Occupants in simplicial complexes

TL;DR

This paper generalizes manifold calculus to simplicial complexes, providing a homotopical formula for the complement of a complex in a manifold, extending previous work on smooth submanifolds.

Contribution

It introduces a homotopical formula for the complement of a simplicial complex in a manifold, broadening the scope of manifold calculus methods.

Findings

Derived a homotopical formula for $M\setminus K$ in terms of finite subsets

Extended previous work from smooth submanifolds to simplicial complexes

Utilized functor calculus methods for the generalization

Abstract

Let $M$ be a smooth manifold and $K\subset M$ be a simplicial complex of codimension at least 3. Functor calculus methods lead to a homotopical formula of $M\setminus K$ in terms of spaces $M\setminus T$ where $T$ is a finite subset of $K$. This is a generalization of the author's previous work with Michael Weiss where the subset $K$ is assumed to be a smooth submanifold of $M$ and uses his generalization of manifold calculus adapted for simplicial complexes.