# Duality between K-theory and K-homology in the non-spin^c case.

Let M be a closed manifold. Then there is a cap product $K^\ast(M) \times K_\ast(M) \to K_\ast(M)$ between the K-theory of M and its K-homology. For a definition of it one could see my prior question about it: Cap product between K-Theory and K-Homology - take there A = C(M).

Now if M is spin$^c$, it has a fundamental class $[M] \in K_\ast(M)$ and it is well-known that the cap product with $[M]$ induces the Poincare duality $K^\ast(M) \stackrel{\cong}\to K_\ast(M)$. (See also my question on MO about it: http://mathoverflow.net/questions/131378/.)

In the book "Spin Geometry" by Lawson, Michelsohn it is shown (on page 257) that every class in $K_{cpt}(TM) \cong K_0(M)$ is the difference of two Atiyah-Singer operators with coefficients (if M is spin and even dimensional), i.e., this translates to saying that the Poincare duality map is surjective onto $K_0(M)$ in this case.

Then it is written: "For non-spin manifolds, one can argue similarly by using the signature operator with coefficients". This means that there is some class $[D] \in K_\ast(M)$ (the class of the signature operator) such that the cap product with [D] is onto on $K_0(M)$.

Now the question is, if this statement generalizes:

Is it crucial that it is the signature operator? Could we also take the Euler characteristic operator?

Is there always (i.e., in the non-spin$^c$ case and not only for even dimensional manifolds) a class $[D] \in K_\ast(M)$ such that the cap product with $[D]$ is onto on $K_\ast(M)$ (and not only on $K_0(M)$)?

Maybe even if M is not orientable? Or even when M is not a manifold?

Are there other sufficient conditions besides spin$^c$ such that such a map is injective?

If one of the statements above is true, it would be nice to have some references. Thanks!

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Posted this question on MO: mathoverflow.net/questions/132966 –  AlexE Jun 6 '13 at 16:02