Algebra/Topology Seminar

Speaker: Andrea Bianchi

Title: Hurwitz spaces and Moduli spaces of Riemann surfaces
Let S_d be a fixed symmetric group, with d at least 2. For k>=0, the
classical Hurwitz space hur(k,S_d) parametrises d-fold branched covers of
the complex plane C with precisely k branch points. We will consider a
construction that amalgamates all Hurwitz spaces all values of k into a
single space Hur(S_d).

The motivation for the construction is the following. For all g>=0 and
n>=1, let M_{g,n} denote the moduli space of Riemann surfaces of genus g
with n boundary components. If d is large enough (with respect to g and
n), there exists a connected component of Hur(S_d) which is homotopy
equivalent to M_{g,n}.

The space Hur(S_d) carries a natural structure of topological monoid
graded by natural numbers h>=0, and we will briefly sketch the computation
of the stable, rational cohomology of its subspaces Hur(h,S_d). The result
is very explicit in degrees up to roughly d/2. Letting d go to infinity,
one can in particular recover the Mumford conjecture on the stable,
rational cohomology of moduli spaces of Riemann surfaces, originally
proved by Madsen and Weiss.