From homological bounds to ordinary Morse theory

Conley and Zehnder proved the nondegenerate bound on the standard symplectic torus, where the ordinary Morse number equals the total Betti number (Conley and Zehnder 1983). Floer’s pseudoholomorphic-curve approach established homological Morse inequalities for Hamiltonian fixed points, including the monotone setting (Floer 1989). Virtual-moduli constructions of Fukaya–Ono, Liu–Tian and Ruan extended the rational Betti-number bound to general closed symplectic manifolds (Fukaya and Ono 1999; Liu and Tian 1998; Ruan 1999). The integral bounds of Bai–Xu retain torsion information as well (Bai and Xu 2022, Theorem A), (Bai and Xu 2026, Corollary 1.2). These homological bounds do not identify the number of generators of a Floer complex with the minimum number of critical points of an ordinary Morse function.

Stabilization explains one source of this distinction. The stable Morse number minimizes the number of critical points of Morse functions \(F:M\times\mathbb R^r\to\mathbb R\) that are almost quadratic at infinity, allowing the integer \(r\geq0\) to vary. Here this means that \(\|dF-dQ\|\) is uniformly bounded in a product metric for some nondegenerate quadratic form \(Q\) on \(\mathbb R^r\) (Dimitroglou Rizell and Golovko 2017, sec. 2.1). Arnold already considered a related stable critical-point minimum and asked whether it equals the minimum on the original manifold (Arnol’d 1986, sec. 5). Eliashberg and Gromov developed the relationship with simple chain homotopy over the fundamental-group ring and augmentation-ideal generation (Eliashberg and Gromov 1998, sec. 0.2.2 and 0.2.5). Damian proved that ordinary and stable Morse numbers of closed manifolds can differ (Damian 2002). A relative realization theorem after sufficiently large stabilization is given in (Dimitroglou Rizell and Golovko 2018, Proposition 2.9).

The algebraic distinction used here is classical and particularly sharp for direct powers of nontrivial perfect finite groups. Cossey, Gruenberg and Kovács proved that the augmentation-ideal generator number remains constant under these direct powers, while the group generator number is unbounded (Cossey et al. 1974, Theorems 1–2); see also (Gruenberg and Linnell 2008, secs. 1–2). Fundamental-group refinements of Floer theory reflect this distinction: Ono–Pajitnov obtain augmentation-ideal bounds under weak monotonicity (Ono and Pajitnov 2016, Theorem 5.7), whereas, in the symplectically aspherical or monotone setting, Barraud’s reconstruction of group generators counts Floer trajectories with multiplicities (Barraud 2018, Theorem 1.3 and Remark 3). The trajectory multiplicities are part of the bound; these results do not yield the unrestricted ordinary Morse-number inequality.

For closed symplectic manifolds on which both \([\omega]\) and \(c_1(TM)\) vanish on \(\pi_2(M)\), Dimitroglou Rizell and Golovko prove the stable Morse number as a lower bound for nondegenerate Hamiltonian fixed points (Dimitroglou Rizell and Golovko 2017, Corollary 1.3). The sphere factors in our examples have positive symplectic area, so this hypothesis does not hold. The construction below instead realizes a short relative Morse complex in exactly eight additional variables, with the function equal to a prescribed quadratic form outside a compact set. Its small-time return map is then inserted into a closed product of spheres. The relative handle realization and the exclusion of all additional fixed points are the two geometric steps needed to turn the ordinary/stable distinction into the stated closed Hamiltonian example.

The integral homology of our examples is torsion-free and concentrated in even degrees, with total rank \(\chi(M)\), as verified in Section 6. Their fixed-point count therefore exceeds the homological quantities above.

Ma’s preprint asserts that the minimum number of fixed points of a nondegenerate Hamiltonian diffeomorphism always equals the ordinary Morse number (Ma 2013, Theorem 1.5). Theorem 1 contradicts that unrestricted assertion.

The construction and its two counts

For a finitely generated group \(G\), let \(d(G)\) be the least number of group generators. Set \(b=2\) and consider the finite groups \(G=A_5^k\), where \(A_5\) has order \(60\). Two elementary algebraic facts will drive the proof: \(d(A_5^k)\) is unbounded as \(k\) increases, whereas the right augmentation ideal \[I_G=\ker(\mathop{\mathrm{aug}}:\mathbb Z[G]\longrightarrow\mathbb Z)\] has a generating list of length \(b\), independent of \(k\). The distinction is between generation of a group and generation of a module over its group ring.

