Sharpness of the Cyclic Integral Floer Bound Below the Stable Morse Number

Proof. Write Z_i=\ker\partial_i and B_i=\operatorname{im}\partial_{i+1}, which are free abelian. The exact sequences defining cycles, boundaries, and homology give

Smith normal form for the inclusion B_i\subset Z_i shows that \Tor(Z_i/B_i) has at most \rank B_i nontrivial cyclic invariant factors. Hence t_i\leq\rank B_i, proving both inequalities. This argument also applies verbatim with cyclic indices.

For attainment, decompose each prescribed homology group into its free part and nontrivial cyclic invariant factors. A free generator in degree i is represented by an isolated copy of \Z in that degree. A cyclic factor \Z/n, n>1, in degree i is represented by

in degrees i+1 and i, with all other differentials zero. Take the direct sum of these complexes. In the cyclic grading the source and target lie in the two opposite parities, with the reverse differential zero. Each cyclic factor therefore costs exactly two free generators in either grading.□

Proof. Let F\colon W\times\R^k\to\R be Morse and equal to a fixed nondegenerate quadratic form Q outside a compact subset of the product. We construct a finite free integral complex with one generator per critical point and homology equal to that of W, up to a uniform grading shift. For k=0 the ordinary Morse complex suffices. Suppose k>0.

After a linear change of auxiliary coordinates, write

Choose R_0>0 such that F=Q for |(u,v)|\geq R_0 and all critical points lie in W\times B_{R_0}. This is possible because dQ vanishes only at the origin. Choose

and set

The core W\times\overline B_{R_0} is excluded at level -A and included at level A, for both F and Q. Outside that core the two functions agree. Consequently

We explain why this compact truncation introduces no extra cells in the level band [-A,A]. On the side boundary W\times\partial B_T, the restriction of F is Q. Its restricted differential can vanish only when u=0 or v=0, so its only possible critical values are T^2 and -T^2. Neither is in the band. Thus a gradient-like vector field for F on the band can be chosen tangent to the side boundary: take a tangential gradient there, extend across a collar, and combine it with interior gradient-like fields by a partition of unity. Away from the critical points all these fields increase F strictly.

The usual Morse attachment argument now applies to this compact manifold with boundary. Between critical levels its deformation flow preserves the side boundary, and each interior critical point attaches exactly one cell of its Morse index. This is the standard attachment theorem [11, Section 3], with the tangential field supplying the boundary-preserving deformation. The endpoint level hypersurfaces meet the side boundary transversely. Hence the endpoint sublevels are compact manifolds with corners, or empty, and have finite CW type. Replacing K_- by a finite CW model and using cellular approximation on the finitely many attaching maps gives a finite free relative cellular complex C_* satisfying

Critical values may first be separated by a small perturbation supported near the critical points; this leaves the endpoint pair and all critical-point counts unchanged.

By (2.1), shrinking v to zero preserves both members of the pair: it decreases Q and the Euclidean norm. The resulting deformation retraction identifies its relative homology with that of

The disk–annulus pair has integral relative homology \Z in degree \nu and zero otherwise. When \nu=0 it is the pair (\{0\},\varnothing), with the same assertion in degree zero. The relative Künneth theorem therefore gives

This includes positive definite, negative definite, and indefinite quadratic forms. Applying Lemma 2.1 to (2.2) proves \#\operatorname{Crit}(F)\geq\lambda(W). Taking the minimum over all admissible functions proves the claim.□

Proof. The existence assertion is the torsion-inclusive form of the classical geometric Morse theorem of Smale [13, Theorem 2.4 and the following paragraph]; see also [5, Theorem 2 and Section 6] for its statement and handle proof. In these statements the torsion count is the number of cyclic invariant factors, equivalently the minimum number of generators of the torsion subgroup. Thus summing the prescribed index counts gives \Morse(W)\leq\lambda(W). An ordinary Morse function is admissible in the definition of \SM by choosing k=0; on the compact space W the exterior condition is then vacuous. Proposition 2.2 yields

which proves both equalities.□

Proof. Choose finitely many coordinate balls, each relatively compact in a larger coordinate chart, and smaller balls that cover P. Uniform speed bounds give \delta>0 such that a trajectory starting in a smaller ball stays in its corresponding coordinate ball for time \delta. Choose a common Lipschitz constant L>0 for the coordinate vector fields on these balls.

