The Homotopy Groups of the Simplicial Mapping Space between Algebras

  • Emanuel Darío Rodríguez Cirone

    Dep. de Matemática-IMAS, F. Cs. Exactas y Naturales, Univ. de Buenos Aires, Ciudad Universitaria Pab 1, 1428 Buenos Aires, Argentina
The Homotopy Groups of the Simplicial Mapping Space between Algebras cover
Download PDF

This article is published open access.

Abstract

Let \ell be a commutative ring with unit. To every pair of \ell-algebras AA and BB one can associate a simplicial set Hom(A,BΔ)\text{Hom}(A,B^\Delta) so that π0Hom(A,BΔ)\pi_0\text{Hom}(A,B^\Delta) equals the set of polynomial homotopy classes of morphisms from AA to BB. We prove that πnHom(A,BΔ)\pi_n\text{Hom}(A,B^\Delta) is the set of homotopy classes of morphisms from AA to BSnB^{\mathfrak{S}_n}_\bullet, where BSnB^{\mathfrak{S}_n}_\bullet is the ind-algebra of polynomials on the nn-dimensional cube with coefficients in BB vanishing at the boundary of the cube. This is a generalization to arbitrary dimensions of a theorem of Cortiñas-Thom, which addresses the cases n1n\leq 1. As an application we give a simplified proof of a theorem of Garkusha that computes the homotopy groups of his matrix-unstable algebraic KKKK-theory space in terms of polynomial homotopy classes of morphisms.

Cite this article

Emanuel Darío Rodríguez Cirone, The Homotopy Groups of the Simplicial Mapping Space between Algebras. Doc. Math. 24 (2019), pp. 251–270

DOI 10.4171/DM/680