The homotopy groups of the simplicial mapping space between algebras

TL;DR

This paper generalizes a theorem relating homotopy groups of simplicial mapping spaces between algebras to classes of morphisms into polynomial ind-algebras, extending previous results to all dimensions.

Contribution

It extends the Cortiñas-Thom theorem to all dimensions, connecting homotopy groups with polynomial homotopy classes for arbitrary n.

Findings

Homotopy groups of the simplicial mapping space correspond to classes of morphisms into polynomial ind-algebras.

Provides a simplified proof of Garkusha's theorem on algebraic KK-theory.

Generalizes known low-dimensional results to higher dimensions.

Abstract

Let $\ell$ be a commutative ring with unit. To every pair of $\ell$-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 $B^{S_n}$, where $B^{S_n}$ 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 Corti\~nas-Thom, which addresses the cases $n\leq 1$. As an application we give a simplified proof of a theorem of Garkusha that computes the homotopy groups of his matrix-unstable algebraic KK-theory space in terms of polynomial homotopy classes of morphisms.