On formal group laws over the quotients of Lazard’s ring

We present a formal power series {\sum A_{ij}x^{i}y^{j}} over the Lazard ring Λ and the formal group laws {F_{n}} , {n\geq 2} , over the quotient rings of Λ. For each {F_{n}} , we construct a complex cobordism theory with singularities with the coefficient ring {\mathbb{Q}[p_{1},\dots,p_{2n}]} , with parameters {p_{i}} , {|p_{i}|=2i} .

HILBERT STRATIFOLDS AND A QUILLEN TYPE GEOMETRIC DESCRIPTION OF COHOMOLOGY FOR HILBERT MANIFOLDS

In this paper we use tools from differential topology to give a geometric description of cohomology for Hilbert manifolds. Our model is Quillen’s geometric description of cobordism groups for finite-dimensional smooth manifolds [Quillen, ‘Elementary proofs of some results of cobordism theory using steenrod operations’, Adv. Math., 7 (1971)]. Quillen stresses the fact that this construction allows the definition of Gysin maps for ‘oriented’ proper maps. For finite-dimensional manifolds one has a Gysin map in singular cohomology which is based on Poincaré duality, hence it is not clear how to extend it to infinite-dimensional manifolds. But perhaps one can overcome this difficulty by giving a Quillen type description of singular cohomology for Hilbert manifolds. This is what we do in this paper. Besides constructing a general Gysin map, one of our motivations was a geometric construction of equivariant cohomology, which even for a point is the cohomology of the infinite-dimensional space $BG$, which has a Hilbert manifold model. Besides that, we demonstrate the use of such a geometric description of cohomology by several other applications. We give a quick description of characteristic classes of a finite-dimensional vector bundle and apply it to a generalized Steenrod representation problem for Hilbert manifolds and define a notion of a degree of proper oriented Fredholm maps of index $0$.

Quillen's work on formal group laws and complex cobordism theory

In 1969 Quillen discovered a deep connection between complex cobordism and formal group laws which he announced in [Qui69]. Algebraic topology has never been the same since. We will describe the content of [Qui69] and then discuss its impact on the field. This paper is a writeup of a talk on the same topic given at the Quillen Conference at MIT in October 2012. Slides for that talk are available on the author's home page.

Algebraic cobordism theory attached to algebraic equivalence

Based on the algebraic cobordism theory of Levine and Morel, we develop a theory of algebraic cobordism modulo algebraic equivalence.We prove that this theory can reproduce Chow groups modulo algebraic equivalence and the semi-topological K0-groups. We also show that with finite coefficients, this theory agrees with the algebraic cobordism theory.We compute our cobordism theory for some low dimensional varieties. The results on infinite generation of some Griffiths groups by Clemens and on smash-nilpotence by Voevodsky and Voisin are also lifted and reinterpreted in terms of this cobordism theory.

Variations on a theme of rationality of cycles

We prove certain weak versions of some celebrated results due to Alexander Vishik comparing rationality of algebraic cycles over the function field of a quadric and over the base field. The original proofs use Vishik’s symmetric operations in the algebraic cobordism theory and work only in characteristic 0. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2. Our weak versions are still sufficient for existing applications. In particular, Vishik’s construction of fields of u-invariant 2r + 1, for r ≥ 3, is extended to arbitrary characteristic ≠ 2.

Around rationality of cycles

We prove certain results comparing rationality of algebraic cycles over the function field of a quadric and over the base field. These results have already been obtained by Alexander Vishik in the case of characteristic 0, which allowed him to work with algebraic cobordism theory. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2.