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

TL;DR

This paper develops an algebraic model to determine the rational homotopy type of a blow-up of a manifold along an embedding with a complex normal bundle, depending only on the embedding's rational homotopy class and Chern classes.

Contribution

It introduces a method to compute the rational homotopy type of blow-ups using algebraic models derived from embeddings and Chern classes, extending understanding in complex and symplectic geometry.

Findings

Constructed an algebraic model for the rational homotopy type of blow-ups.

Showed the model depends only on the embedding's rational homotopy class and Chern classes.

Applicable to simply connected spaces, simplifying rational homotopy classification.

Abstract

Suppose that 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 dim(W)>2.dim(V)+2 and that H^1(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.