Parametrized Borsuk-Ulam problem for projective space bundles

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 ${\pi }^{{}^{\prime }}:E^{{}^{\prime }}\to B$ be vector bundle such that ${\mathbb{Z}}_2$ acts fiber preserving and freely on $E$ and $E^{{}^{\prime }}-0$, where 0 stands for the zero section of the bundle ${\pi }^{{}^{\prime }}:E^{{}^{\prime }}\to B$. For a fiber preserving ${\mathbb{Z}}_2$-equivariant map $f:E\to E^{{}^{\prime }}$, 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 $f:E\to E^{{}^{\prime }}$.

14 pages, to appear in Fundamenta Mathematicae

Cited by in corpus (1)

• Some remarks on the parametrized Borsuk-Ulam theorem