Unitary $S^1$-Manifolds with Four Fixed Points and Kosniowski's Conjecture

Zhi Lü; Jun Ma; Qiangbo Tan

Home/Archive/arXiv

arXiv · 2610.23855

Unitary $S^1$-Manifolds with Four Fixed Points and Kosniowski's Conjecture

Zhi Lü·Jun Ma·Qiangbo Tan

View original source ↗Read source document ↗Export metadata ↓

Abstract

We study unitary $S^1$-manifolds with four isolated fixed points in relation to Kosniowski's conjectural inequality $4χ(M)>\dim M$. For every integer $m\ge3$, we show that there exists a unitary $S^1$-manifold $M_m^{18}$ with exactly four isolated fixed points, and these manifolds are pairwise distinct and nonzero in ordinary complex bordism. Since $χ(M_m^{18})=4$, they give an infinite family of counterexamples to $4χ(M)>\dim M$. By gluing two blocks formed by $G_2/SU(3)\cong S^6$ and the unit disk $D(V)$ of 12- dimensional real $S^1$-representation $V$ along their equivariantly identified free boundaries, we explicitly construct such a manifold $M^{18}_3$. We also prove that the four-point case is null-bordant in real dimension $16$ and in every real dimension greater than $18$; hence $18$ is the smallest dimension in which such a counterexample occurs.

Explore related subjects

math.AT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Zhi Lü, Jun Ma, Qiangbo Tan. 2026-10-16. Unitary $S^1$-Manifolds with Four Fixed Points and Kosniowski's Conjecture. https://arxiv.org/abs/2610.23855

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Extra-fine sheaves and interaction decompositions

We introduce an original notion of (upper) extra-fine sheaf on a topological space, for which Čech cohomology in strictly positive degree vanishes. We apply this notion when the topological space is a partially ordered set (poset) equipped with the Alexandroff topology and the sheaves are made of duals of decomposable functors of vector spaces. The decomposition of the functor is known as interaction decomposition and can be expressed by a novel sum-intersection condition. We use these results to compute the dimension of the space of global sections when the sheaves are derived from those freely generated over a presheaf (of finite sets); this generalizes and makes topological classical counting formulae for the number of solutions of the linearized marginal problem (Kellerer and Matúš). We finish with a comparison theorem between the Čech cohomology associated to a covering and the topos cohomology of the poset with coefficients in the presheaf, which is also the cohomology of a cosimplicial local system over the nerve of the poset. For that, we give a detailed treatment of cosimplicial local systems on simplicial sets. The appendixes present presheaves, sheaves relative Čech cohomology, and their application to the marginal problem.

math.AT↗

arXiv2026

Parametrised functor calculus: excision, spheres, and semiadditivity

We lay down the foundations of a theory of parametrised functor calculus, generalising parts of the functor calculus of Goodwillie. We introduce the notion of excisable posets and develop a theory of excisive approximations in this context. As an application, we introduce two different excisable posets when parametrising over an atomic orbital category. By comparing the notions of excisiveness for these two posets, we relate the invertibility of certain spheres with Nardin's notion of parametrised semiadditivity, generalising Wirthmüller's classical result in equivariant homotopy theory for finite groups.

math.AT↗

arXiv2026

Continuous Comparison of Free Simplicial Prounipotent and Pro-$p$ Resolutions

We give a continuous version of the comparison theorem for free simplicial resolutions in the categories of prounipotent groups over a field of characteristic zero and of pro-$p$ groups. The resolutions carry free bases compatible with degeneracies; their ranks may be infinite. In characteristic zero the lifting step follows from continuous linear splittings and the universal property of completed free Lie algebras. In the pro-$p$ case it follows from the classical projectivity of free pro-$p$ groups. A relative lifting argument and an explicit simplicial cylinder establish homotopy uniqueness. We formulate the free bases using actual surjections in the simplex category, so degeneracy words related by simplicial identities are identified from the outset, and we spell out the full matching objects used in the lifting argument. The question arose naturally in our preceding work on Bousfield--Kan completions of subcontractible presentations.

math.AT↗