2025/01/10 by Bludov, Mikhail V. · 2 citations
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2501.05799
In this paper, we study a construction of homotopy invariants of open or closed covers, where the homotopy class is defined relative to a pair (V,r), with V a finite set of points in ℝd and r a point in the interior of their convex hull. We show that the simplicial complex of non-balanced subsets associated with (V,r) has the homotopy type of a sphere, and use this to develop a theory of homotopy invariants of covers relative to balanced sets. A key result is that the homotopy class of a cover depends only, up to an involution, on the balanced-equivalence class of (V,r). As applications, we obtain extension theorems for covers in this setting and derive the KKMS lemma, its analogues, and related combinatorial fixed-point results.