Homotopy equivalence in unbounded KK-theory

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 $σ$-unital C*-algebras, we can then associate a semigroup U\!K\!K̄(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 U\!K\!K̄(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.