The Mumford conjecture states that for each integer $n$, we have: the map $\mathbb{Q}[x\_1,x\_2,\dots] \to H^\ast(M\_g ; \mathbb{Q})$ sending $x\_i$ to the kappa class $\kappa\_i$, is an isomorphism in degrees less than $n$, for sufficiently large $g$. Here $M\_g$ denotes the moduli of genus $g$ curves, and the degree of $x\_i$ is the degree of the kappa class $\kappa\_i$. This conjecture was proved by Madsen-Weiss a few years ago.


1. What are the heuristic or moral reasons for the conjecture? (EDIT: I am particularly interested in algebraic geometric reasons, if there are any. Though algebraic topologial reasons are very welcome too.) What lead Mumford to formulating the conjecture in the first place?
2. I know very little about the Madsen-Weiss proof, but I know that it mainly uses algebraic topology methods. Are there any approaches to the conjecture which are more algebraic-geometric?
3. Is there any analogous theorem or conjecture regarding the (topological) $K$-theory of $M\_g$? Or the Chow ring of $M\_g$? etc.



Mumford conjecture: Heuristic reasons? Generalizations? ... Algebraic geometry approaches?
All current proofs of Mumford's conjecture in fact prove a far stronger result, the "Strong Mumford conjecture", first formulated by Ib Madsen. This says the following (where by "moduli space" in the following we mean a homotopy type classifying concordance classes of surface bundles with perhaps some extra structure): there is a stable moduli space 
$$\mathcal{M}\_\infty := \mathrm{colim} \,\, \mathcal{M}\_{g, 1}$$
where $\mathcal{M}\_{g, 1}$ denotes the moduli space of genus $g$ surfaces with a single boundary component, and the colimit is formed by gluing on a torus with a single boundary component using the "pair of pants" product.


There is also a space, usually called $\Omega^\infty MTSO(2)$, which classifies cobordism classes of "formal surface bundles": that is, codimension -2 submersions with an orientation of the (stable) vertical tangent bundle. As a surface bundle is a formal surface bundle, there is a map
$$\alpha: \mathcal{M}\_\infty \to \Omega^\infty MTSO(2).$$
The strong Mumford conjecture says that this is an integral homology equivalence. For the record, there are currently four distinct known proofs, due to:


1. [The stable moduli space of Riemann surfaces: Mumford's conjecture](http://arxiv.org/abs/math/0212321), Madsen and Weiss,
- [The homotopy type of the cobordism category](http://arxiv.org/abs/math/0605249) , Galatius, Madsen, Tillmann and Weiss,
- [Monoids of moduli spaces of manifolds](http://arxiv.org/abs/0905.2855), Galatius and myself,
- [Madsen-Weiss for geometrically minded topologists](http://arxiv.org/abs/0907.4226), Eliashberg, Galatius and Mishachev.


For part 1) of your question, from this point of view (I do not know what Mumford had in mind): the map $\alpha$ can be thought of as comparable to the map which compares holonomic sections to formal sections in the statement of Gromov's $h$-principle for a sheaf. The idea is then that given a "formal surface bundle" one may begin improving it to be more and more like a bundle, but in this process one cannot really control the genus of the fibres one ends up with. This is why it gives the infinite genus moduli space. This is more or less the approach to proving the conjecture that Eliashberg, Galatius and Mishachev take. The other approaches are less direct and use more algebraic topological machinery.


So, for part 3) of your question: the map $\alpha$ is also an equivalence in any other (co)homology theory (by the Atiyah-Hirzebruch spectral sequence, for example). Thus the topological K-theory of $\mathcal{M}\_\infty$ is "known" in the sense that it is the K-theory of the infinite loop space of a well-understood spectrum, which fits into various simple cofibration sequences and so on. On the other hand, it is "not known" in the sense that I don't think anyone knows what $K^0(\mathcal{M}\_\infty)$ is as a group, though I once tried to compute it without success. On the other hand, even knowing this group, it does not necessarily tell you anything about what you are really interested in, $K^0(\mathcal{M}\_g)$, because stability for ordinary homology does not imply stability in non-connective (co)homology theories.


