Notre Dame Topology Seminar

Fall 2009 Schedule

September 22 at 10AM in HH125: Vigleik Angeltveit, University of Chicago; Uniqueness of Morava K-theory
October 6 at 10AM in HH125: Qayum Khan, University of Notre Dame; The Nil-Nil theorem in algebraic K-theory
October 13 at 10AM in HH125: Daniel Cibotaru, University of Notre Dame; The odd Chern character and index localization formulae
October 27 at 11:30AM in H258: Megan Guichard Shulman, University of Chicago; RO(Z/p)-graded cohomology of some classifying spaces
November 3 at 2PM in DBRT 206: Zhixu Su, Rose-Hulman Institute of Technology; Rational analog of projective planes
November 10 at 2PM in DBRT 206: Kyle Ormsby, University of Michigan; Computations in stable motivic homotopy theory
November 24 at 10:30AM in H 258: Sunil Chebolu, Illinois State University; Progress report on the Freyd's generating hypothesis
December 8 at 2PM in DBRT 206: Bert Guillou, University of Illinois, Urbana-Champaign; Models for equivariant spectra

2008-2009 Topology Seminar
Graduate Topology/Geometry Seminar
Notre Dame Department of Mathematics

Questions? Contact Kate Ponto

Fall 2009 Abstracts

September 22, 2009: Vigleik Angeltveit

Uniqueness of Morava K-theory.

Any cohomology theory is represented by what is called a spectrum. For each prime p and positive integer n there is a cohomology theory, or spectrum, called the n'th Morava K-theory K(n). When n=1 this is essentially mod p complex K-theory. I will discuss multiplicative structure on K(n). In particular I will discuss how many associative multiplications K(n) has. The main result is that there is essentially only one. To analyze this question I will discuss the notion of an A-infinity structure, a multiplication which is associative up to homotopy and higher homotopies in a precise sense.

Slides from the talk.

October 6, 2009: Qayum Khan

The Nil-Nil theorem in algebraic K-theory

Bass defined an exotic Nil-summand of the algebraic K-theory of a polynomial extension. Later, Waldhausen extended the definition to tensor algebras and defined an exotic Nil-summand of the algebraic K-theory of an injective amalgam of groups. The Nil-Nil theorem states, under a certain finiteness condition, that there is a natural isomorphism from the amalgam Nil to a tensor Nil. An important application is that the Farrell-Jones conjecture in algebraic K-theory can be sharpened from the family of virtually cyclic subgroups to the family of finite-by-cyclic subgroups of a discrete group G.

This is joint work with J.F. Davis and A.A. Ranicki.

October 13, 2009: Daniel Cibotaru

The odd Chern character and index localization formulae

A family of self-adjoint, Fredholm operators, parametrized by a manifold M determines a sequence of odd-cohomology classes in M . The first such class, when seen as a representation of the fundamental group of M, associates to every path of operators its spectral flow, which is, roughly, a count with sign of the 0-eigenvalues. I use symplectic topology techniques to describe geometric representatives of the Poincare duals of these classes as certain degeneracy loci. This allows the computation of these classes in terms of the spectral data, in the spirit of the spectral flow.

October 27, 2009: Megan Guichard Shulman

RO(Z/p)-graded cohomology of some classifying spaces

When dealing with G-spaces for a finite group G, there are many reasons to think that RO(G)-graded Bredon cohomology is the ``correct'' equivariant cohomology theory to consider. Unfortunately, it is also very difficult to compute with. Gaunce Lewis calculated the RO(Z/p)-graded cohomology of complex projective spaces in the 1980s, and William Kronholm calculated the RO(Z/2)-graded cohomology of some real projective spaces in his 2008 thesis, but to date no other calculations have been done. In this talk, I will describe an equivariant spectral sequence which can be used in conjunction with the equivariant Serre spectral sequence and the equivariant cohomology of complex projective spaces to identify the RO(Z/p)-graded cohomology of the equivariant classifying space B_{Z/p} O(2).

November 3, 2009: Zhixu Su

Rational analog of projective planes

In terms of the dimension, there are only 4 kinds of projective planes: real, complex, quaternionic, and octonionic. There does not exist any closed manifold as a higher dimensional example due to the non-existence of a Hopf invariant 1 map. However, one can ask the existence of a rational analog in higher dimensions: a smooth closed 4k dimensional manifold whose rational cohomology is rank one in dimension 0, 2k and 4k and is zero otherwise. Applying rational surgery theory, the problem is reduced to finding possible Pontryagin classes satisfying the Hirzebruch signature formula and a number of congruence relations determined by the Riemann-Roch integrality condi- tions. And this is eventually equivalent to finding possible solutions to a system of Diophantine equations.

November 10, 2009: Kyle Ormsby

Computations in stable motivic homotopy theory

The Morel-Voevodsky motivic homotopy category provides fertile ground in which the tools of computational algebraic topology can reap algebro-geometric results. I will discuss computations of motivic stable homotopy groups of spheres over p-adic fields. Along the way, I'll show how the motivic Adams spectral sequence produces simple computations of algebraic K-theory groups and novel results about algebraic cobordism.

November 24, 2009: Sunil Chebolu

Progress report on the Freyd's generating hypothesis

Freyd's generating hypothesis is a very fundamental and deep statement about the category of finite spectra. It is a conjecture due to Peter Freyd (1965) which states that the stable homotopy functor on the category of finite spectra is faithful. An unbelievable consequence of this conjecture is that it reduces the study of finite CW spectra to that of graded modules over the homotopy ring of the sphere spectrum. Therefore this conjecture stands as a central problem in the field which is still open. To the best of my knowledge there hasn't been any progress on this conjecture in the recent years. However, there has been lots of developments in analogues and variations of this conjecture on other axiomatic stable homotopy categories including equivariant stable homotopy categories, derived categories, and the stable module categories of finite groups. This talk will be a survey of these results with particular emphasis on the stable module categories which is joint work with Carlson and Minac.

December 8, 2009: Bert Guillou

Models for equivariant spectra

For a compact Lie group G, I will describe G-spectra as enriched diagrams on a small spectral category. When the group is moreover finite, I will discuss some work in progress that improves on this model.