Ainfinity-algebra Structure in Cohomology and its Applications

TL;DR

This paper explores the $A()$ and $C()$ algebra structures in cohomology, revealing richer information about topological spaces and their rational homotopy types through multioperations.

Contribution

It introduces the minimality theorem for $A()$ and $C()$ algebras in cohomology, connecting these structures to topological and rational homotopy properties.

Findings

Cohomology of topological spaces admits $A()$ algebra structures.

Rational cohomology algebra admits $C()$ algebra structures that determine rational homotopy type.

Multioperations provide more detailed topological information than traditional cohomology algebra.

Abstract

In these lectures we present our minimality theorem by which in cohomology of a topological space appear multioperations which turn it ot Stasheff $A(\infty)$ algebra. This rich structure carries more information than just the structure of cohomology algebra, particularly it allows to define cohomologies of the loop space. We present also the notion of $C(\infty)$ algebra and the commutatitive version of the minimality theorem by which in rational cohomology algebra appear multioperations which form a $C(\infty)$ algebra structure which completely determines the rational homotopy type.