Borsuk-Ulam Theorems and Their Parametrized Versions for $\mathbb {F}P^m\times \mathbb {S}^3$

Abstract

Let $G=\mathbb {Z}_p,$ $p>2$ a prime, act freely on a finitistic space X with mod p cohomology ring isomorphic to that of $\mathbb {F}P^m\times \mathbb {S}^3$, where $m+1\not \equiv 0$ mod p and $\mathbb {F}=\mathbb {C}$ or $\mathbb {H}$. We wish to discuss the nonexistence of G-equivariant maps $\mathbb {S}^{2q-1}\rightarrow X$ and $X\rightarrow \mathbb {S}^{2q-1}$, where $\mathbb {S}^{2q-1}$ is equipped with a free G-action. These results are analogues of the celebrated Borsuk-Ulam theorem. To establish these results first we find the cohomology algebra of orbit spaces of free G-actions on X. For a continuous map $f\!:\! X\rightarrow \mathbb {R}^n$, a lower bound of the cohomological dimension of the partial coincidence set of f is determined. Furthermore, we approximate the size of the zero set of a fibre preserving G-equivariant map between a fibre bundle with fibre X and a vector bundle. An estimate of the size of the G-coincidence set of a fibre preserving map is also obtained. These results are parametrized versions of the Borsuk-Ulam theorem.