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The Circle Transfer and Cobordism Categories

Part of: Fiber spaces and bundles Homology and cohomology theories

Published online by Cambridge University Press:  11 January 2019

Jeffrey Giansiracusa
Jeffrey Giansiracusa*
Affiliation:
Department of Mathematics Swansea University, Singleton Park, Swansea SA2 8PP, UK (j.h.giansiracusa@swansea.ac.uk)
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Abstract

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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 𝒞1(X) denote the one-dimensional cobordism category and let Circ(X) ⊂ 𝒞1(X) denote the subcategory whose objects are disjoint unions of unparametrized circles. Multiplication in S1 induces a functor Circ(X) → Circ(LX), and the composition of this functor with the inclusion of Circ(LX) into 𝒞1(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 𝒞2(X) and find that it is null-homotopic when X is a point.

Keywords

cobordismcircle transferfree loop spacePontrjagin-Thomequivariant homotopy

MSC classification

Primary: 55R12: Transfer
Secondary: 55N22: Bordism and cobordism theories, formal group laws
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 3 , August 2019 , pp. 719 - 731
Copyright
Copyright © Edinburgh Mathematical Society 2019 

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