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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

Suppose f : V 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 W˜ which is the blow-up of W along V . Assume that dimW 2dimV + 3 and that H1(f) is injective. We construct an algebraic model of the rational homotopy type of the blow-up 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 W˜ depends only on the rational homotopy class of f and on the Chern classes of the normal bundle.

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