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Rigidity of cohomology automorphisms of homogeneous spaces and coincidence theory

Manas Mandal, Divya Setia

Abstract

We obtain a rigidity phenomena of rational cohomology automorphisms of certain homogeneous spaces, in the presence of external cohomology classes arising from spaces with trivial cup product in rational cohomology algebra. We classify graded endomorphisms of the rational cohomology algebra of the product of a sphere and a complex Grassmannian, whose images are nonzero in the second cohomology of the Grassmannian. We also derive necessary conditions for the generalized Dold spaces to satisfy the coincidence property, in particular the fixed-point property. As an application of our results, we obtain several sufficient conditions for the existence of a point of coincidence between a pair of continuous functions on certain generalized Dold spaces.

Rigidity of cohomology automorphisms of homogeneous spaces and coincidence theory

Abstract

We obtain a rigidity phenomena of rational cohomology automorphisms of certain homogeneous spaces, in the presence of external cohomology classes arising from spaces with trivial cup product in rational cohomology algebra. We classify graded endomorphisms of the rational cohomology algebra of the product of a sphere and a complex Grassmannian, whose images are nonzero in the second cohomology of the Grassmannian. We also derive necessary conditions for the generalized Dold spaces to satisfy the coincidence property, in particular the fixed-point property. As an application of our results, we obtain several sufficient conditions for the existence of a point of coincidence between a pair of continuous functions on certain generalized Dold spaces.
Paper Structure (10 sections, 28 theorems, 82 equations)

This paper contains 10 sections, 28 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cohomology of complex Grassmannians
  4. Cohomology of homogeneous spaces
  5. Graded endomorphisms on $\mathbf{H^*_{\mathbb{C}G}}$
  6. Generalized Dold spaces
  7. Rational cohomology of $\mathbf{P(\mathbb S^m,\mathbb CG_{n,k})}$
  8. Rigidity of Cohomology Automorphisms
  9. Graded endomorphism of $H^*(\mathbb S^m\times \mathbb CG_{n,k};\mathbb Q)$
  10. Coincidence theory of $P(m,n,k)$

Key Result

Theorem 1.1

Let $X$ be a path-connected finite CW complex with trivial cup product in its rational cohomology algebra. Let $G$ be a compact connected Lie group and $H$ be a closed subgroup of maximal rank. Consider a graded endomorphism such that $i^*\circ\phi\circ p^*$ is an automorphism Then, $\phi(z)=\phi_0(z)$ for all $z\in H^*(G/H;\mathbb Q).$

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 2.1
  • Theorem 2.2: shiga-tezuka, Theorem $A^{'}$
  • Theorem 2.3: glover-homer, Theorem 1, hoffman, Theorem 1.1
  • Theorem 2.4: mandal-sankaran2
  • ...and 41 more