  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
$n>1$.

Algebraic Topology;Algebraic Geometry;K-Theory and Homology