Some examples of non-tidy spaces

Takahiro Matsushita

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