Gompf’s existence theorem supplies a closed connected symplectic four-manifold \(X\) with \(\pi_1(X)=G\) (Gompf 1992, Theorem 3.1); see also (Gompf 1995). We construct a Morse function on \(X\times\mathbb R^8\) with precisely \(\chi(X)+4b\) critical points, equal outside a compact set to the quadratic form of index four \[Q(u,v)=\frac{|v|^2-|u|^2}{2},\qquad u,v\in\mathbb R^4.\] The eight variables are fixed throughout: the proof performs handle moves on one twelve-dimensional cobordism, rather than appealing to a stabilization of unspecified dimension.

The closed symplectic manifold is \(M=X\times(S^2)^4\). Sixteen disjoint product-cap neighborhoods correspond to the north/south choices on the four sphere factors. We place the same small-time return map of the constructed Morse function in each cap. The main gluing issue is to match the local endpoint with product rotations at both poles while excluding fixed points outside the caps. The resulting smooth one-periodic Hamiltonian satisfies \[ \#\mathop{\mathrm{Fix}}(\phi_H^1)=16\bigl(\chi(X)+4b\bigr) =\chi(M)+128. \tag{1}\]

On the other hand, the fundamental group forces every Morse function on \(M\) to have at least \(d(G)\) critical points in each of indices one and eleven. Euler’s identity gives \[ \mathop{\mathrm{Morse}}(M)\ge\chi(M)+4d(G). \tag{2}\] Since \(d(A_5^k)\) is unbounded while \(b\) is fixed, these two estimates give the strict inequality and the arbitrary deficits. Figure 1 separates the two roles of the fundamental group.

Proof organization

Section 2 proves the two group-ring facts and the exact row operation needed for geometric cancellation. Section 3 constructs the relative Morse model and its handle complex. Section 4 reduces the handle counts while preserving the quadratic collars. Section 5 realizes the local map on the closed product. Section 6 compares the two counts and proves the theorem and its deficit corollary.

An elementary row reduction

The next lemma supplies an entry equal to \(1\) using only elementary basis additions. This is the form needed for handle slides. It is a special case of Bass’s stable-range results applied to the opposite ring (Bass 1964, Lemma 6.4 and Theorems 4.2(a), 11.1); we include a direct proof to retain control of the multiplication sides and the row length.

The domain and its collars

Define \[ \begin{gathered} E=\{(u,v):r^2\leq 9,\ -1\leq Q\leq 1\}, \qquad E_{\pm}=E\cap\{Q=\pm1\},\\ W=X\times E,\qquad A=X\times E_-,\qquad B=X\times E_+. \end{gathered} \tag{7}\] The incoming and outgoing faces are \(A\) and \(B\); the side is the face \(r=3\). At its corners, both \(u\) and \(v\) are nonzero and \(dr,dQ\) are linearly independent. In a neighborhood of the side, the coordinates \[(r,Q,u/|u|,v/|v|)\] give a product collar with interval coordinate \(Q\). Since \(dQ\neq0\) on \(A\cup B\), these faces also have collars with coordinate \(Q\). To make the collars compatible, choose a vector field with \(dQ\)-value one which is the product coordinate field near the side, and extend it over the ends by a partition of unity. All handle constructions below take place away from a fixed smaller side collar.

Radial contraction preserves \(E\), so \(E\) is contractible. On \(E_-\), the parametrization \[(\xi,v)\longmapsto \bigl(\sqrt{2+|v|^2}\,\xi,v\bigr), \qquad \xi\in S^3,\quad |v|^2\leq\frac72,\] identifies \(E_-\) with \(S^3\times D^4\). Interchanging \(u\) and \(v\) gives the same description of \(E_+\). Consequently both inclusions \(A\hookrightarrow W\) and \(B\hookrightarrow W\) induce isomorphisms on fundamental groups.

