Linked spaces and exit paths

Abstract

Conically smooth spaces (CSSs), introduced by Ayala, Francis and Tanaka, constitute a large class of singular spaces including Whitney-stratified spaces. We reduce the stratified topology of CSSs over depth-$1$ posets to the ordinary topology of linked smooth manifolds, i.e., spans $M\xleftarrow{\pi} L\xrightarrow{\iota}N$ of smooth manifolds where $\pi$ is a fibre bundle and $\iota$ is a closed embedding. To that end, we introduce explicit exit path quasi-categories (EPCs) for linked spaces and prove that this induces a fully faithful functor from a quasi-category of linked spaces to the quasi-category of all quasi-categories whose essential image includes the Lurie--MacPherson EPCs of CSSs over depth-$1$ posets. We use linked smooth manifolds to resolve various weaker versions of a conjecture of Ayala--Francis--Rozenblyum in the negative by exhibiting quasi-categories with conservative functors to $\{0<1\}$ satisfying certain finiteness conditions but which are not equivalent to EPCs of CSSs. In a sequel, we develop a tangential theory for linked smooth manifolds and reduce the classification conically smooth bundles over depth-$1$ posets to that of ordinary bundles on linked smooth manifolds.