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

TL;DR

This paper extends the Borsuk--Ulam theorem to cyclic p-groups using connective K-theory, providing lower bounds on the dimension of zero-sets of equivariant maps between spheres of complex representations.

Contribution

It introduces a new Borsuk--Ulam type theorem for cyclic p-groups within connective K-theory, generalizing classical results to a broader class of group actions.

Findings

Establishes a lower bound on the covering dimension of zero-sets for equivariant maps.

Connects fixed point subspace structure with topological properties of zero-sets.

Provides a formula involving representation dimensions and fixed subspaces.

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 $k\geq 0$. For $0\leq l\leq k$, 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 $\delta (V)=\sum_{l=0}^k p^l dim_C (V_l/V_{l+1})-(p^k-1)$. 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 $2(dim_C U - \delta (V)-1)$, if $dim_C U> \delta (V)$.