Geometry & Topology

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

Pascal Lambrechts and Donald Stanley

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513800118

Digital Object Identifier
doi:10.2140/gt.2008.12.1921

Mathematical Reviews number (MathSciNet)
MR2431012

Zentralblatt MATH identifier
1153.55010

Subjects
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

Keywords
blow-up shriek map rational homotopy symplectic manifold

Citation

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. https://projecteuclid.org/euclid.gt/1513800118


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