I posted this question on Math.SE (<https://math.stackexchange.com/questions/409444/>), but got no answer. So I repost it here.


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 on Math.SE about it: <https://math.stackexchange.com/questions/402170/> - 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 other question about it: [Duality between K-theory and K-homology in the non-compact, spin$^c$ case](https://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!



Duality between K-theory and K-homology in the non-spin^c case.
This is nicely understood by duality (in the sense of [dualizable objects](http://ncatlab.org/nlab/show/dualizable%20object) in monoidal categories) in the monoidal category [KK](http://ncatlab.org/nlab/show/KK-theory). The relevant results are collected and referenced on the nLab at *[Poincaré duality algebra](http://ncatlab.org/nlab/show/Poincar%C3%A9+duality+algebra)*:


For $X$ a closed manifold equipped with a twist $\chi$ (the class of a [circle 2-bundle](http://ncatlab.org/nlab/show/circle%202-bundle), realized as a $U(1)$bundle gerbe etc., let $C\_\chi(X)$ be the corresponding [twisted groupoid convolution C\*-algebra](http://ncatlab.org/nlab/show/category+algebra), well defined up to Morita equivalence and hence KK-equivalence. This is the $C^\ast$-algebra such that


$$
 KK\_\bullet(\mathbb{C}, C\_\chi(X)) \simeq K^{\bullet+\chi}(X)
$$


is the $\chi$-[twisted K-theory](http://ncatlab.org/nlab/show/twisted%20K-theory) of $X$. 


Then here is the statement: The dual object of $C\_\chi(X)$ in $(KK, \otimes)$ is


$$
 \left(
 C\_\chi\left(X\right)
 \right)^\vee
 \simeq
 C\_{-\chi - W\_3(T X)}(X)
 \,,
$$


where $W\_3(T Q)$ is the [third integral Stiefel-Whitney class](http://ncatlab.org/nlab/show/integral%20Stiefel-Whitney%20class) of the tangent bundle of $X$.


This formula implies everything in this business. 


For instance it implies that $C\_\chi(X)$ is self-dual, and hence (since duality in KK is duality between K-homology and K-cohomology) that $X$ exhibits K-Poincaré duality, precisely if $\chi = 0$ and $W\_3(T X) = 0$. The latter is precisely the condition of [spin^c structure](http://ncatlab.org/nlab/show/spin%5Ec%20structure).


But you get much more by playing with the formula, and this is I think what you are after. For instance given a map of closed manifolds $i \colon Q \to X$ with $X$ carrying a twist $\chi$ and $Q$ carrying the twist $i^\ast \chi$, we get the corresponding pullback map of twisted convolution algebras


$$
 i^\ast \colon C\_\chi(X) \to C\_{i^\ast \chi}(Q)
 \,.
$$


Now since both objects have dualty, we can form the corresponding [dual morphism](http://ncatlab.org/nlab/show/dual%20morphism) by the general formula in monoidal categories:


$$
 i\_! := (i^\ast)^\vee : (C\_{i^\ast \chi}(Q))^\vee \to (C\_\chi(X))^\vee
 \,,
$$


hence by the above


$$
 i\_! : C\_{-i^\ast \chi - W\_3(T Q)}(Q) \to C\_{-\chi-W\_3(T X)}(X)
 \,.
$$


By just relabelling the original twist for transparency as $\chi \mapsto - \chi - W\_3(T X)$ this is equivalently a morphism


$$
 i\_! : C\_{i^\ast \chi + W\_3(N\_i Q)}(Q) \to C\_{\chi}(X)
 \,.
$$


(Here $N\_i Q$ denotes the normal bundle of $Q$ in $X$ via $i$.)


By postcomposition in KK and using the general relation that $KK(\mathbb{C}, A) \simeq K(A)$ this yields a homomorphism


$$
 i\_! : K^{\bullet + i^\ast \chi + W\_3(N\_i Q)} \to K^{\bullet + \chi}(X)
$$


from the $(i^\ast \chi + W\_3(N\_i Q))$-twisted K-theory of $Q$ to the $\chi$-twisted K-thory of $X$.


This is the general **twisted Umkehr map** in twisted K-theory. Notice again that it is simply the [dual morphism](http://ncatlab.org/nlab/show/dual%20morphism) of $i^\ast$ in $(KK,\otimes)$.


If now the normal bundle has $Spin^c$-structure then that term drops out, and so on. But in general it is like this.


If here we think of $X$ as the target spacetime for a type II superstring, of $\chi$ as the instanton sector of the [B-field](http://ncatlab.org/nlab/show/B-field), and of $Q \hookrightarrow X$ as the worldvolume of a D-brane, then the $(i^\ast \chi + W\_3(N\_i Q))$-twisted K-cocycles on $Q$ are the (underlying instanton sectors of) the [Chan-Paton gauge fields](http://ncatlab.org/nlab/show/Chan-Paton%20gauge%20field) subject to the 
[Freed-Witten-Kapustin anomaly cancellation condition](http://ncatlab.org/nlab/show/Freed-Witten+anomaly) and their image under the above twisted Umkehr map is the [D\_brane charge](http://ncatlab.org/nlab/show/D-brane+charge) of the D-brane with that Chan-Paton bundle. 


All this just means really: the dual map of $i^\ast$ picks up twists as above. See at *[Poincaré duality algebra](http://ncatlab.org/nlab/show/Poincar%C3%A9+duality+algebra)* for citations. All these results go through also for $G$-equivariant twisted K-theory.


