Parametrized Borsuk-Ulam problem for projective space bundles

TL;DR

This paper extends the Borsuk-Ulam theorem to parametrized settings involving projective space bundles, providing cohomological estimates for zero and coincidence sets under fiber-preserving maps with group actions.

Contribution

It introduces a parametrized Borsuk-Ulam type theorem for projective space bundles with group actions, offering new cohomological bounds for zero and coincidence sets.

Findings

Cohomological bounds for zero sets of equivariant maps

Estimates for the dimension of coincidence sets

Application to fiber bundles with projective space fibers

Abstract

Let $\pi: E \to B$ be a fiber bundle with fiber having the mod 2 cohomology algebra of a real or a complex projective space and let $\pi^{'}: E^{'} \to B$ be vector bundle such that $\mathbb{Z}_2$ acts fiber preserving and freely on $E$ and $E^{'}-0$, where 0 stands for the zero section of the bundle $\pi^{'}:E^{'} \to B$. For a fiber preserving $\mathbb{Z}_2$-equivariant map $f:E \to E^{'}$, we estimate the cohomological dimension of the zero set $Z_f = \{x \in E | f(x)= 0\}.$ As an application, we also estimate the cohomological dimension of the $\mathbb{Z}_2$-coincidence set $A_f=\{x \in E | f(x) = f(T(x)) \}$ of a fiber preserving map $f:E \to E^{'}$.