6
$\begingroup$

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 $\Delta_N$?

$\endgroup$
1
  • $\begingroup$ It would be nice if a definition (or a construction) of this $\Delta_M$ could be included. $\endgroup$
    – user2015
    Oct 7 '15 at 14:20
6
$\begingroup$

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).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.