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The stable mapping class group of simply connected 4-manifolds / Jeffrey Giansiracusa

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

Swansea University Author: Giansiracusa, Jeffrey

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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
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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 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.
Keywords: 4-manifolds, isotopy, mapping class group, cobordism category, pseudo-isotopy, quadratic forms, Hermitian K-theory
College: College of Science
Issue: 617
Start Page: 215
End Page: 235