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The circle transfer and cobordism categories
Jeffrey Giansiracusa 
Proceedings of the Edinburgh Mathematical Society, Volume: 62, Issue: 3, Pages: 1 - 13
Swansea University Author:
Jeffrey Giansiracusa
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DOI (Published version): 10.1017/S0013091518000615
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...
| Published in: | Proceedings of the Edinburgh Mathematical Society |
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| ISSN: | 0013-0915 1464-3839 |
| Published: |
Cambridge, UK
Cambridge University Press
2019
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| Online Access: |
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| URI: | https://cronfa.swan.ac.uk/Record/cronfa44755 |
| 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. |
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| Keywords: |
transfer, stable homotopy, cobordism categories, circle equivariant |
| College: |
Faculty of Science and Engineering |
| Issue: |
3 |
| Start Page: |
1 |
| End Page: |
13 |