**
Localizing to get Relations in Cohomology
Localizing to get Relations in Cohomology**
Continuous maps of compact manifolds generate cohomology classes by pushing
forward and pulling back. For example, an embedding:

generates the class f_{*}1 (``the class of X'') in
the cohomology of Y, and more mysteriously, f^{*}f_{*}1 (``the Euler class
of X in Y'') in the cohomology of X. If the manifolds are also algebraic
varieties, and the maps are regular, one can also push forward coherent
sheaves and take Chern classes to generate even more (and more mysterious)
cohomology classes.

Moduli spaces in algebraic geometry tend to have lots of maps among them,
giving rise to lots of cohomology classes (the ``tautological classes'').
So here's a relevant question. When are two classes generated by maps
cohomologous? That is, what are the relations among tautological classes?

One standard method for finding relations involves hard calculations with
the Grothendieck Riemann-Roch theorem. Here I want to look at another
method, and convince you of its utility. The idea is to use the Atiyah-Bott
localization theorem. In its most crude form, this involves taking a
compact manifold Z with a group action and a ``forgetful'' map, that is
an equivariant map from Z to the manifold X (equipped with the trivial
group action):

Localization then gives relations among certain classes generated by the
maps F ® X from the fixed loci of the group action.
Surprisingly, this method (or rather a refinement of it) gives a wealth of
relations among tautological classes on moduli spaces of pointed
curves. I'll illustrate this with the case of the (compactified) moduli
space of elliptic curves, and the famous relation:

on that space.