分野別セミナー

The classical Borsuk-Ulam theorem states that, if $n ¥geq k$ then for every continuous map $f:S^{n}¥to ¥mathbb{R}^k$ there exists a point $x ¥in S^{n}$ such that $f(x) = f(-x)$.

Jaworowski, Dold and Nakaoka extended this problem to a fibre-wise setting, by considering maps $f : S(E) ¥to E^{¥prime}$ which preserve fibres, where $S(E)$ denotes the total space of the sphere bundle associated over $B$ to a vector bundle $E ¥to B$, and $E^{¥prime}$ is another vector bundle over $B$. This can be viewed as the parametrization of the Borsuk-Ulam theorem (parametrized by the base space).

In this seminar, we will discuss recent results related to parametrized Borsuk-Ulam problem.