Write \(\widetilde X\) for the universal cover of \(X\) and use tildes for the corresponding covers of \(W,A,B\). The relative homology of \((E,E_-)\) is \(\mathbb Z\) in degree four and zero in every other degree. The relative Künneth formula therefore gives, equivariantly for the deck action, \[ H_{4+i}(\widetilde W,\widetilde A;\mathbb Z) \cong H_i(\widetilde X;\mathbb Z). \tag{8}\] There is no torsion term because the relative homology of \((E,E_-)\) is free. The same formula holds with \(\widetilde B\) in place of \(\widetilde A\). In particular, \[ H_4(\widetilde W,\widetilde A;\mathbb Z)\cong\mathbb Z, \qquad H_5(\widetilde W,\widetilde A;\mathbb Z)=0, \tag{9}\] where the action on \(\mathbb Z\) is trivial. Here we used connectedness and simple connectedness of \(\widetilde X\).

An initial height function

Choose a Morse function \(f:X\to\mathbb R\) with one minimum and one maximum. For completeness, this follows from the usual handle description of a connected manifold. After ordering the handles, the zero- and one-handles form a connected graph. Cancelling the edges of a spanning tree against all but one zero-handle leaves one minimum. Attachments above each cancelling pair are transferred through the resulting boundary identification. Applying the same procedure to the dual handle decomposition removes the extra maxima without changing the minimum. This uses the handle decomposition theorem and the geometric cancellation criterion, in their elementary zero/one-handle cases (Ranicki 2002, Theorem 2.22 and Lemma 8.24).

Let \(\eta:[0,\infty)\to[0,1]\) be smooth, equal to one near zero and supported in \([0,1)\). For \(\sigma>0\) sufficiently small, set \[ h_0(x,u,v)=Q(u,v)+\sigma\eta(r^2)f(x). \tag{10}\] Choose \(\sigma\) so that \[\sigma\|f\|_{\infty}<\frac12, \qquad 2\sigma\|f\|_{\infty}\|\eta'\|_{\infty}<1.\] The perturbation is supported where \(r<1\) and hence \(|Q|<1/2\). Thus \(h_0(W)\subset[-1,1]\), and \(h_0=Q\) near every face of \(W\). For \(r>0\), its derivative in the extra variables has norm at least \[r\bigl(1-2\sigma\|f\|_{\infty}\|\eta'\|_{\infty}\bigr)>0.\] At \(r=0\), the critical-point equation is \(df=0\), and the Hessian is the direct sum of \(\sigma\operatorname{Hess}f\) and \(\operatorname{Hess}Q\). Therefore \(h_0\) is Morse, with exactly the critical points of \(f\), each shifted upward by four in index. In particular it has one critical point of index four and one of index eight.

Levels and the relative handle complex

Use gradient-like fields which are product fields on the side collar. The Morse–handle correspondence then describes \((W;A,B)\) by handles attached in the interiors of its regular levels. The standard local handle traces and their product regions give this correspondence relative to the side collar; see (Ranicki 2002, Proposition 2.20 and Theorem 2.22) for the handle construction. Only handles of indices \(4,\ldots,8\) will occur.

We may order these handles by index. Indeed, if an index-\(j\) handle is followed by an index-\(\ell\) handle with \(\ell\leq j\), the latter attaching sphere has dimension \(\ell-1\), and the former belt sphere has dimension \(11-j\) in the intervening eleven-dimensional level. Their dimension sum is \[(\ell-1)+(11-j)=10+\ell-j<11.\] General position makes the attaching sphere disjoint from that belt sphere. After shrinking its framed neighborhood, it can be moved back through the surgery, interchanging the two handles. All later attachments are transferred through the induced identification of the upper level. The same argument permits arbitrary ordering within an equal-index block, with all moves supported in the interior.

Every regular level \(L\) is connected, and its inclusion into \(W\) induces \[ \pi_1(L)\cong\pi_1(W)\cong G. \tag{11}\] To see this directly, surgery for an index-\(j\) handle removes a neighborhood of \(S^{j-1}\) and inserts one of \(S^{11-j}\). For \(4\leq j\leq8\), both spheres have dimension at least three and codimension at least four in the level. General position and van Kampen show that these surgeries preserve connectedness and the fundamental group. Start with \(A\), whose inclusion into \(W\) is a fundamental-group isomorphism. Alternatively, from any intermediate level both the upper part and the reversed lower part are built with handles of index at least four, so neither part changes the fundamental group. The interior of \(L\) has the same fundamental group as \(L\).

