Number of fixed points for unitary Tⁿ⁻¹-manifold

Abstract

Let M be a 2n-dimensional closed unitary manifold with a Tⁿ⁻¹-action with only isolated fixed points. In this paper, we first prove that the equivariant cobordism class of a unitary Tⁿ⁻¹-manifold M is just determined by the equivariant Chern numbers c_{\omega }^{T^{n-\mathrm{1}}}[M],where ω= (i₁, i₂, ..., i₆) are the multi-indexes for all i₁, i₂, ..., i₆∈\mathbb{N}. Then we show that if Mdoes not bound equivariantly, then the number of fixed points is greater than or equal to \lceil n/\mathrm{6}\rceil +\mathrm{1}, where \lceil n/\mathrm{6}\rceil denotes the minimum integer no less than n/6.

Key words： Unitary torus manifold; equivariant Chern number; cobordism; localization theorem

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