The circle transfer and cobordism categories

Abstract

The circle transfer $Q\Sigma (LX_{hS^1})_+ \to QLX_+$ has appeared in several contexts in topology. In this note we observe that this map admits a geometric re-interpretation as a morphism of cobordism categories of 0-manifolds and 1-cobordisms. Let $C_1(X)$ denote the 1-dimensional cobordism category and let $Circ(X) \subset C_1(X)$ denote the subcategory whose objects are disjoint unions of unparametrised circles in $\mathbb{R}^\infty$. Multiplication in $S^1$ induces a functor $Circ(X) \to Circ(LX)$, and the composition of this functor with the inclusion of $Circ(LX)$ into $C_1(LX)$ is homotopic to the circle transfer. As a corollary, we describe the inclusion of the subcategory of cylinders into the 2-dimensional cobordism category $C_2(X)$ and find that it is null-homotopic when $X$ is a point.