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