The Stolz--Teichner program proposes a deep connection between geometric field theories and certain cohomology theories. In this paper, we extend by developing theory of power operations for restricted to closed bordisms. These satisfy relations analogous the ones exhibited their homotopical counterparts. We also provide computational tools identify geometrically defined with usual on complexif...

We study cohomology theories of strongly homotopy algebras, namely A∞, C∞ and L∞-algebras and establish the Hodge decomposition of Hochschild and cyclic cohomology of C∞-algebras thus generalising previous work by Loday and Gerstenhaber-Schack. These results are then used to show that a C∞-algebra with an invariant inner product on its cohomology can be uniquely extended to a symplectic C∞-alge...

1. Grothendieck Topologies 1 2. The Category of Sheaves on a Site 3 3. Operations on presheaves and sheaves 6 4. Stalks of étale sheaves; the mapping cylinder 7 4.1. Stalks 7 4.2. The mapping cylinder 10 5. Cohomology 12 6. Flabby sheaves and Čech cohomology 13 7. Excision and cohomology with supports 17 8. Comparison theorems 19 8.1. Comparison of étale and Zariski cohomologies 20 8.2. Compari...

This is the first of two interconnected parts: Part I contains the geometric theory of generalized modular forms and their connections with the cooperation algebra for elliptic cohomology, E``∗E``, while Part II is devoted to the more algebraic theory associated with Hecke algebras and stable operations in elliptic cohomology. We investigate the structure of the stable operation algebra E``∗E``...

Let G be an exceptional Lie group with a maximal torus T ⊂ G. We present the integral cohomology ring H∗(G/T ) by a minimal set of Schubert classes on G/T . This completes the program of determining the integral cohomology of all complete flag manifolds G/T in the context of Schubert calculus. The results have been applied in [DZ3] to construct the integral cohomology of a simple Lie group G in...

In these lecture notes we discuss a body of work in which Morse theory is used to construct various homology and cohomology operations. In the classical setting of algebraic topology this is done by constructing a moduli space of graph flows, using homotopy theoretic methods to construct a virtual fundamental class, and evaluating cohomology classes on this fundamental class. By using similar c...

The study of the action of the Steenrod algebra on the mod p cohomology of spaces has many applications to the topological structure of those spaces. In this paper we present combinatorial formulas for the action of Steenrod operations on the cohomology of Grassmannians, both in the Borel and the Schubert picture. We consider integral lifts of Steenrod operations, which lie in a certain Hopf al...

Suppose E is a commutative S–algebra, in the sense of Elmendorf, Kriz, Mandell and May [6], and A is a commutative E–algebra. We want to capture the properties and underlying structure of the homotopy groups π∗A = A∗ of A, by studying operations associated to the cohomology theory that E represents. An important family of cohomology operations, called power operations, is constructed via the ex...

We explore differential and algebraic operations on the exterior product of spinor representations and their twists that give rise to cohomology, the spin cohomology. A linear differential operator d is introduced which is associated to a connection ∇ and a parallel spinor ζ, ∇ζ = 0, and the algebraic operators D(p) are constructed from skew-products of p gamma matrices. We exhibit a large numb...

Abstract We determine the mod $p$ cohomological invariants for several affine group schemes $G$ in characteristic $p$. These are of $G$-torsors with values étale motivic cohomology, or equivalently Kato’s version Galois cohomology based on differential forms. In particular, we find 2 symmetric groups and orthogonal 2, which Serre computed not 2. also all operations fields, extending Vial’s comp...