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27
votes
3answers
2k views

Mumford conjecture: Heuristic reasons? Generalizations? … Algebraic geometry approaches?

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 ...
13
votes
9answers
5k views

Teichmuller Theory introduction

What is a good introduction to Teichmuller theory, mapping class groups etc., and relation to moduli space of curves or Riemann surfaces?
21
votes
2answers
4k views

Explanation for the Thom-Pontryagin construction (and its generalisations)

In 1950, Pontryagin showed that the n-th framed cobordism group of smooth manifolds was equal to n-th stable homotopy group of spheres: $$ \lim_{k \to \infty} \pi_{n+k}(S^k) \cong ...