A Borsuk--Ulam theorem for cyclic $p$-groups

Abstract

We describe a connective $K$-theory Borsuk--Ulam/Bourgin--Yang theorem for cyclic groups of order a power of a prime $p$. Consider two finite dimensional complex representations $U$ and $V$ of the cyclic group $Z/p^{k+1}$ of order $p^{k+1}$, where $k\ge 0$. For $0\le l\le k$, we write $V_l$ for the subspace of $V$ fixed by the cyclic subgroup of order $p^l$, and require that the fixed subspace, $V_{k+1}$, be zero and that $V_k$ be non-zero. Put $\delta (V)={\sum }_{l=0}^k{p}^{l}di{m}_C(V_l/V_{l+1})-(p^k-1)$. Then the zero-set of any $Z/p^{k+1}$-map $S(U)\to V$ from the unit sphere in $U$ (for some invariant inner product) has covering dimension greater than or equal to $2(di{m}_{C}U-\delta (V)-1)$, if $di{m}_{C}U>\delta (V)$.