There is a homological Chern character $ch\_\ast \colon K\_\ast(X) \to H\_\ast(X)$ for $X$ a smooth, compact manifold.


I found only one definition of it (in the paper "K-Homology and Index Theory" by Baum, Douglas), where it is defined (i) by the usual cohomological Chern character map and Poincare duality between $H^\ast(X)$ and $H\_\ast(X)$, and (ii) by using the geometric definition of the K-homology groups $K\_\ast(X)$ of $X$.


(The geometric definition of K-homology uses cycles $(M, E, \phi)$, where $M$ is a closed Spin$^c$ manifold, $E$ a complex vector bundle on it and $\phi \colon M \to X$ a continuous map. Then $ch\_\ast (M, E, \phi) := \phi\_\ast(ch^\ast(E) \cup Td(TM) \cap [M])$, where $ch^\ast$ is the cohomological Chern character.)


Especially the use of Poincare duality to reduce the definition to the usual cohomological Chern character bothers me.



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With "direct" I mean not using Poincare duality and then the cohomological Chern character. It would also be nice if this direct definition would not use the geometric description of K-homology but either the one by Fredholm modules or through the dual algebra $\mathfrak{D}(C\_0(X))$.


Maybe one has to go through cyclic (co-)homology or Hochschild homology, e.g., first define $K\_\ast(X) \to HC$ and then $HC \to H\_\ast(X)$. That would be ok for me.



Definition of the homological Chern character
I like this question! I think this problem (the Chern character of K-homology) has been studied and solved by Alain Connes in his paper "Noncommutative differential geometry" in 1985. He indeed used cyclic cohomology etc. Unfortunately he uses analytic description of K-homology instead of dual algebras but they are essentially the same.


You can also read the book "Elements of noncommutative geometry" by Gracia-Bondia, Varilly and Figueroa. In particular Chapter 10 of that book gives a detailed description of Connes' construction.


