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The stable mapping class group of simply connected 4-manifolds

Jeffrey Giansiracusa Orcid Logo

Journal für die reine und angewandte Mathematik (Crelles Journal), Volume: 2008, Issue: 617, Pages: 215 - 235

Swansea University Author: Jeffrey Giansiracusa Orcid Logo

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DOI (Published version): 10.1515/CRELLE.2008.031

Abstract

We consider mapping class groups \Gamma(M) = pi_0 Diff(M fix \partial M) of smooth compact simply connected oriented 4-manifolds M bounded by a collection of 3-spheres. We show that if M contains CP^2 (with either orientation) as a connected summand then \Gamma(M) is independent of the number of bou...

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Published in: Journal für die reine und angewandte Mathematik (Crelles Journal)
ISSN: 0075-4102 1435-5345
Published: De Gruyter 2008
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URI: https://cronfa.swan.ac.uk/Record/cronfa7888
first_indexed 2013-07-23T11:59:43Z
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spelling 2015-07-31T17:08:59.0604729 v2 7888 2012-02-23 The stable mapping class group of simply connected 4-manifolds 03c4f93e1b94af60eb0c18c892b0c1d9 0000-0003-4252-0058 Jeffrey Giansiracusa Jeffrey Giansiracusa true false 2012-02-23 MACS We consider mapping class groups \Gamma(M) = pi_0 Diff(M fix \partial M) of smooth compact simply connected oriented 4-manifolds M bounded by a collection of 3-spheres. We show that if M contains CP^2 (with either orientation) as a connected summand then \Gamma(M) is independent of the number of boundary components. By repackaging classical results of Wall, Kreck and Quinn, we show that the natural homomorphism from the mapping class group to the group of automorphisms of the intersection form becomes an isomorphism after stabilization with respect to connected sum with CP^2 # \bar{CP^2}. We next consider the 3+1 dimensional cobordism 2-category of 3-spheres, 4-manifolds (as above) and enriched with isotopy classes of diffeomorphisms as 2-morphisms. We identify the homotopy type of the classifying space of this category as the Hermitian algebraic K-theory of the integers. We also comment on versions of these results for simply connected spin 4-manifolds. Finally, we observe that a related 4-manifold operad detects infinite loop spaces. Journal Article Journal für die reine und angewandte Mathematik (Crelles Journal) 2008 617 215 235 De Gruyter 0075-4102 1435-5345 4-manifolds, isotopy, mapping class group, cobordism category, pseudo-isotopy, quadratic forms, Hermitian K-theory 13 5 2008 2008-05-13 10.1515/CRELLE.2008.031 http://www.degruyter.com/view/j/crll.2008.2008.issue-617/crelle.2008.031/crelle.2008.031.xml COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University 2015-07-31T17:08:59.0604729 2012-02-23T17:02:22.0000000 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Jeffrey Giansiracusa 0000-0003-4252-0058 1
title The stable mapping class group of simply connected 4-manifolds
spellingShingle The stable mapping class group of simply connected 4-manifolds
Jeffrey Giansiracusa
title_short The stable mapping class group of simply connected 4-manifolds
title_full The stable mapping class group of simply connected 4-manifolds
title_fullStr The stable mapping class group of simply connected 4-manifolds
title_full_unstemmed The stable mapping class group of simply connected 4-manifolds
title_sort The stable mapping class group of simply connected 4-manifolds
author_id_str_mv 03c4f93e1b94af60eb0c18c892b0c1d9
author_id_fullname_str_mv 03c4f93e1b94af60eb0c18c892b0c1d9_***_Jeffrey Giansiracusa
author Jeffrey Giansiracusa
author2 Jeffrey Giansiracusa
format Journal article
container_title Journal für die reine und angewandte Mathematik (Crelles Journal)
container_volume 2008
container_issue 617
container_start_page 215
publishDate 2008
institution Swansea University
issn 0075-4102
1435-5345
doi_str_mv 10.1515/CRELLE.2008.031
publisher De Gruyter
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 http://www.degruyter.com/view/j/crll.2008.2008.issue-617/crelle.2008.031/crelle.2008.031.xml
document_store_str 0
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description We consider mapping class groups \Gamma(M) = pi_0 Diff(M fix \partial M) of smooth compact simply connected oriented 4-manifolds M bounded by a collection of 3-spheres. We show that if M contains CP^2 (with either orientation) as a connected summand then \Gamma(M) is independent of the number of boundary components. By repackaging classical results of Wall, Kreck and Quinn, we show that the natural homomorphism from the mapping class group to the group of automorphisms of the intersection form becomes an isomorphism after stabilization with respect to connected sum with CP^2 # \bar{CP^2}. We next consider the 3+1 dimensional cobordism 2-category of 3-spheres, 4-manifolds (as above) and enriched with isotopy classes of diffeomorphisms as 2-morphisms. We identify the homotopy type of the classifying space of this category as the Hermitian algebraic K-theory of the integers. We also comment on versions of these results for simply connected spin 4-manifolds. Finally, we observe that a related 4-manifold operad detects infinite loop spaces.
published_date 2008-05-13T12:15:23Z
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score 11.048149

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