2022/01/02 by Donghoon Jang, Jang, Donghoon
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2201.00352
openalex publication_date 2022/01/02 · openalex created_date 2022/05/05 · openalex updated_date 2026/08/04
Let a k-dimensional torus Tk act on a 2n-dimensional compact connected almost complex manifold M with isolated fixed points. As for circle actions, we show that there exists a (directed labeled) multigraph that encodes weights at the fixed points of M. This includes the notion of a GKM graph as a special case that weights at each fixed point are pairwise linearly independent. If in addition k=n, i.e., M is an almost complex torus manifold, the multigraph is a graph; it has no multiple edges. We show that the Hirzebruch χy-genus χy(M)=∑i=0n ai(M) ⋅ (-y)i of an almost complex torus manifold M satisfies ai(M) > 0 for 0 ≤ i ≤ n. In particular, the Todd genus of M is positive and there are at least n+1 fixed points. Petrie's conjecture asserts that if a homotopy \mathbbCPn admits a non-trivial circle action, its Pontryagin class agrees with that of \mathbbCPn. Petrie proved this conjecture if instead it admits a Tn-action. We prove that if a 2n-dimensional almost complex torus manifold M only shares the Euler number with the complex projective space \mathbbCPn, an associated graph agrees with that of a linear Tn-action on \mathbbCPn; consequently M has the same weights at the fixed points, Chern numbers, equivariant cobordism class, Hirzebruch χy-genus, Todd genus, and signature as \mathbbCPn. If furthermore M is equivariantly formal, the equivariant cohomology and the Chern classes of M and \mathbbCPn also agree.