Index and Sectional Category

2023/12/19 by Zapata, Cesar Augusto Ipanaque, Gonçalves, Daciberg Lima
#Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2312.11957

Abstract

Let G be a finite group with order |G|=ℓ and 2≤ q≤ ℓ. For a free G-space X, we introduce a notion of q-th index of (X,G), denoted by ind_q(X,G). Our concept is relevant in the Borsuk-Ulam theory. We draw general estimates for the q-th index in terms of the sectional category of the quotient map X→ X/G, denoted by secat(X→ X/G). This property connects a standard problem in Borsuk-Ulam theory to current research trends in sectional category. Under certain hypothesis we observed that secat(X→ X/G)=ind₂(X,G)+1. As an application of our results, we present new results in Borsuk-Ulam theory and sectional category.

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