paragraph]{Smale1963}.
Appendix~\ref{app:handles} supplies a handle proof in the stated range,
including the cancellation of excess handles in degrees $2$ and $3$.
Summing the prescribed index counts gives an ordinary Morse function with
$\lambda(W)$ critical points. Proposition~\ref{prop:stable-lower} and
the inclusion of ordinary Morse functions among the admissible functions
then give
\[
\lambda(W)\leq\SM(W)\leq\Morse(W)\leq\lambda(W).
\]
\end{proof}
\begin{remark}\label{rem:torsion-generators}
The number $t_i(W)$ counts cyclic invariant factors, so it can combine
torsion at different primes. For example, for distinct primes $p,q$,
the minimum number of generators of
$(\Z/p)^a\oplus(\Z/q)^b$ is $\max(a,b)$.
Reduction modulo $p$ and modulo $q$ gives the lower bound; pairing
$\min(a,b)$ summands by the Chinese remainder theorem gives the upper
bound. This dependence on the degreewise distribution of primes is the
feature of $\lambda$ used below.
\end{remark}
==== END Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//morse.tex ====
==== BEGIN Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//references.tex ====
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==== END Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number//references.tex ====