Choose an orientation and a lift of each handle. Regard the deck action as a right action: if it was originally written on the left as \(D_g\), use \(c\cdot g=D_{g^{-1}}c\). Let \(C_i\) be the based free right \(\Lambda\)-module generated by the handles of index \(4+i\), and let \(r_i\) be their number. The resulting relative handle complex is \[ 0\longrightarrow C_4\xrightarrow{\partial_4}C_3 \xrightarrow{\partial_3}C_2\xrightarrow{\partial_2}C_1 \xrightarrow{\partial_1}C_0\longrightarrow0, \qquad C_i\cong\Lambda^{r_i}. \tag{12}\] Its degree-\(i\) homology is \(H_{4+i}(\widetilde W,\widetilde A;\mathbb Z)\). One way to obtain the complex is to filter the handle decomposition by index; each successive relative group is freely generated by the lifted handles in that index. Boundary coefficients are signed intersections of lifted attaching spheres with lifted belt spheres one index lower. Thus, writing \[\partial e_j=\sum_i e_i a_{ij},\] the entries \(a_{ij}\) lie in \(\Lambda\) and multiply the basis elements on the right. Replacing a basis vector \(e_j\) by \(e_j+e_i q\), with \(i\neq j\), adds column\(_i\,q\) to column\(_j\) in its outgoing differential. If \(T\) is the elementary matrix of this basis change, the adjacent differentials change coherently as \(D\mapsto DT\) and \(D'\mapsto T^{-1}D'\). These conventions will be used for the handle slides in Section 4.

We have now constructed a height on the required fixed cobordism and identified the algebra that its handles must satisfy. To finish Proposition 6, it remains to reduce the index-five and index-seven counts to \(b\) while retaining the single index-four and index-eight handles and the prescribed collars. Then Lemma 7 will force \(r_2=\chi(X)-2+2b\) and the total count \(\chi(X)+4b\).

Introducing the augmentation generators

A slide of one index-\(j\) handle over another replaces a basis vector \(e_j\) by \(e_j+e_iq\) in the associated free right \(\Lambda\)-module. To realize this, first take \(q=\pm g\), with \(g\in G\): choose a band whose path represents \(g\), and choose its orientation to give the sign. The slide arcs can be chosen embedded and disjoint from the other attaching cores in an eleven-dimensional level; their tubular neighborhoods can then be shrunk as needed. Repeated slides realize an arbitrary finite integral combination of group elements. The framing is carried through the band construction. This is the usual labelled handle slide; see (Morgan 2020, proof of Theorem 4.2, pp. 9–10). The two adjacent differentials change together as described in Section 3.

Write \(e_1,\ldots,e_s\) for the existing index-five handles and \(a_i=\partial_1e_i\in I\). By Lemma 7, the \(a_i\) generate \(I\) as a right ideal. Fix a generating list \(y_1,\ldots,y_b\) for \(I\), padded with zeros if necessary.

Introduce \(b\) cancelling pairs of indices five and six, with new lower handles \(n_1,\ldots,n_b\) satisfying \(\partial_1n_j=0\). Here is a relative placement of the births. In the level just after the index-four handle, choose disjoint small balls missing its belt sphere and all old index-five attaching cores. These cores and the belt sphere have dimensions four and seven, respectively, so their finite union has empty interior. After choosing the balls, shrink the old attaching neighborhoods to miss them. The balls then survive the old index-five attachments. Their product traces support ordinary local five/six births. When ordering the resulting handles by index, pull the new lower attaching spheres back along these product traces. They still miss the index-four belt sphere, so their boundary columns are zero as asserted.

Choose coefficients in \(\Lambda\) such that \[y_j=\sum_{i=1}^s a_iq_{ij}, \qquad a_i=\sum_{j=1}^b y_jt_{ji}.\] First slide each new handle to replace \(n_j\) by \(n_j+\sum_i e_iq_{ij}\). Its new column is \(y_j\). Next replace each old handle by \(e_i-\sum_j n_jt_{ji}\), using the new bases. Every old column is now zero. The new \(b\) columns still generate \(I\).

