Cohomology algebra of the orbit space of some free actions on spaces of cohomology type (a, b)

TL;DR

This paper computes the mod 2 cohomology algebra of orbit spaces resulting from free Z_2 and S^1 actions on spaces with specific cohomology structures, revealing restrictions on possible group actions based on cohomology type.

Contribution

It determines the cohomology ring of orbit spaces for free actions on spaces with cohomology type (a, b), especially when a and b are even, and explores restrictions on group actions based on cohomology.

Findings

Cohomology ring of X/G for free Z_2 actions with even a and b.

No equivariant map S^m --> X exists for m > 3n under certain conditions.

Z_2 cannot act freely on spaces of type (a, b) with odd a and even b.

Abstract

Let X be a finitistic space with non-trivial cohomology groups H^in(X;Z)=Z with generators v_i, where i = 0, 1, 2, 3. We say that X has cohomology type (a, b) if v_1^2 = av_2 and v_1v_2 = bv_3 . In this note, we determine the mod 2 cohomology ring of the orbit space X/G of a free action of G = Z_2 on X, where both a and b are even. In this case, we observed that there is no equivariant map S^m --> X for m > 3n, where S^m has the antipodal action. Moreover, it is shown that G can not act freely on space X which is of cohomology type (a, b) where a is odd and b is even. We also obtain the mod 2 cohomology ring of the orbit space X/G of free action of G = S^1 on the space X of type (0, b).