On the Hopf Conjectures with Symmetry

ON THE HOPF CONJECTURES WITH SYMMETRY Lee Kennard Wolfgang Ziller, Advisor We prove results related to the classical Hopf conjectures about positively curved manifolds under the assumption of a logarithmic symmetry rank bound. The main new tool is the action of the Steenrod algebra on cohomology. Along the way, we formulate a certain notion of periodicity in cohomology and obtain a generalization in this context of Adem’s theorem on singly generated cohomology rings.