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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...

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

URI: https://cronfa.swan.ac.uk/Record/cronfa44755
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last_indexed 2021-02-24T04:05:58Z
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spelling 2021-02-23T14:44:51.9581370 v2 44755 2018-10-04 The circle transfer and cobordism categories 03c4f93e1b94af60eb0c18c892b0c1d9 Jeffrey Giansiracusa Jeffrey Giansiracusa true false 2018-10-04 FGSEN 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. Journal Article Proceedings of the Edinburgh Mathematical Society 62 3 1 13 Cambridge University Press Cambridge, UK 0013-0915 1464-3839 transfer, stable homotopy, cobordism categories, circle equivariant 31 12 2019 2019-12-31 10.1017/S0013091518000615 https://arxiv.org/abs/1711.09433 COLLEGE NANME Science and Engineering - Faculty COLLEGE CODE FGSEN Swansea University 2021-02-23T14:44:51.9581370 2018-10-04T06:46:52.9197404 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Jeffrey Giansiracusa 1 0044755-04102018064819.pdf 1711.09433.pdf 2018-10-04T06:48:19.0800000 Output 204311 application/pdf Accepted Manuscript true 2019-07-11T00:00:00.0000000 true eng
title The circle transfer and cobordism categories
spellingShingle The circle transfer and cobordism categories
Jeffrey Giansiracusa
title_short The circle transfer and cobordism categories
title_full The circle transfer and cobordism categories
title_fullStr The circle transfer and cobordism categories
title_full_unstemmed The circle transfer and cobordism categories
title_sort The circle transfer and cobordism categories
author_id_str_mv 03c4f93e1b94af60eb0c18c892b0c1d9
author_id_fullname_str_mv 03c4f93e1b94af60eb0c18c892b0c1d9_***_Jeffrey Giansiracusa
author Jeffrey Giansiracusa
author2 Jeffrey Giansiracusa
format Journal article
container_title Proceedings of the Edinburgh Mathematical Society
container_volume 62
container_issue 3
container_start_page 1
publishDate 2019
institution Swansea University
issn 0013-0915
1464-3839
doi_str_mv 10.1017/S0013091518000615
publisher Cambridge University Press
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
url https://arxiv.org/abs/1711.09433
document_store_str 1
active_str 0
description 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.
published_date 2019-12-31T03:56:08Z
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score 10.998093

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