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.

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    $\begingroup$ Well, de Rham cohomology is also defined in terms of smooth stuff and doesn't end up depending on it. $\endgroup$ Oct 6 '15 at 3:44

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.


This is more or less an addendum to AlexE's answer:

Even though the $K$-homology groups themselves do not depend on the smooth structure there are classes in $K$-theory, which can tell apart some of the smooth structures:

Let $\mathcal{S}(M)$ denote the group of isotopy classes of smooth structures on $M$ and $M_{\alpha}$ denote $M$ equipped with the smooth structure $\alpha$. The identity map induces a map on the corresponding sphere bundles $\varphi \colon SM_{\alpha} \to SM$. There is a unit $u \in K^0(SM)$, such that $\varphi_*([SM_{\alpha}]) = u \cap [SM]$, where $[SM] \in K_1(SM)$ denotes the fundamental class of the spin$^c$-manifold $SM$ in $K$-homology. The class $u$ descends to a unit $\theta(\alpha) \in K^0(M)$, such that $\pi^*(\theta(\alpha)) = u$, where $\pi \colon SM \to M$ is the bundle projection. It was proven by Jerome Kaminker that $\theta \colon \mathcal{S}(M) \to K^0(M)$ is a homomorphism and that there are smooth manifolds $M$ for which $\theta$ is nontrivial (see "Pseudo-differential Operators and differential structures").

This implies for example that the algebra of order $0$ pseudodifferential operators $\mathcal{P}_M$ depends on the smooth structure. To see this, note that it fits into an exact sequence $$ 0 \to \mathcal{K} \to \mathcal{P}_M \to C(SM) \to 0 $$ (where $\mathcal{K}$ are the compact operators) giving an element in $K_1(SM)$ by BDF-theory. This class agrees with the fundamental class $[SM] \in K_1(SM)$.


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