Manchester Geometry Seminar 2002/2003

We study finite group actions on manifolds and global invariants of manifolds with such actions from the point of view of cobordism theory. The case of our particular concern is Z/p-actions (with odd prime p), however some results may be generalised to other finite groups. There are two main approaches to calculating cobordism invariants (e.g. characteristic numbers, Todd genus, signature, elliptic genus, and other Hirzebruch genera) of Z/p-manifolds in terms of local invariants of the action: the equivariant cobordism theory (ascending to the works of Conner--Floyd and Buchstaber--Novikov), and the index theory (after Atiyah--Bott and Atiyah--Singer). The two methods are of different power, and the resulting formulae usually are not similar at the first glance. They are related by the so-called Conner--Floyd equations, which in some circumstances gives rise to nice polynomial identities (e.g. in the case of elliptic genus the corresponding Conner--Floyd equations produce identities for Legendre polynomials). On the other hand the existence of a non-trivial Z/p action puts restrictions on the cobordism class of manifold. In some particular cases (such as the actions with only isolated fixed points or with fixed submanifolds having trivial normal bundle) it is possible to give an explicit description of cobordism classes of manifolds admitting such an action, say in terms of characteristic numbers. These results extrapolate some classical facts from the cobordism theory, such as the Hattori--Stong theorem.