# Homotopy equivalence in unbounded KK-theory

## TL;DR

This paper introduces a new framework for unbounded KK-theory, defining a homotopy group of cycles that aligns with Kasparov's KK-theory for separable algebras, enhancing the understanding of unbounded KK-classes.

## Contribution

It proposes a new notion of unbounded KK-cycle, establishes a homotopy group isomorphic to KK-theory, and clarifies the relation between homotopy, operator-homotopies, and degenerate cycles.

## Key findings

- The semigroup of homotopy classes forms an abelian group.
- For separable algebras, the group is isomorphic to KK-theory.
- Homotopy coincides with operator-homotopies plus degenerate cycles.

## Abstract

We propose a new notion of unbounded $K\!K$-cycle, mildly generalising unbounded Kasparov modules, for which the direct sum is well-defined. To a pair $(A,B)$ of $\sigma$-unital $C^{*}$-algebras, we can then associate a semigroup $\overline{U\!K\!K}(A,B)$ of homotopy equivalence classes of unbounded cycles, and we prove that this semigroup is in fact an abelian group. In case $A$ is separable, our group $\overline{U\!K\!K}(A,B)$ is isomorphic to Kasparov's $K\!K$-theory group $K\!K(A,B)$ via the bounded transform. We also discuss various notions of degenerate cycles, and we prove that the homotopy relation on unbounded cycles coincides with the relation generated by operator-homotopies and addition of degenerate cycles.