Let \(\ell\) be a commutative ring with unit. To every pair of \(\ell\)-algebras \(A\) and \(B\) one can associate a simplicial set \(\text{Hom}(A,B^\Delta)\) so that \(\pi_0\text{Hom}(A,B^\Delta)\) equals the set of polynomial homotopy classes of morphisms from \(A\) to \(B\). We prove that \(\pi_n\text{Hom}(A,B^\Delta)\) is the set of homotopy classes of morphisms from \(A\) to \(B^{\mathfrak{S}_n}_\bullet\), where \(B^{\mathfrak{S}_n}_\bullet\) 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 Cortiñas-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.

55Q52, 19K35

homotopy theory of algebras, bivariant algebraic \(K\)-theory

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Dep. de Matemática-IMAS, F. Cs. Exactas y Naturales, Univ. de Buenos Aires, Ciudad Universitaria Pab 1, 1428 Buenos Aires, Argentin