THE HOMOTOPY GROUPS OF THE SIMPLICIAL MAPPING SPACE BETWEEN ALGEBRAS

Let l be a commutative ring with unit. To every pair of l-algebras A and B one can associate a simplicial set Hom(A, B-Delta) so that pi(0)Hom(A, B-Delta) equals the set of polynomial homotopy classes of morphisms from A to B. We prove that pi(n) Hom(A, B-Delta) is the set of homotopy classes of morphisms from A to B-.(Gn), where B-.(Gn) is the ind-algebra of polynomials on the n -dimensional cube with coefficients in B vanishing at the boundary of the cube. This is a generalization to arbitrary dimensions of a theorem of Cortinas-Thom, which addresses the cases n <= 1. As an application we give a simplified proof of a theorem of Garkusha that computes the homotopy groups of his matrix-unstable algebraic KK-theory space in terms of polynomial homotopy classes of morphisms.