Rigidity of cohomology automorphisms of homogeneous spaces and coincidence theory

TL;DR

This paper studies the rigidity of rational cohomology automorphisms of certain homogeneous spaces and explores conditions for coincidence points of continuous functions on generalized Dold spaces.

Contribution

It classifies graded endomorphisms of cohomology algebras of specific spaces and establishes conditions for the coincidence property.

Findings

Classified graded endomorphisms of cohomology of sphere and Grassmannian products.

Derived necessary conditions for generalized Dold spaces to have the coincidence property.

Provided sufficient conditions for the existence of coincidence points between functions.

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.