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The description of the real $KO$-homology groups $KO_n(X)$ can be given abstractly as maps to the real K-theory spectrum $KO$ smash $X$, or via triples $(M,x,\phi)$ where $M$ is a closed manifold, $\phi:M\to X$, and $x\in KO^n(M)$. In the article "A bordism-type theory description of homology" by Martin Jakob, the two points of view are identified via the correspondence sending such a triple to $\phi_*(x\cap [M]_{KO})$, where $[M]_{KO}$ denotes the Thom class of $TM$ w.r.t. the homology theory $KO_*$. In the context of complex $K$-theory, we have (according to that article) a homological Chern character $$ch: K_0(X)\to \bigoplus_{i\in\mathbb{Z}}H_{2i}(X)$$ sending $[M,x,f]\mapsto f_*(ch(x)\cup Todd(TM)\cap [M])$. Rationally this should be an isomoprhism. My question is:

how is this map defined in the case of real $KO$-homology?

My guess is $[M,x,f]\mapsto f_*(ph(x)\cup L(TM)\cap [M])$, where $ph$ is the Pontryagin character, but I haven't found any reference for this. I would also like to know explicitly a triple which generates $KO_{4n}(\bullet)\otimes \mathbb{Q}=\pi_{4n}(KO)\otimes\mathbb{Q}=\mathbb{Q}$. I think that $[\mathbb{C}P^2,\gamma,\bullet]\otimes 1/3$ should correspond to $1\in\mathbb{Q}$, where $\gamma$ is the tautological bundle.

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    $\begingroup$ This article arxiv.org/abs/hep-th/0610177 should be the answer to the question. Section 4.2 gives an explicit map, using the $\hat{A}$-genus instead of the $L$-genus $\endgroup$ Nov 14, 2015 at 10:43

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