What is an example of a (compact) manifold, which has two non-equivalent differential structures such that the K-homology groups are non-isomorphic? If no such example exists, i.e. "K-homology does not see the differential structure", then can someone give a heuristic explanation of why an object (originally at least) defined in terms of pseudo-differential operators does not depend on a choice of differential structure.



Differential structures and K-homology groups
K-homology is usually not defined in terms of pseudodifferential operators. In fact, I don't even know which definition you mean.


K-homology is either defined as the dual of K-theory (i.e., it is defined as the generalized homology theory associated to the K-theory spectrum), which means that it only depends on the weak homotopy type of the space.


Or it is defined using Fredholm modules over the C\*-algebra $C\_0(X)$ of continuous functions vanishing at infinity, which means that a priori it depends only on the homeomorphism type of the space.


Or it is defined using a geometric picture (cycles are compact spin$^c$-manifolds with a Hermitian vector bundle and a continuous map to the space).


All three definitions can be applied to topological spaces, i.e., do not need a smooth structure on the space or the space being a manifold at all. All three definitions coincide for finite complexes.