Producing a cancellable pair

A coefficient equal to one. Select an old index-five handle \(e\), and place it last among the index-five handles. Since \(\partial_1e=0\), exactness in Lemma 7 gives \(e\in\operatorname{im}\partial_2\). If \(h_1,\ldots,h_m\) are the index-six handles and \(x_j\) is the \(e\)-coefficient in \(\partial_2h_j\), it follows that \[x_1\Lambda+\cdots+x_m\Lambda=\Lambda.\] While an old handle remains, \(r_1\geq b+1\); hence \(m=r_2\geq r_1\geq b+1\geq3\) by (13). Lemma 5 therefore applies. Slides among the index-six handles produce one whose coefficient into \(e\) is exactly \(1\). Place this handle first among the index-six handles, immediately after \(e\).

Removing the extra intersections. Let \(L\) be the intervening regular level, \(T\cong S^5\) the upper handle’s attaching core sphere, and \(D\cong S^6\) the lower handle’s belt sphere. Their signed, group-labelled intersection is \(1\). For each group label, pair intersection points with opposite signs, leaving exactly one positive point with identity label. This pairing is possible because all nonidentity coefficients vanish and the identity coefficient is one. In particular, the argument uses the specific coefficient \(1\), rather than an arbitrary unit of \(\Lambda\).

For each pair, choose arcs joining its points on \(T\) and on \(D\). The two spheres are simply connected. Equality of the group labels means that the resulting Whitney loop is null-homotopic in \(W\). By (11), it is null-homotopic in \(L\). It therefore bounds a disk in the interior of \(L\). General position makes this disk embedded, with interior disjoint from \(T\), \(D\), the other disks, and the other attaching or belt cores that must be retained. The relevant inequalities are \[2+5<11,\qquad 2+6<11,\qquad 2+2<11, \qquad 2+7<11.\] The last inequality covers the largest other core that occurs in our index range. Boundary arcs can likewise be chosen to miss the other cores and intersection points. All of these are finite, compact constructions in \(\operatorname{int}L\).

The high-dimensional Whitney theorem now removes each paired intersection by an isotopy of \(T\). Its hypotheses are complementary sphere dimensions at least three, opposite intersection signs, and the null-homotopic Whitney loop just established; simple connectedness of \(L\) is not required. The framing step is part of this theorem: the normal splitting along the boundary of the Whitney disk extends across it. In the present dimensions the frame-extension space is \(V_4(\mathbb R^9)=O(9)/O(5)\), which is simply connected. See (Milnor 1965, Theorem 6.6 and Lemma 6.13), or the group-ring formulation in (Ranicki 2002, Theorem 7.27(i) and Corollary 7.30). Transport the existing normal framing of \(T\) through its isotopy. We do not require that this entire handle framing extend over the Whitney disk.

For support, first arrange the Whitney disks and their compact move neighborhoods to miss the other cores; then shrink the other tubular attaching neighborhoods to miss those supports. A dimension argument alone would not give disjointness from preassigned full-dimensional neighborhoods. The selected belt sphere remains the reference sphere against which intersections are counted; the isotopy of the attaching sphere removes intersections with it. After the finite sequence of Whitney moves, \(T\) meets \(D\) transversely in exactly one point.

Cancellation relative to the boundary. The geometric handle cancellation criterion now applies (Ranicki 2002, Lemma 8.24). We spell out its relative support here because the quadratic collars must survive. Isolate the two adjacent critical points in a slab containing no others, and use a descending gradient-like field that is a product on the side collar. The single transverse intersection gives precisely one connecting trajectory from the index-six point to the index-five point. Choose a regular level \(c\) between the slab’s bottom and the lower critical value. Every other nonconstant trajectory in the upper point’s unstable manifold crosses \(c\). Indeed, a trajectory that remained above \(c\) could not escape through the product side; compactness and strict decrease of the height would force convergence to the only lower critical point, contrary to uniqueness of the connecting trajectory.

