Introduction to smooth finite transformation groups I

I will give an introduction to the theory of smooth finite transformation groups. My choice of material will be guided by the problem of determining which finite groups admit smooth effective actions on a given manifold. A complete answer to this problem is at present out of reach, but many partial results are available. Among these, I will explain: (i) the basics of Smith theory, (ii) a classical result of Mann and Su on actions of finite abelian groups, and (iii) some of the recent results on the Jordan property for diffeomorphism groups. With this motivation in mind I will explain basic facts on differential topology, group cohomology and equivariant cohomology. I will only assume that the audience is familiar with standard notions in algebraic topology such as singular (co)homology and spectral sequences, basic differential and Riemannian geometry, and very basic notions on finite groups.