University of Chicago
Department of Mathematics

The group theoretic structure of the "universal operations" on certain families of single loop spaces is analyzed, and identified as a group of natural transformations. With some additional topological features of double loop spaces, consequences are

1. bounds on the order of the torsion in the homotopy groups of certain spaces,
   as well as
2. constructions of specific torsion of "higher order" in the homotopy groups of certain finite complexes.

This talk will describe joint work with Mike Mandell in which we streamline the multiplicative features of the classical K-theory machinery that produces a spectrum output from a permutative category input. The structural framework we use in our results is the concept of multicategory, which is familiar to category theorists and computer scientists, but perhaps less so to topologists. We also end up using multicategories as objects of study in their own right, since every permutative category has an underlying multicategory, and the K-theory machinery turns out to depend only on this underlying multicategory. The joke is that while permutative, and more generally symmetric monoidal categories merely form the objects of a multicategory, multicategories form the objects of a symmetric monoidal category. We conjecture that this symmetric monoidal category, which is also closed and bicomplete, is a model for the connective stable homotopy category.

This is a report on joint in progress with Nitu Kitchloo and Miguel Abreu. I will describe the topology of the classifying space of the group of symplectomorphisms of a rational ruled surface in terms of the deformation theory of complex structures compatible with the symplectic form.

For every commutative ring $A$ we define the 2-typical de Rham-Witt complex of $A$. This construction generalizes the de Rham-Witt complex of Bloch-Deligne-Illusie, which is defined for $F_2$-algebras, and the de Rham-Witt complex of Hesselholt-Madsen for $Z_{(p)}$-algebras, with $p$ an odd prime. We indicate the new phenomena that arise when $p$ equals 2. We then describe the structure of the de Rham-Witt complex if $A$ is the ring of integers.

We prove that if X is a loop space and the integral homology of X is finitely generated as an abelian group, then X is homotopy equivalent to a closed, smooth, parallellisable manifold. This is joint work with Tilman Bauer, Nitu Kitchloo and Dietrich Notbohm.

We show how methods from algebraic topology for studying finite group actions on spaces, such as equivariant cohomology and Smith theory, can be adapted to apply to group actions on algebraic varieties, even in finite characteristic.

Hurwitz posed the problem of determining all possible "sums-of-squares" formulas over the complex numbers, and one can ask the same question over arbitrary fields. The problem is far from being solved, but in characteristic 0 certain necessary conditions have been provided over the years using topological methods. I will talk about how to generalize these conditions to characteristic p fields using motivic homotopy theory.

Consider the two self-maps of \Omega S²ⁿ⁺¹ given by
(1)\Psi the H-space squaring map, and
(2)\Omega[2], the loop of the degree two map.
A classical result of Adams implies these two maps are homotopic iff 2n+1= 1, 3, or 7. We have reason to conjecture that the two maps become homotopic after looping 2n times. A natural question to ask is, what is the smallest i such that \Omega^i-1\Psi \simeq \Omegaⁱ[2]? The purpose of this talk is to give lower bounds on i for these two maps to be homotopic. For example, we prove the following theorem.
Theorem [Cohen - J.] Let j-4 &ge k &ge 1. If the maps \Omega^i-1\Psi and \Omegaⁱ[2]: \OmegaⁱS^{2^j+2^k-1} \rightarrow \Omegaⁱ S^{2^j+2^k-1} are homotopic, then i>2^j+2^k-2^j-3
We consider other odd spheres S²ⁿ⁺¹ for $2n+1 \neq 2^j -1$.
Some of the tools we use are a secondary operation of Brown and Peterson along with factorizations of the Steenrod Squares. This gives information about factoring the Whitehead product which leads to information about the two maps in question.

Formal spaces are those whose rational homotopy type is completely determined by their cohomology; this has proved a very useful concept. I will discuss the various approaches to adapting this idea to the equivariant case, and compare several alternate definitions of equivariant formality.

FIRST QUARTERLY AllChicago TOPOLOGY SEMINAR (joint between UIC, NU, and UC)
Abstract: One may construct an S^1-equivariant cohomology theory EC associated to a complex curve C of genus g with additional structure, so that the value of the cohomology theory on spheres of representations is equal to the sheaf cohomology of suitable line bundles on the curve. Having got the theory, it is natural to attempt to use theta functions to give Thom isomorphisms for suitable bundles. In the fairly familiar case when C is an elliptic curve (ie g=1), an extension of Ando's work shows that this works precisely if the Borel Chern classes c_1=c_2=0, and hence one obtains a rational S^1-equivariant sigma orientation MString--->EC, giving the Ando-Hopkins-Strickland sigma orientation non-equivariantly.

Fix a smooth manifold M and consider the space of pairs (S,f) where S is a Riemann surface and f is a smooth map S -> M. This space modulo an equivalence relation is the moduli space of smooth surfaces in M, MTop(M). MTop(M) is closely related to the moduli space of holomorphic curves studied in algebraic geometry.
A rational combinatorial model for MTop(M) is obtained from Igusa's Fat category. This category whose objects are fat graphs is a (rational) model for the moduli space of Riemann surfaces. Using Anderson and Serre spectral sequence, this combinatorial model for MTop(M) gives a combinatorial chain complex computing the rational homology of MTop(M) which generalizes the Kontsevich graph complex.