A product side collar has an invariant inner boundary, which the unstable manifold cannot cross. Retain a strictly smaller side collar outside that boundary. The closure of the unstable manifold above \(c\) then lies in the compact part of the slab between \(c\) and the upper critical value, away from the retained collar and the slab’s ends. This closure can include limiting trajectories through the lower critical point; it need not be a smooth disk. The local Morse cancellation theorem gives a deformation supported in any sufficiently small neighborhood of this closure, fixed elsewhere (Laudenbach 2013, Theorem and Corollary 1). One can apply its closed-manifold formulation by first extending the height and field to a closed enlargement; they agree on the slab, and the deformation restricts to the same \(W\). Thus the cancellation changes neither the earlier handles nor the side and end collars. The upper attaching maps are transferred through the induced identification of levels.

For clarity, the chosen index-six column may also have coefficients into other lower handles. In bases beginning with the selected pair, its differential has the block form \[\partial_2= \begin{pmatrix}1&a\\ v&B\end{pmatrix}, \qquad \partial_1=(0\ \ d),\qquad dv=0.\] Cancellation leaves \(B-va\) as the next differential and leaves \(d\) unchanged. This records algebraically why one coefficient equal to one suffices and why the other old zero columns of \(\partial_1\) survive. Geometrically this is the preservation of the lower handles and transfer of the upper attachments just described.

Iteration. We can therefore repeat the construction until every old index-five handle has been removed. Each step cancels one handle of index five and one of index six. The identities of Lemma 7 continue to hold for the same cobordism, and the \(b\) new lower handles remain. At the end, \[r_1=b.\]

The reverse reduction and the quadratic collars

Turn the cobordism around. Near its faces the new height is \(-Q\), and an original handle of index \(4+i\) has reversed index \(8-i\). The relative covering homology for \((W,B)\) is the same as for \((W,A)\), so the preceding construction applies without changing the dimension or adding any variables. It reduces the reversed index-five count to \(b\), that is, the original index-seven count to \(b\). Reversed five/six births and cancellations are original seven/six births and cancellations. They preserve the number \(r_1=b\) of original index-five handles; the upper attaching data in the reversed presentation are transferred as needed. Both endpoint counts remain one. Lemma 7 now gives \[ (r_0,r_1,r_2,r_3,r_4) =\bigl(1,b,\chi(X)-2+2b,b,1\bigr). \tag{14}\]

The handle traces and their intervening product regions supply an actual smooth Morse height on the same \(W\). Since only finitely many compactly supported moves were used, a common smaller side collar still has height \(Q\). We may also prescribe height \(-1\) and \(1\) on the two end faces and make its inward derivative strictly positive at the incoming end and strictly negative at the outgoing end. It remains to ensure equality with \(Q\) on whole end collars, compatible with the side collar.

At \(A\), use the inward coordinate \(s=Q+1\). On a sufficiently small collar the height has the form \[-1+s\,a(y,s),\qquad a(y,s)>0,\] and \(a=1\) near the side. Replace it by \[-1+s\bigl[1+\theta(s)(a(y,s)-1)\bigr],\] where \(0\leq\theta\leq1\), with \(\theta=0\) close to \(s=0\) and \(\theta=1\) farther inside. Put \(c_0=\min(1,\inf a)>0\). Its normal derivative is \[1+\theta(a-1)+s\theta\,\partial_sa+s\theta'(a-1).\] The first two terms together are at least \(c_0\). Choose the collar so small that \(|s\partial_sa|<c_0/4\), and choose the transition so slowly in \(\log s\) that \(|s\theta'|\,\|a-1\|_\infty<c_0/4\). For example, a fixed smooth transition applied to \(\log(s/s_0)/L\), with \(L\) large and then \(s_0\) sufficiently small, has these properties and is constant near both endpoints. The normal derivative is then at least \(c_0/2\). This introduces no critical points and leaves the side collar fixed. Apply the identical argument at \(B\) to one minus the height with inward coordinate \(1-Q\).

The final height equals \(Q\) near every face and corner of \(W\). Extend it by \(Q\) to \(X\times\mathbb R^8\) and call the result \(F\). This extension is smooth across all seams. Its exterior has no critical points because \(dQ=0\) only on the zero section, which lies in the interior of \(W\). It equals \(Q\) for \(r\geq3\) and, by (14), has exactly \[1+b+\bigl(\chi(X)-2+2b\bigr)+b+1=\chi(X)+4b\] critical points. This completes the proof of Proposition 6.