Occupants in simplicial complexes (Q2311438)

Let $M$ be a smooth manifold, and let $K\subset M$ be a simplicial complex of codimension at least $3$. The author applies manifold calculus adapted for simplicial complexes to study the homotopy type of the complement $M\setminus K$. He considers a topological poset consisting of the sets $M\setminus V_K(T,\rho)$, where $T$ runs through the finite subsets of $K$ and $V_K(T,\rho)$ denotes a thickening of $T$. The main result states that the canonical map \[ M\setminus K\to \text{holim}\ M\setminus V_K(T,\rho) \] is a weak equivalence. The results in this paper generalize the results in [\textit{S. Tillmann} and \textit{M. S. Weiss}, Contemp. Math. 682, 237--259 (2017; Zbl 1369.57031)].