Torus and Z/p actions on manifolds

Let G be either a finite cyclic group of prime order or S^1. We find new relations between cohomology of a manifold (or a Poincare duality space) M with a G-action on it and cohomology of the fixed point set, M^G. Our main tool is the notion of Poincare duality on the Leray spectral sequence of the map M_G -> BG. We apply our results to study group actions on 3-manifolds.Comment: To appear in Topology, 25 page

Skein theory for SU(n)-quantum invariants

For any n>1 we define an isotopy invariant, _n, for a certain set of n-valent ribbon graphs Gamma in R^3, including all framed oriented links. We show that our bracket coincides with the Kauffman bracket for n=2 and with the Kuperberg's bracket for n=3. Furthermore, we prove that for any n, our bracket of a link L is equal, up to normalization, to the SU_n-quantum invariant of L. We show a number of properties of our bracket extending those of the Kauffman's and Kuperberg's brackets, and we relate it to the bracket of Murakami-Ohtsuki-Yamada. Finally, on the basis of the skein relations satisfied by _n, we define the SU_n-skein module of any 3-manifold M and we prove that it determines the SL_n-character variety of pi_1(M).Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-36.ab