The transverse singular complex

TL;DR

This paper proves that the singular simplicial set of a smooth manifold deformation retracts onto a subset of simplices that are transverse to a given collection of submanifolds with corners.

Contribution

It establishes a deformation retraction of the singular set onto a transverse subset, extending the understanding of smooth singular simplices in manifolds.

Findings

The singular simplicial set deformation retracts onto the transverse subset.

Transversality conditions can be incorporated into the singular complex.

The result applies to manifolds with corners and countable collections of submanifolds.

Abstract

Let $M$ be a smooth manifold without boundary and let $\mathcal{T}$ be a countable collection of manifolds with corners, each equipped with a smooth map to $M$. We show that the singular simplicial set $\mathrm{Sing}(M)$ of $M$ deformation retracts onto the simplicial subset $\mathrm{Sing}^{\mathcal{T}}\!(M)$ of smooth singular simplices that are transverse to every element of $\mathcal{T}$.