Fundamental class in $KO[1/2]$

Question

Let $M^m$ be an oriented Riemannian manifold. The signature operator associates to $M$ a class $\Delta_M\in KO_m(M)[1/2]$. I have two questions about this class $\Delta_M$:

1. Rationally, $\Delta_M$ is computed by the Chern character, i.e. $$\text{ch}(\Delta_M)=L(M)\cap [M].$$ Where could I find details of the proof of this formula?

   Here $\text{ch}$ denotes the Chern character, $L$ denotes the $L$-class and $[M]$ the fundamental class in the ordinary homology theory.

2. Is there a formula for $\Delta_{M\times N}$ in terms of $\Delta_M$ and $\Delta_N$?

Answer 1

Concerning your first question, I think a good place to learn about it is Markus Banagl's book "Topological invariants of stratified spaces" (chapter 6) or have a look at Paul Siegel's paper "$KO$-homology at odd primes" Amer. J. Math. 105 (1983), 1067-1105 (in particular the appendix where Sullivan's approach is explained).

In Siegel's paper you have a bordism theory $\Omega^{Witt}_*(-)$ build from cobordisms of singular spaces called Witt spaces. A Witt space $X$ carries rational homology L-classes $l_i\in H_{dim(X)-4i}(X,\mathbb{Q})$. This homology $L$-classes were defined by Goresky and McPherson in this singular context extending the homology $L$-classes of a manifold $M$ which are poincaré dual to Hirzebruch $L$-classes $L_i\in H^{4i}(M,\mathbb{Q})$: $$l_i(M)=L_i(M)\cap [M]$$

• In Witt bordism any Witt space has a fundamental class $[X]\in \Omega^{Witt}_{dim(X)}(X)$ represented by $id:X\rightarrow X$.

• Moreover we have a natural transformation $$\Phi:\Omega^{Witt}_k(X)\rightarrow \oplus_{i} H_{k-4i}(X,\mathbb{Q})$$such that for a Witt space we have $\Phi([X])=l_0+l_1+\ldots$, for a manifold we get that $\Phi([M])=L(M)\cap [M]$.

• And this bordism theory when tensored by $\mathbb{Z}[1/2]$ is naturally isomorphic to $ko_*(-)\otimes \mathbb{Z}[1/2]$. Siegel used Sullivan's construction of $ko_*(-)\otimes \mathbb{Z}[1/2]$ to get a natural transformation $$\mu: \Omega^{Witt}_*(-)\rightarrow ko_*(-)\otimes \mathbb{Z}[1/2].$$ such that $\mu([M])=\Delta_M.$

Concerning your second question you can have a look at theorem 11.1 "Multiplicativity of the L-theory fundamental class" of this paper: "The L-homology fundamental class for IP-spaces and the stratified Novikov conjecture" by Markus Banagl, Gerd Laures and Jim McClure (available on Banagl's homepage).