Suppose \psi^T(x)=x with T<\delta. Write the trajectory in one of these charts as y:[0,T]\to\R^d, with \dot y=v(y). Set

and choose t_0 realizing the maximum. Since y(T)=y(0), \int_0^T v(y(t))\,dt=0. Moreover, |y(t)-y(t_0)|\leq Tm. Consequently

For T<\min\{\delta,L^{-1}\} this forces m=0.□

Proof. Average a Riemannian metric over the involution. At a fixed point x, the exponential map for this metric intertwines g with d_xg. Thus the fixed locus is a smooth closed submanifold whose tangent space at x is

The local description and compactness give finitely many connected components. Since (d_xg)^2=I, there is a direct-sum decomposition

For u\in E_x^+ and v\in E_x^-, symplectic invariance gives \omega_x(u,v)=\omega_x(d_xg(u),d_xg(v))=-\omega_x(u,v). The summands are therefore symplectically orthogonal, and the restriction of \omega_x to each is nondegenerate. In particular, each F_j is symplectic and E_x^- is its symplectic normal space.

Extend the h_j simultaneously to a smooth function \widetilde f on P, using disjoint tubular neighborhoods of the components and cutoff functions. Set

Then f\circ g=f and f|_{F_j}=h_j. Because g is symplectic, invariance implies g_*X_f=X_f, so g commutes with \psi^s. At x\in F_j, invariance also gives d_xf(v)=0 for v\in E_x^-. It follows that X_f(x)\in T_xF_j and that X_f|_{F_j} is the Hamiltonian vector field of h_j for \omega|_{F_j}. In particular,

Write \Phi_\epsilon=\psi^\epsilon\circ g. Commutation gives

Choose 2\epsilon<\tau, where \tau is supplied by Lemma 3.1 for X_f. If \Phi_\epsilon(x)=x, then \psi^{2\epsilon}(x)=x, so X_f(x)=0. Thus \psi^\epsilon(x)=x, and the original fixed-point equation implies g(x)=x. Equation (3.1) identifies x as a critical point of one of the h_j. Conversely, every such critical point is fixed by both g and \psi^\epsilon. This proves the asserted equality of fixed sets.

We next check nondegeneracy in the entire tangent space. At one of these equilibria let A_x be the linearization of X_f. Then d_x\psi^\epsilon=\exp(\epsilon A_x). Equivariance implies that A_x preserves E_x^+ and E_x^-; denote its restrictions by A_x^+ and A_x^-. On E_x^+=T_xF_j, the defining identity for X_f gives

Thus A_x^+ is invertible. The limits

show that both blocks of d_x\Phi_\epsilon-I are invertible for all sufficiently small positive \epsilon. A single choice of \epsilon works at the finite set of critical points.

Finally, let K_s, 0\leq s\leq1, be a smooth Hamiltonian whose time-one map is g. Choose nonnegative smooth functions a,b, each of integral one, supported in successive disjoint subintervals of (0,1), and put

The first interval traverses the Hamiltonian path of K_s, and the second traverses \psi^s from s=0 to s=\epsilon. Hence the time-one map is \psi^\epsilon\circ g. Since H_\epsilon vanishes near t=0 and t=1, its periodic extension is smooth. On a simply connected P, every loop traced by a fixed point along this path is contractible.□

Proof. The blow-up is a smooth projective surface and therefore admits a Kähler form. On affine coordinates the diagonal action is

It preserves the four centers and lifts holomorphically to Y. Averaging a Kähler form over this action gives an invariant Kähler form \omega_Y.

Removing four points from the simply connected real four-manifold \CP^1\times\CP^1 preserves simple connectivity: paths and their contracting disks can be moved off the points by general position. Every loop in Y can similarly be moved off the exceptional curves, which have real codimension two. The complement is identified by blow-down with the punctured original surface, so the loop contracts. Hence Y is simply connected. If V is the infinitesimal generator of the lifted circle action, Cartan’s formula yields

Since H^1(Y;\R)=0, there is a smooth k:Y\to\R with dk=\omega_Y(V,\cdot). The circle action is therefore Hamiltonian, and the autonomous Hamiltonian k/2 generates its half-turn at time one.

