Real homotopy theory of semi-algebraic sets

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. P A 6.1. Poincaré Lemma for Ω * P A 6.2. Sheaf propery of Ω * P A 6.3. Extendability of the simplicial set Ω P A (∆ • ) 6.4. The weak equivalence Ω * P A ≃ A P L 7. Monoidal equivalences 8. Oriented semi-algebraic bundles and integration along the fiber 8.1. Properties of SA bundles 8.2. Properties of integration along the fiber

Cyclic operad formality for compactified moduli spaces of genus zero surfaces

“…This is a slight generalisation of [11,Prop 8.11] and has a similar proof. Given any form ξ ∈ Ω * min (B), one has the push-pull formula ( [11,Prop.…”

“…For the proof of Theorem A, we will need to work in the category of semi-algebraic spaces. The theory of semi-algebraic spaces and its corresponding de Rham theory have been developed by Hardt, Lambrechts, Turchin and Volic in [11].…”

“…The first map is a semi-algebraic bundle by [17, Appendix A], the second map is a trivial bundle, and the composite of bundle projections is a bundle projection by [11,Prop. 8.5].…”

“…Suppose π : E → B is a semi-algebraic fibre bundle with n-dimensional compact fibres and equipped with a fibrewise orientation. Integration of forms fibrewise defines a map (see [11,Section 8]…”

“…As the theory is developed in [11], the pushforward of a minimal form is a P A form by definition, and the pushforward of a P A form is not defined. Note that the minimal forms constitute a sub-CDGA of all P A forms.…”