On the Gap Between the Co-Indices of a Free Z_2-Space and Its Suspension

For a free $\mathbb{Z}_2$-space $X$, the co-index $\mathrm{coind}(X)$ is the largest integer $m$ for which there exists a $\mathbb{Z}_2$-equivariant map $S^m \to X$, where $S^m$ carries the antipodal action.

Since suspension sends such a map to a $\mathbb{Z}_2$-equivariant map $S^{m+1}\to S(X),$ one always has $$\mathrm{coind}(S(X)) \geq \mathrm{coind}(X)+1.$$ We prove that the excess over this lower bound can be arbitrarily large.

More precisely, for every $n \geq 2$, we construct a finite free $n$-dimensional simplicial $\mathbb{Z}_2$-complex $\mathcal{K}$ such that $\mathrm{coind}(\mathcal{K})=1$ and $\mathrm{coind}(S(\mathcal{K}))=n+1$.

This answers a question of Simonyi, Tardos, and Vrécica on the possible growth of co-index under suspension and, equivalently, shows that the co-index lower bound on the chromatic number of a graph $G$ obtained from $B_0(G)$ can exceed the corresponding bound obtained from the box complex $B(G)$ by an arbitrarily large amount.