The Morse-number Arnold conjecture for nondegenerate Hamiltonian diffeomorphisms

Theorem 1.1: there are a closed symplectic 12-manifold and a one-periodic Hamiltonian with all fixed points nondegenerate and contractible and $\#\mathrm{Fix}=\chi(M)+128<\mathrm{Morse}(M)$; Corollary 1.2 makes $\mathrm{Morse}(M)-\#\mathrm{Fix}$ arbitrarily large (the manifold depends on the deficit). The mechanism: the augmentation ideal of $A_5^k$ needs two generators while $d(A_5^k)$ is unbounded, realized by relative handle cancellation on $X\times\mathbb R^8$. The fixed-point count still exceeds the homological (Floer) bounds. The examples are not simply connected; the October 5 companions give simply connected Kahler examples below the stable Morse number.

The release says: 'Ma [Theorem 1.5] asserted that the minimum number of fixed points over all Hamiltonian diffeomorphisms always equals Crit(M). Theorem 1.1 contradicts that assertion' (degenerate paper) and 'Ma's preprint asserts that the minimum number of fixed points of a nondegenerate Hamiltonian diffeomorphism always equals the ordinary Morse number ... Theorem 1.1 contradicts that unrestricted assertion' (Morse-number paper). The other side: R. Ma, Proofs on Arnold conjectures, arXiv 0808.0613v7 (2013), Theorem 1.5, claims these lower bounds hold in general. Ma's preprint is unpublished; no referee has ruled between them. Theorem 1.1: there are a closed symplectic 12-manifold and a one-periodic Hamiltonian with all fixed points nondegenerate and contractible and $\#\mathrm{Fix}=\chi(M)+128<\mathrm{Morse}(M)$; Corollary 1.2 makes $\mathrm{Morse}(M)-\#\mathrm{Fix}$ arbitrarily large (the manifold depends on the deficit). The mechanism: the augmentation ideal of $A_5^k$ needs two generators while $d(A_5^k)$ is unbounded, realized by relative handle cancellation on $X\times\mathbb R^8$. The fixed-point count still exceeds the homological (Floer) bounds. The examples are not simply connected; the October 5 companions give simply connected Kahler examples below the stable Morse number.