A rather simple natural outer derivation of the graded Lie algebra of all vector valued differential forms with the Frölicher-Nijenhuis bracket turns out to be a differential and gives rise to a cohomology of the manifold, which is functorial under local diffeomorphisms. This cohomology is determined as the direct product of the de Rham cohomology space and the graded Lie algebra of ”traceless”...

Let A be a differential graded algebra with cohomology ring HA. A graded module over HA is called realisable if it is (up to direct summands) of the form HM for some differential graded A-module M . Benson, Krause and Schwede have stated a local and a global obstruction for realisability. The global obstruction is given by the Hochschild class determined by the secondary multiplication of the A...

String theorists (notably Witten [W]) recently introduced the notion of a “quantum” deformation of the cohomology ring of a smooth projective variety X. This quantum deformation, or quantum cohomology ring, as it is often called, is an algebra over a formal-power-series ring which specializes to the ordinary cohomology ring, and which is defined in terms of intersection data (the Gromov-Witten ...

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 generalizati...

Introduction The theory of motives has always had two faces. One is the geometric face where a universal cohomology theory for varieties is cooked up from geometric objects like cycles. The other one is the linear algebra face where restricting conditions are put on objects of linear algebra like vector spaces with an operation of the Galois group. The ideal theorem would be an equivalence of t...

We explore the relationship between de Rham and Lie algebra cohomologies in the finite dimensional and affine settings. In particular, given a ĝκmodule that arises as the global sections of a twisted D-module on the affine flag manifold, we show how to compute its BRST reduction modulo n(K) using the de Rham cohomology of the restrictions to N(K) orbits. A similar relationship holds between the...

We introduce the concept of extended Hopf algebras and define their cyclic cohomology in the spirit of Connes-Moscovici cyclic cohomology for Hopf algebras. Extended Hopf algebras are closely related to, but different from, Hopf algebroids. Their definition is motivated by attempting to define a cyclic cohomology theory for Hopf algebroids in general. We show that many of Hopf algebraic structu...