Geometry & Topology

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

Pascal Lambrechts and Donald Stanley

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Suppose f:VW 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 dimW2dimV+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.

Article information

Geom. Topol., Volume 12, Number 4 (2008), 1921-1993.

Received: 25 January 2006
Accepted: 26 March 2008
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P62: Rational homotopy theory
Secondary: 14F35: Homotopy theory; fundamental groups [See also 14H30] 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.) 53D05: Symplectic manifolds, general

blow-up shriek map rational homotopy symplectic manifold


Lambrechts, Pascal; Stanley, Donald. The rational homotopy type of a blow-up in the stable case. Geom. Topol. 12 (2008), no. 4, 1921--1993. doi:10.2140/gt.2008.12.1921.

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