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

Abstract

Suppose $f:V\to W$ 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 $\tilde{W}$ 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 $\tilde{W}$ 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 $\tilde{W}$ depends only on the rational homotopy class of $f$ and on the Chern classes of the normal bundle.