A nondegenerate counterexample to the Morse-number Arnold bound

1 The Morse-number question

Arnold’s fixed-point conjectures compare Hamiltonian fixed points with critical points of functions on the underlying symplectic manifold [1, Section 2]. For a sufficiently C^2-small autonomous Hamiltonian, the fixed points of its time-one map are precisely its critical points. The question is how much of this finite-dimensional lower bound survives for an arbitrary Hamiltonian path. We address the formulation using the ordinary Morse number.

Let (M,\omega) be a closed connected symplectic manifold and let H\in C^\infty((\R/\Z)\times M,\R). We use the convention

for the Hamiltonian vector field and its flow. A fixed point x of \phi_H^1 is nondegenerate if 1\notin\Spec(d_x\phi_H^1). Write \Fix_0(\phi_H^1;H) for the fixed points whose loops t\mapsto\phi_H^t(x) are contractible in M. Contractibility here refers to this Hamiltonian path.

The ordinary Morse number is

The Morse-number form of the Arnold conjecture asks whether \#\Fix_0(\phi_H^1;H)\ge\Morse(M) whenever the time-one map is nondegenerate [14, Conjecture 1.1]. We construct examples in which even the total number of fixed points is smaller than \Morse(M), and every fixed-point trajectory is contractible.

The same construction gives a quantitative separation in fixed dimension.

1.1 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 [6]. Floer’s pseudoholomorphic-curve approach established homological Morse inequalities for Hamiltonian fixed points, including the monotone setting [12]. Virtual-moduli constructions of Fukaya–Ono, Liu–Tian and Ruan extended the rational Betti-number bound to general closed symplectic manifolds [13, 19, 25]. The integral bounds of Bai–Xu retain torsion information as well [2, Theorem A], [3, 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\R^r\to\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 \R^r [9, Section 2.1]. Arnold already considered a related stable critical-point minimum and asked whether it equals the minimum on the original manifold [1, Section 5]. Eliashberg and Gromov developed the relationship with simple chain homotopy over the fundamental-group ring and augmentation-ideal generation [11, Sections 0.2.2 and 0.2.5]. Damian proved that ordinary and stable Morse numbers of closed manifolds can differ [8]. A relative realization theorem after sufficiently large stabilization is given in [10, 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 [7, Theorems 1–2]; see also [17, Sections 1–2]. Fundamental-group refinements of Floer theory reflect this distinction: Ono–Pajitnov obtain augmentation-ideal bounds under weak monotonicity [23, Theorem 5.7], whereas, in the symplectically aspherical or monotone setting, Barraud’s reconstruction of group generators counts Floer trajectories with multiplicities [4, 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 [9, 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 [20, Theorem 1.5]. Theorem 1.1 contradicts that unrestricted assertion.

1.2 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

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 [15, Theorem 3.1]; see also [16]. We construct a Morse function on X\times\R^8 with precisely \chi(X)+4b critical points, equal outside a compact set to the quadratic form of index four

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

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

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.

1.3 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.

2 Group generators and augmentation ideals

We need finite groups whose group generators are numerous but whose augmentation ideals have uniformly short generating lists. The distinction between these two numbers is classical; see [17, Sections 1–2]. We give the estimates needed here, then prove the elementary row reduction used in the handle construction. All modules in this section are right modules.

For a finitely generated group G, let d(G) be its minimum number of group generators.

We shall use

For now k\ge1 is arbitrary. The group S is nonabelian simple, hence perfect, and the direct product G is also perfect. Every finite group is finitely presented: one may take symbols t_g for its elements and relations t_1=1 and t_gt_h=t_{gh}. Thus these groups satisfy the finite-presentation condition needed for the symplectic realization below.

Write \Lambda=\Z[G] and let

be the augmentation map. Its kernel I is the augmentation ideal, regarded as a right \Lambda-module.

The uniform bound below is a special case of the direct-product theorem of Cossey, Gruenberg and Kovács [7, Theorem 2]. For finite groups, that theorem expresses the augmentation-ideal generator number of a direct product as the maximum of the corresponding factor numbers and the generator number of its abelianization. We give an elementary proof for the present family, including the passage from modular to integral generators.

In particular, choosing any finite k>60^{16b} gives

We always pad such a list by zeros if necessary. Increasing k makes d(G) unbounded by Lemma 2.1, while Lemma 2.2 keeps the same bound b.

