Topological transversals to a family of convex sets
L. Montejano, R.N. Karasev

TL;DR
This paper introduces a topological concept of transversals to convex sets in Euclidean space, linking it to classical transversality and deriving geometric consequences using algebraic topology and combinatorial theorems.
Contribution
It establishes a connection between topological and classical transversals for convex sets and applies advanced topological tools to derive geometric results.
Findings
Topological $ ho$-transversals imply classical $ ho$-transversals under certain conditions.
Uses Schubert cocycles and Lusternik-Schnirelmann category to derive geometric consequences.
Extends colorful Helly theorem to topological transversality context.
Abstract
Let be a family of compact convex sets in . We say that has a \emph{topological -transversal of index } (, ) if there are, homologically, as many transversal -planes to as -planes containing a fixed -plane in . Clearly, if has a -transversal plane, then has a topological -transversal of index for and . The converse is not true in general. We prove that for a family of compact convex sets in a topological -transversal of index implies an ordinary -transversal. We use this result, together with the multiplication formulas for Schubert cocycles, the Lusternik-Schnirelmann category of the Grassmannian, and different versions of the colorful Helly…
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