A Borsuk--Ulam theorem for cyclic p-groups

2022/11/15 by M. C. Crabb, Crabb, M. C.
Mathematics · #55M25 #55N15 #55R25 #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2211.08087

openalex publication_date 2022/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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≥ 0. For 0≤ l≤ 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 δ(V)=∑_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) → V from the unit sphere in U (for some invariant inner product) has covering dimension greater than or equal to 2(dim_C U - δ(V)-1), if dim_C U> δ(V).

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