The mapping space of unbounded differential graded algebras

TL;DR

This paper provides a concrete description of the higher homotopy groups of the mapping space between unbounded differential graded algebras, linking them to Hochschild cohomology in the connective case.

Contribution

It offers a detailed description of the higher homotopy groups of the mapping space for unbounded DGA's and relates them to Hochschild cohomology in the connective case.

Findings

Explicit description of higher homotopy groups of Map_{Alg}(R,S)

Connection between higher homotopy groups and Hochschild cohomology

Applicable to unbounded differential graded algebras over a ring

Abstract

In this paper, we give a concrete description of the higher homotopy groups (n>0) of the mapping space Map_{Alg}(R,S) for R and S unbounded differential graded algebras (DGA) over a commutative ring k. In the connective case, we describe the relation between the higher (negative) Hochschild cohomology $HH^{-n+1}(R,S)$ and higher homotopy groups $\pi_{n} Map_{Alg}(R,S)$, when $n>1$.