2.1 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 [5, 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.

3 A relative Morse model in twelve dimensions

We now turn a short generating list for an augmentation ideal into a Morse function with few critical points. Let G be a finite group, put \Lambda=\Z[G], and let I\subset\Lambda be its right augmentation ideal. Assume that I has a generating list of length b\geq2. We will use this list and the elementary-row lemma established above. Let X be a closed connected oriented smooth four-manifold with \pi_1(X)\cong G. Since G is finite, rational homology and Poincaré duality give

No symplectic structure is needed for this part of the argument. The relation between quadratic stabilization, group-ring complexes and augmentation generators is developed by Eliashberg and Gromov [11, Sections 0.2.2 and 0.2.5]. We carry out the required realization on a specified relative cobordism, so that the number of variables and the exterior function remain fixed.

For u,v\in\R^4, put

We also write Q for its pullback to X\times\R^8.

The number of added variables is part of the assertion. We will prove the proposition by changing handles on one fixed compact twelve-dimensional cobordism. This section constructs the cobordism and its handle complex; Section 4 performs the reduction and completes the proof.

3.1 The domain and its collars

Define

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

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

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 \Z in degree four and zero in every other degree. The relative Künneth formula therefore gives, equivariantly for the deck action,

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,

where the action on \Z is trivial. Here we used connectedness and simple connectedness of \widetilde X.

3.2 An initial height function

Choose a Morse function f:X\to\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 [24, 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

Choose \sigma so that

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

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.

3.3 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 [24, 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

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

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

Its degree-i homology is H_{4+i}(\widetilde W,\widetilde A;\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

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 3.1, 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 3.2 will force r_2=\chi(X)-2+2b and the total count \chi(X)+4b.

4 Geometric reduction of the handles

We continue with the fixed cobordism (W;A,B) and the conventions of Section 3. The aim is to retain just b handles in each of indices five and seven. There are two distinct steps: elementary operations produce a coefficient exactly equal to 1, and Whitney moves turn that coefficient into one geometric intersection. All moves below occur in the interior of W.

4.1 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 [22, 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 3.2, 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

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.

4.2 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 3.2 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

While an old handle remains, r_1\geq b+1; hence m=r_2\geq r_1\geq b+1\geq3 by (13). Lemma 2.3 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

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(\R^9)=O(9)/O(5), which is simply connected. See [21, Theorem 6.6 and Lemma 6.13], or the group-ring formulation in [24, 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 [24, 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 [18, 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

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 3.2 continue to hold for the same cobordism, and the b new lower handles remain. At the end,

4.3 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 3.2 now gives

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

and a=1 near the side. Replace it by

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

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\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

critical points. This completes the proof of Proposition 3.1.

5 Closing the Hamiltonian model

We now turn a Morse function with eight quadratic variables into a Hamiltonian map on a closed manifold. Each of the sixteen products of north and south sphere caps will contain one copy of its critical set. The rotations outside the caps must be chosen with care: their endpoint maps agree with the quadratic model at both poles, although their paths can differ by full turns. We track this difference to identify the homotopy classes of the resulting fixed-point loops as well.

We first record the local short-period estimate used below. Yorke proved the sharp lower bound 2\pi/L for a nonconstant periodic solution of a Euclidean vector field with Lipschitz constant L [26]. The weaker coordinate estimate here suffices on a compact invariant set.

6 The Morse lower bound and the deficit

We finish by comparing the count from Proposition 5.1 with the ordinary Morse number. The lower bound uses the number of generators of the fundamental group, not the augmentation ideal.

In these examples the integral homology is also easy to identify to the extent needed for the comparison with homological bounds. The group G is perfect, so H_1(X;\Z)=0. Poincaré duality gives H_3(X;\Z)=H^1(X;\Z)=0; the universal coefficient theorem and duality give H_2(X;\Z)\cong H^2(X;\Z)\cong\Hom(H_2(X;\Z),\Z), so H_2(X;\Z) is free abelian. Hence the integral homology of M=X\times(S^2)^4 is torsion-free and concentrated in even degrees. Its total rank is \chi(M), and the fixed-point count in (22) exceeds this number. Thus the total fixed-point count exceeds both the rational Betti number and the rank-and-torsion quantity discussed in the introduction. The obstruction detected by the ordinary Morse number is the additional cost of the fundamental group.