## The Homotopy Groups of the Simplicial Mapping Space between Algebras

##### Doc. Math. 24, 251-270 (2019)
DOI: 10.25537/dm.2019v24.251-270

### Summary

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

### Keywords/Phrases

homotopy theory of algebras, bivariant algebraic $K$-theory

### References

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### Affiliation

Rodríguez Cirone, Emanuel Darío
Dep. de Matemática-IMAS, F. Cs. Exactas y Naturales, Univ. de Buenos Aires, Ciudad Universitaria Pab 1, 1428 Buenos Aires, Argentina