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The circle transfer and cobordism categories / Jeffrey Giansiracusa

Proceedings of the Edinburgh Mathematical Society, Pages: 1 - 13

Swansea University Author: Giansiracusa, Jeffrey

Abstract

The circle transfer QΣ(LXhS1)+→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 1-dimensional cobordism category and let Circ(X)⊂C1...

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Published in: Proceedings of the Edinburgh Mathematical Society
ISSN: 0013-0915 1464-3839
Published: Cambridge, UK Cmbridge University Press 2019
Online Access: Check full text

URI: https://cronfa.swan.ac.uk/Record/cronfa44755
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Abstract: The circle transfer QΣ(LXhS1)+→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 1-dimensional cobordism category and let Circ(X)⊂C1(X) denote the subcategory whose objects are disjoint unions of unparametrised circles in ℝ∞. 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 2-dimensional cobordism category C2(X) and find that it is null-homotopic when X is a point.
Keywords: transfer, stable homotopy, cobordism categories, circle equivariant
College: College of Science
Start Page: 1
End Page: 13