  We consider Hilsum's notion of bordism as an equivalence relation on
unbounded $KK$-cycles and study the equivalence classes. Upon fixing two
$C^*$-algebras, and a $*$-subalgebra dense in the first $C^*$-algebra, a
$\mathbb{Z}/2\mathbb{Z}$-graded abelian group is obtained; it maps to the
Kasparov $KK$-group of the two $C^*$-algebras via the bounded transform. We
study properties of this map both in general and in specific examples. In
particular, it is an isomorphism if the first $C^*$-algebra is the complex
numbers (i.e., for $K$-theory) and is a split surjection if the first
$C^*$-algebra is the continuous functions on a compact manifold with boundary
when one uses the Lipschitz functions as the dense $*$-subalgebra.
