Some examples of non-tidy spaces

TL;DR

The paper constructs specific free $bZ_2$-manifolds with unique topological properties, demonstrating the existence of manifolds where certain equivariant maps do not exist despite non-trivial Stiefel-Whitney classes.

Contribution

It introduces new examples of non-tidy free $bZ_2$-manifolds with particular cohomological properties and equivariant map obstructions.

Findings

Constructed manifolds with $w_1^n eq 0$

Proved non-existence of equivariant maps from $S^2$

Provided new insights into non-tidy spaces

Abstract

We construct a free $\mathbb{Z}_2$-manifold $X_n$ for a positive integer $n$ such that $w_1(X_n)^n \neq 0$, but there is no $\mathbb{Z}_2$-equivariant map from $S^2$ to $X_n$.