## Homological perturbations, equivariant cohomology, and Koszul duality

Citations: | 10 - 6 self |

### BibTeX

@MISC{Huebschmann_homologicalperturbations,,

author = {Johannes Huebschmann},

title = {Homological perturbations, equivariant cohomology, and Koszul duality},

year = {}

}

### OpenURL

### Abstract

Dedicated to the memory of V.K.A.M. Gugenheim Abstract. Our main objective is to demonstrate how homological perturbation theory (HPT) results over the last 40 years immediately or with little extra work give some of the Koszul duality results that have appeared in the last decade. Higher homotopies typically arise when a huge object, e. g. a chain complex defining various invariants of a certain geometric situation, is cut to a small model, and the higher homotopies can then be dealt with concisely in the language of sh-structures (strong homotopy structures). This amounts to precise ways of handling the requisite additional structure encapsulating the various coherence conditions. Given e. g. two augmented differential graded algebras A1 and A2, an sh-map from A1 to A2 is a twisting cochain from the reduced bar construction BA1 of A1 to A2 and, in this manner, the class of morphisms of augmented differential graded algebras is extended to that of sh-morphisms. In the present paper, we explore small models for equivariant (co)homology via differential homological algebra techniques including homological perturbation theory which, in turn, is a standard tool to handle