Downstairs, the half-turn fixes exactly the four centers. At any center the circle weights are (\alpha,\beta) with \alpha,\beta\in\{1,-1\}, so the half-turn is (z_1,z_2)\mapsto(-z_1,-z_2). The local blow-up consists of pairs (\ell,v) with \ell\subset\C^2 a complex line and v\in\ell. The lifted half-turn is

It fixes exactly the exceptional zero section, acts identically on its tangent bundle, and acts by -1 on its normal line. Equivariant blow-down excludes any further fixed points. The symplectic normal description follows from the eigenspace calculation in Proposition 3.2.

Each point blow-up replaces a four-ball by a disk bundle over \CP^1 with common boundary S^3. Mayer–Vietoris therefore adds one free generator to H_2 and leaves H_1 unchanged. Together with simple connectivity and integral Poincaré duality, this gives the displayed homology groups from the ranks (1,2,1) of \CP^1\times\CP^1. Finally, the normal bundle of an exceptional curve is \mathcal O_{\CP^1}(-1). The tangent-normal exact sequence gives

□

Proof. The Hamiltonian involution g=\mathrm{id}_X\times g_Y has precisely the four fixed components X\times E_j, each diffeomorphic to C. Transport h to these components and apply Proposition 3.2. The product M is simply connected, so all fixed-point loops are contractible. On the sphere \{x\}\times E_j, the X-directions form a trivial bundle and Proposition 3.3 gives

Thus c_1(TM)(\pi_2(M))=\Z.□

Proof. We first give an embedding with an explicit ambient dimension. A degree-three line bundle on an elliptic curve has a three-dimensional space of sections. Riemann–Roch gives dimension two or one after subtracting an effective divisor of degree one or two. Its complete linear system consequently has no base points and separates points and tangent vectors. It embeds the curve into \CP^2. Applying this to both factors and taking the Segre embedding yields

Regard a nonzero vector in \C^9 as a linear polynomial on its dual. Multiplying representatives of the p points

defines a projective degree-p polynomial. This construction is holomorphic and independent of the representatives. It is unchanged by \sigma_p, so it descends to a holomorphic map

Unique factorization recovers the unordered orbit from its product of linear factors; hence F_p is injective.

To check immersion, choose local nonzero representatives L_j of the orbit points. If a tangent vector has zero image under the projective differential of F_p, its derivatives satisfy

for a scalar a. Freeness makes the L_j pairwise nonproportional. Restricting this identity to \{L_j=0\} gives \dot L_j=0 there: the other factors have nonzero product in the polynomial ring of that hyperplane, which is an integral domain. Thus each \dot L_j is proportional to L_j. Since \iota_p is immersive and the quotient map is a local biholomorphism, the original tangent vector vanishes. Compactness now makes F_p a holomorphic embedding. The dimensions

give the stated projective spaces.

For the integral homology, use the universal covering \C^2\to S_p. Its deck group is generated by

Indeed, a^p is translation by 1 in the first coordinate, and these generators therefore include the full lattice of E\times E_p and a lift of \sigma_p. The commuting transformations a,d generate \Z^2, act on \Lambda_p by \zeta_p and the identity, and have trivial intersection with its translation subgroup. Consequently

The lattice endomorphism \zeta_p-1 is -2I for p=2. For p=3, its matrix in the basis (1,\zeta_3) is

whose Smith normal form is \operatorname{diag}(1,3). This proves the asserted description of H_1.

The complex orientation gives integral Poincaré duality. In particular, H_3(S_p;\Z)\cong H^1(S_p;\Z)\cong\Z^2, while the universal coefficient theorem gives

Finally, the degree-p covering by E\times E_p implies p\chi(S_p)=\chi(E\times E_p)=0. The ranks already computed force \rank H_2(S_p;\Z)=2, completing (4.1).□

Proof. Write \nu for the complex normal bundle of Z in A. The local blow-up model replaces v\in\C^c by pairs (\ell,v) with v\in\ell. It gives a smooth complex manifold, a proper map \pi that is an isomorphism off Z, and an exceptional divisor

The normal line bundle of E is \mathcal O_E(-1). Properness and the local model give compactness and connectedness.

