Let G be a finitely generated group and let Z2G denote the group ring of G with coefficients in the field Z2. The first cohomology group of G with coefficients in Z2G is essentially the reduced cohomology of the space of ends of G; by a theorem of Hopf and Freudenthal from the 40's, it is a Z2 vector space of dimension 0, 1, or infinity. The meaning/utility of the second cohomology with Z2G coefficients is not as transparent. In the lecture I will explain how two geometric lemmas about H2(G;Z2G) can be used to:

• Strengthen a 1974 theorem of Farrell which gave a partial analog of Hopf-Freudenthal for the second cohomology.
• Give a new proof of the Seifert fibered space conjecture for 3-manifolds.
• Give a new proof of the recent theorem of Papasoglu on quasi-isometry invariance of splittings over 2-ended subgroups.
• Prove a conjecture of Dicks-Dunwoody characterizing 2-dimensional Poincare duality groups over a commutative ring R.