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The stable mapping class group of simply connected 4-manifolds
Journal für die reine und angewandte Mathematik (Crelles Journal), Volume: 2008, Issue: 617, Pages: 215 - 235
Swansea University Author: Jeffrey Giansiracusa
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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...
| Published in: | Journal für die reine und angewandte Mathematik (Crelles Journal) |
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| ISSN: | 0075-4102 1435-5345 |
| Published: |
De Gruyter
2008
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| Online Access: |
Check full text
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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. |
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| Keywords: |
4-manifolds, isotopy, mapping class group, cobordism category, pseudo-isotopy, quadratic forms, Hermitian K-theory |
| College: |
Faculty of Science and Engineering |
| Issue: |
617 |
| Start Page: |
215 |
| End Page: |
235 |