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Parametrized Borsuk–Ulam problem for projective space bundles

Tom 211 / 2011

Fundamenta Mathematicae 211 (2011), 135-147 MSC: Primary 55M20, 55M35; Secondary 55R91, 55M10, 55R25. DOI: 10.4064/fm211-2-2

Streszczenie

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 a 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 $Z_f = \{x \in E \mid f(x)= 0\}.$ As an application, we also estimate the cohomological dimension of the $\mathbb{Z}_2$-coincidence set $A_f=\{x \in E\mid f(x) = f(T(x)) \}$ of a fiber preserving map $f:E \to E'$.

Autorzy

• Mahender SinghInstitute of Mathematical Sciences
C I T Campus
Taramani, Chennai 600113, India
e-mail
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