I am reading a paper which uses some $K$-homology which is the homology theory dual to $K$-theory can be defined using the homotopy theoretic formulation: $$ K_\ast(X)\cong\pi_\ast(K\wedge X). $$ Alternatively, the homology cycles can be defined by the triples $(M,E,f:M\rightarrow X)$, where $M$ is a $\text{Spin}^c$ manifold and $E$ is a complex vector bundle over $M$. I am trying to understand how do these two definitions agree. Is there a source that someone could direct me to that makes this equivalence explicit? Or even a short black-box explanation would be appreciated.
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1$\begingroup$ I heard about a related construction and recorded a sketch of it in the first section of chromotopy.org/latex/papers/mu2k.pdf (with some myopia on technical details). Based on that, I think a sketch for yours could look like first writing "pi_* K ^ X" as "pi_* colim_k Omega^2k (BU ^ X) = Omega^fr(colim_k Susp^(-2k) BU ^ X)". Kochman then gives a geometric chains model in terms of {M, f: M --> X ^ BU} with M framed. Finally, you might use the orientation S --> MSpin^c --> K to reduce from Omega^fr to Omega^(Spin^c) and hence the structure group of M. Just a guess! $\endgroup$– Eric PetersonCommented Sep 15 at 15:51
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Unfortunately I don't think this is as well documented in the literature as it should be. Let me try to sketch my (far from complete) understanding of the story. I would welcome any corrections or comments on my description of the situation!
There are really three players here, the homotopy-theoretical K-homology you define above, the geometric definition you indicate, and a third analytical construction $E_*$ due to Brown, Douglas, and Fillmore (in the paper I'll reference below).
The link between the homotopy-theoretical definition of K-homology and the analytical definition $E_*$ is found in the article:
Kahn, Daniel S.; Kaminker, Jerome; Schochet, Claude Generalized homology theories on compact metric spaces. Michigan Math. J.24(1977), no.2, 203–224.
The link between the analytical and geometric definitions seems to have been folklore for a very long time, and the following article was written to give a complete account:
Baum, Paul(1-PAS); Higson, Nigel(1-PAS); Schick, Thomas(D-GTN) On the equivalence of geometric and analytic K-homology.(English summary)Pure Appl. Math. Q.3,Special Issue: In honor of Robert D. MacPherson. Part 3(2007), no.1, 1–24.
(Here's the arXiv preprint: https://arxiv.org/pdf/math/0701484.)
Note: There is some discussion of the homotopy-theoretic definition of K-homology in
L. Brown, R. Douglas, and P. Fillmore. Extensions of $C^∗$-algebras and K-homology. Annals of Math., 105:265–324, 1977.
It's probably worth quoting from Schocet's review of that article in Math Reviews (MR0431136): "Suppose that X is a finite complex. Then a homology $K$-theory $K_∗ (X)$ may be defined by $K_∗(X)=K_∗(DX)$, where $DX$ is an appropriate Spanier-Whithead dual... Slanting with a duality class in $K_0(DX×X)$ yields natural transformations $E_j(X)\rightarrow K_{−j}(DX)\simeq K_j(X)$ for each $j$. The authors assert that these maps yield a natural equivalence of homology theories $E_∗\simeq K_∗$ on finite complexes. (In fact they only claim $E_0\simeq K_0$ (a far weaker result) but it would seem that they have the former in mind). The argument given is very brief and it has troublesome gaps. The assertion may be proved by using earlier results of their paper to show that there is a transformation of homology theories $E_∗ (X)\rightarrow K_∗(X)$ on finite complexes, checking that the transformation is an isomorphism on spheres (a direct calculation), and then using the Atiyah-Hirzebruch spectral sequence or an induction argument on cells. Alternately, one may appeal to the much more general result of D. S. Kahn, Kaminker and the reviewer [Michigan Math J. 24 (1977), no. 2, 203–224], who showed that $E_∗$ is naturally equivalent to $sK_∗$ (Steenrod K-homology) on finite-dimensional compact metric spaces."
