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 $\mathrm{c}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{d}(X)$ 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 $$\mathrm{c}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{d}(S(X))\ge \mathrm{c}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{d}(X)+1.$$ We prove that the excess over this lower bound can be arbitrarily large. More precisely, for every $n\ge 2$, we construct a finite free $n$-dimensional simplicial ${\mathbb{Z}}_2$-complex $\mathcal{K}$ such that $\mathrm{c}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{d}(\mathcal{K})=1$ and $\mathrm{c}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{d}(S(\mathcal{K}))=n+1$. 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.