Real Homotopy Theory of Semi-Algebraic Sets

TL;DR

This paper develops a functorial algebraic framework for capturing the real homotopy type of semi-algebraic sets using semi-algebraic differential forms, extending classical de Rham theory.

Contribution

It completes the detailed construction of a graded algebra of semi-algebraic forms that encodes the real homotopy type, crucial for Kontsevich's formality proof.

Findings

Constructed a functorial algebraic model for semi-algebraic sets.

Showed the algebra encodes the real homotopy type.

Applied the theory to prove formality of the little cubes operad.

Abstract

We complete the details of a theory outlined by Kontsevich and Soibelman that associates to a semi-algebraic set a certain graded commutative differential algebra of "semi-algebraic differential forms" in a functorial way. This algebra encodes the real homotopy type of the semi-algebraic set in the spirit of the DeRham algebra of differential forms on a smooth manifold. Its development is needed for Kontsevich's proof of the formality of the little cubes operad.