Volume 12, issue 4 (2008)

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The rational homotopy type of a blow-up in the stable case

Pascal Lambrechts and Donald Stanley

Geometry & Topology 12 (2008) 1921–1993
Abstract

Suppose $f:\phantom{\rule{0.3em}{0ex}}V\to W$ is an embedding of closed oriented manifolds whose normal bundle has the structure of a complex vector bundle. It is well known in both complex and symplectic geometry that one can then construct a manifold $\stackrel{˜}{W}$ which is the blow-up of $W$ along $V$. Assume that $dimW\ge 2dimV+3$ and that ${H}^{1}\left(f\right)$ is injective. We construct an algebraic model of the rational homotopy type of the blow-up $\stackrel{˜}{W}$ from an algebraic model of the embedding and the Chern classes of the normal bundle. This implies that if the space $W$ is simply connected then the rational homotopy type of $\stackrel{˜}{W}$ depends only on the rational homotopy class of $f$ and on the Chern classes of the normal bundle.

Keywords
blow-up, shriek map, rational homotopy, symplectic manifold
Mathematical Subject Classification 2000
Primary: 55P62
Secondary: 14F35, 53C15, 53D05
Publication
Received: 25 January 2006
Accepted: 26 March 2008
Published: 5 July 2008
Proposed: Bill Dwyer
Seconded: Haynes Miller, Tom Goodwillie
Authors
 Pascal Lambrechts Chercheur qualifié au FNRS Université Catholique de Louvain Institut Mathématique Chemin du Cyclotron, 2 B-1348 Louvain-la-Neuve BELGIUM Donald Stanley University of Regina Department of Mathematics College West 307.14 Regina, Saskatchewan Canada S4S 0A2