Abstract
AbstractThe 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 đ1(X) denote the one-dimensional cobordism category and let Circ(X) â đ1(X) denote the subcategory whose objects are disjoint unions of unparametrized circles. Multiplication in S1 induces a functor Circ(X) â Circ(LX), and the composition of this functor with the inclusion of Circ(LX) into đ1(LX) is homotopic to the circle transfer. As a corollary, we describe the inclusion of the subcategory of cylinders into the two-dimensional cobordism category đ2(X) and find that it is null-homotopic when X is a point.
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