University of Chicago Algebraic Topology Seminar

4:30 pm   Tuesday, January 13, 2009   in Eckhart 203

Algebraic Topology: Unstable Periodic Patterns
by Brayton Gray (University of Illinois at Chicago)


The talk will describe how stable periodic families in homotopy behave unstably, as well as unstable periodic families. In order to get a reasonable description, we will discuss appropriate destabilizations of spectra, by which I mean a choice of a suitable sequence of spaces which reflect the unstable issues related to the spectrum. The Moore space spectrum will be examined in detail. The appropriate spaces have some surprising properties.

Pretalk at 3:00 : I will give an historical introduction to periodicity, and recall some of the well know properties of the first two families: the alpha and beta families. Everything in this talk will be at least 20 years old.


4:30 pm   Tuesday, January 20, 2009   in Eckhart 203

Algebraic Topology: Stabilization of Hurwitz spaces
by Craig Westerland (University of Melbourne)


We will describe the Hurwitz space of branched covers of the disc, and study its group completion under "pants multiplication." We will show that this stabilization is a double loop space, and give some evidence that its rational stable homology is very small. If time permits, we will relate this to some number-theoretic conjectures regarding the asymptotic growth of the number of branched covers of the line in finite characteristic.


4:30 pm   Tuesday, January 27, 2009   in Eckhart 203

Algebraic Topology: Rational Cohomology Theories on Free G-Spaces
by John Greenlees (University of Sheffield)


[The talk will describe joint work with Brooke Shipley, and also describe some work of of David Barnes.]

The main result is that if the group G is connected, there is a simple algebraic model for these cohomology theories. Firstly, these cohomology theories are represented by free rational G-spectra. Given this, the more precise statement is that there is a Quillen equivalence

Free-rational-G-spectra = torsion-H^*(BG)-modules

The ingredients are a Morita equivalence, a Koszul duality, Shipley's equivalence between spectral and algebraic algebras and a rigidity theorem. If G is not connected, the representation theory of the component group must be taken into account.


4:30 pm   Tuesday, February 3, 2009   in Eckhart 203

Algebraic Topology: Equivariant Fixed Point Theory
by Kate Ponto (Notre Dame University)


The Lefschetz Fixed Point Theorem associates an integer, the Lefschetz number, to each endomorphism of a compact smooth manifold. The Lefschetz number is zero when the map has no fixed points. For a finite group G, several generalizations of the Lefschetz number and related invariants have been defined for equivariant endomorphisms of compact smooth G-manifolds. I will explain how these invariants are examples of duality and trace in bicategories and how this observation gives simple ways to compare diferent invariants.


4:30 pm   Tuesday, February 10, 2009   in Eckhart 203

Algebraic Topology: A_\infty Structures on Thom Spectra
by Samik Basu (Harvard University)


Let R be an E_\infty ring spectrum. Given a map f:X -> BGL_1(R), we can construct a Thom spectrum X^f. If f is a loop map, then there is an A_\infty R module structure on the Thom spectrum. I will consider various examples of these Thom spectra and construct A_\infty structures on them. I will then use this identification to calculate Topological Hochschild Homology.


4:30 pm   Tuesday, February 17, 2009   in Eckhart 203

Algebraic Topology: No Meeting




4:30 pm   Tuesday, February 24, 2009   in Eckhart 203

Algebraic Topology: The Higher Morava K-theory of Some Infinite Loop Spaces
by Justin Noel (University of Chicago)


There are a number of spectra related to the chromatic filtration on the stable homotopy groups of spheres. These include S^0, sl_1 S^0, HZ, j, coker j and their Bousfield localizations. The j spectrum is designed to capture exactly the 1st stratum in the chromatic filtration. More specifically, there is a surjection (for p odd) pi_* S --> pi_* J which is an isomorphism when restricted to the "height one" elements in pi_* S^0.

With a small, but non-zero, amount of work we can see that K(n)_*(j) is non-trivial if and only if n=1. However, the Morava K-theory of the 0th space of j is never trivial. I will look at the interaction between the K(n)-localization, taking connective covers and taking 0th spaces. Along the way we determine K(2) of the infinite loop space Coker J, the "Rogue Object."

This is joint work with Nick Kuhn.


4:30 pm   Tuesday, March 3, 2009   in Eckhart 203

Algebraic Topology: Finite Topological Spaces and Poincare Duality
by Tathagata Basak (University of Chicago)


We shall define a class of finite topological spaces that model the cell structure of a polyhedral cell complex. We shall develop enough algebraic topology in this setting to prove a Poincare duality theorem for an approprite class of well behaved "orientable" finite spaces. All the arguments will be completely elementary.


4:30 pm   Tuesday, March 10, 2009   in Eckhart 203

Algebraic Topology: Vanishing of Negative K-groups
by Lars Hesselholt (Nagoya University)


A conjecture of Weibel predicts that the Bass negative K-groups of a noetherian scheme X of dimension d vanish in degrees i < - d. A few years ago, the conjecture was verified for schemes separated and essentially of finite type over a field of characteristic 0 by Cortinas-Haesemeyer-Schlichting-Weibel. I will discuss recent work with Thomas Geisser which establishes the conjecture for schemes separated and essentially of finite type over a perfect field of characteristic p > 0 under the assumption of resolution of singularities. The proof in characteristic 0 uses a comparison of K-theory to Connes' cyclic homology. Similarly, the proof in characteristic p > 0 uses a comparison of K-theory to the topological cyclic homology of Bokstedt-Hsiang-Madsen. In both cases, the starting point of the proof is a theorem of Haesemeyer which allows one to show that the difference between K-theory and the respective cyclic theories satisfies descent with respect to Voevodsky's cdh-topology. The proofs are completed by careful analysis of the cyclic theories. A central technical component in the proof is the recent extension from rings to exact categories of the definition of the respective cyclic theories by Keller and Blumberg-Mandell. 


4:30 pm   Tuesday, March 31, 2009   in Eckhart 203

Algebraic Topology: Not meeting this week, we have the Namboodiri Lecture Series