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On Fixed-Point Sets of $\Z_2$-Tori in Positive Curvature

Austin Bosgraaf, Christine Escher, Catherine Searle

Abstract

In recent work of Kennard, Khalili Samani, and the last author, they generalize the Half-Maximal Symmetry Rank result of Wilking for torus actions on positively curved manifolds to $\mathbb{Z}_2$-tori with a fixed point. They show that if the rank is approximately one-fourth of the dimension of the manifold, then fixed point set components of small co-rank subgroups of the $\Z_2$-torus are homotopy equivalent to spheres, real projective spaces, complex projective spaces, or lens spaces. In this paper, we lower the bound on the rank of the $\mathbb{Z}_2$-torus to approximately $n/6$ and $n/8$ and are able to classify either the integral cohomology ring or the $\mathbb{Z}_2$-cohomology ring, respectively, of the fixed point set of the $\mathbb{Z}_2$-torus.

On Fixed-Point Sets of $\Z_2$-Tori in Positive Curvature

Abstract

In recent work of Kennard, Khalili Samani, and the last author, they generalize the Half-Maximal Symmetry Rank result of Wilking for torus actions on positively curved manifolds to -tori with a fixed point. They show that if the rank is approximately one-fourth of the dimension of the manifold, then fixed point set components of small co-rank subgroups of the -torus are homotopy equivalent to spheres, real projective spaces, complex projective spaces, or lens spaces. In this paper, we lower the bound on the rank of the -torus to approximately and and are able to classify either the integral cohomology ring or the -cohomology ring, respectively, of the fixed point set of the -torus.

Paper Structure

This paper contains 12 sections, 36 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Covering spaces
  4. Periodic cohomology
  5. Group Cohomology
  6. Finite group actions
  7. Fixed Point Sets of $\mathbb{Z}_2$-Actions
  8. Positive sectional curvature
  9. The Higher Codimension Lemmas
  10. Restrictions on Chains of Fixed-Point Set Components
  11. The Proof of Theorem A
  12. The proof of Theorem B

Key Result

Proposition 2.1

Hat The map $p_*:\pi_k(\widebar{X}, \widebar{x}_0)\to\pi_k(X, x_0)$ induced by a covering projection $p :(\widebar{X}, \widebar{x}_0)\to(X, x_0)$ is injective if $k=1$, and an isomorphism if $k\geq 2$.

Theorems & Definitions (61)

  • Remark 1.1
  • Remark 1.2
  • Proposition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • Lemma 2.8
  • ...and 51 more