Choose a Kähler form \omega_A and a Hermitian metric on \nu. The induced metric on \mathcal O_E(1) has curvature positive along the projective fibers. Since \mathcal O(-E)|_E\cong\mathcal O_E(1), extend this metric to \mathcal O(-E) and let \eta be its real Chern curvature form. Such an extension is obtained by extending the logarithm of the ratio with any ambient metric. Curvature commutes with restriction to the complex submanifold E, so \eta retains fiberwise positivity. The form \pi^*\omega_A is semipositive, with kernel along E equal to the tangent space of the fiber of \rho, and is positive off E. On these kernel directions \eta is positive. Compactness of the unit tangent bundle therefore implies that

is positive for all sufficiently small \varepsilon>0: near the kernel directions \eta is positive, and on their complement \pi^*\omega_A has a positive lower bound whereas \eta is bounded. This closed real (1,1)-form is Kähler.

If A is simply connected, then so is A\setminus Z. Indeed, general position allows a contracting disk for any loop in that complement to avoid Z, whose real codimension is at least four. A loop in \widehat A can similarly be perturbed off the real codimension-two divisor E. It then contracts in \widehat A\setminus E\cong A\setminus Z.

For the remaining assertion, all cohomology groups have coefficients in \Z. Let \iota:E\hookrightarrow\widehat A and h=c_1(\mathcal O_E(1)). The integral Leray–Hirsch theorem gives

This uses the integral fiber basis 1,h,\ldots,h^{c-1} and does not require torsion-free cohomology of Z. The integral Gysin push-forward \iota_*:H^k(E)\to H^{k+2}(\widehat A) satisfies \iota^*\iota_*(v)=-hv, because the normal bundle of E is \mathcal O_E(-1). Thus the classes

restrict to h^j\rho^*u on E.

We first show that a class v\in H^k(\widehat A) whose restriction to E is \rho^*z lies in \pi^*H^k(A). Collapsing the exceptional divisor and the center gives a homeomorphism \widehat A/E\cong A/Z, so \pi induces an isomorphism H^*(A,Z)\cong H^*(\widehat A,E). In the long exact sequences of these pairs, the connecting image of \rho^*z=\iota^*v is zero. Naturality and this relative isomorphism show that the connecting image of z is zero as well. Hence z is the restriction of some u\in H^k(A). The class v-\pi^*u vanishes on E, so it comes from relative cohomology; the same relative isomorphism expresses it as the pullback of a class on A. This proves the claim.

Given an arbitrary v, subtracting suitable s_j(u_j) removes the positive powers of h in (4.3). The claim then proves surjectivity of

For injectivity, restriction to E first forces every positive-power coefficient u_j to vanish by (4.3). Moreover, \pi^* is injective over \Z. The map \pi has degree one, so cap-product naturality gives, for \alpha\in H^k(A;\Z),

Integral Poincaré duality is an isomorphism on all classes, including torsion, so \pi^*\alpha=0 implies \alpha=0. Thus (4.4) is an isomorphism.

If \dim_{\C}A=m, apply integral Poincaré duality in degree 2m-i. The center summands become H_{i-2c+2j}(Z;\Z); replacing j by c-j gives (4.2).□

Proof of Theorem 1.1. Apply Theorem 2.4 to M and C, whose simple connectivity and dimensions were verified above. It gives \SM(M)=\lambda(M) and an ordinary Morse function h:C\to\R with precisely \lambda(C) critical points. Corollary 3.4 gives a smooth one-periodic Hamiltonian on M with exactly 4\lambda(C) fixed points, all nondegenerate and with contractible fixed-point loops. It also gives c_1(TM)(\pi_2(M))=\Z.

In the explicit formula of Proposition 3.2, the first-stage Hamiltonian can be taken autonomous: it is the pullback of k/2, where k generates the period-one circle action on Y.

Finally, (5.2), (5.5), (5.9), and (5.10) give

Since \dim_{\C}X=1664 and \dim_{\C}Y=2, the dimension is also as asserted.□