Research in Algebraic Topology

[picture of Niles]
\[ E_n \stackrel{\simeq}{\longrightarrow} \Omega E_{n+1} \] \[ \Sigma E_n \to E_{n+1} \]

Structure maps on a representing spectrum.

Ordinary cohomology with coefficients in an abelian group is represented by homotopy classes of maps into the sequence of Eilenberg-Mac Lane spaces for that group. This sequence of spaces is called a (pre-)spectrum, and in fact every generalized cohomology theory can be represented by a (pre)-spectrum. Studying the representing sequence itself, one can reformulate the defining properties of a generalized cohomology theory in terms of structure maps on its corresponding (pre-)spectrum. There are various equivalent choices for what additional structure to assume, and the term spectrum has been used at different times to mean different things. Whatever is meant always entails some kind of object representing a cohomology theory.

This lead topologists to study the algebraic structure of cohomology theories (cup products, Steenrod operations, etc.) in terms of the representing spectra. The early stages of this theory worked "up to homotopy", but the discovery of highly-structured spectra enables topologists to study more rigid "on the nose" algebraic structures which model the underlying homotopy theory. This leads to the so-called brave new algebra of highly-structured spectra.

Complex Oriented Cohomology Theories

Introductory slides

One particular algebraic structure of interest is a complex orientation for a cohomology theory. These slides introduce spectra and complex orientations, then go on to describe some of my joint work with Justin Noel studying power operations for universal complex-oriented spectra, MU and the Brown-Peterson spectrum BP.

Power Operations and p-typicality

This project is joint with Justin Noel.
Topology and its Applications, 2010, 157, 2271-2288. 10.1016/j.topol.2010.06.007.
Slides (2010)
arXiv 0910.3187 (2010): [abs | pdf]

Abstract:

We show, for primes p less than or equal to 13, that a number of well-known MU_(p)-rings do not admit the structure of commutative MU_(p)-algebras. These spectra have complex orientations that factor through the Brown-Peterson spectrum and correspond to p-typical formal group laws. We provide computations showing that such a factorization is incompatible with the power operations on complex cobordism. This implies, for example, that if E is a Landweber exact MU_(p)-ring whose associated formal group law is p-typical of positive height, then the canonical map MU_(p) —> E is not a map of H_infty ring spectra. It immediately follows that the standard p-typical orientations on BP, E(n), and E_n do not rigidify to maps of E_infty ring spectra. We conjecture that similar results hold for all primes.

Norm Operations in Algebraic Topology

This is work in progress.

This project developed as a connection between work on the power operations of complex-oriented cohomology theories and Galois theory for commutative S-algebras. The topological norm is an operation arising from subgroups H ≤ Gal(B/A) for a Galois extension of commutative S-algebras, defined similarly to the classical norm for Galois extensions of fields or commutative rings. The basic definitions were developed by Greenlees-May, and a further application appears in the Kervaire invariant work of Hopkins-Hill-Ravenel.

In the case of {e} ≤ ZZ/p, the norm gives an operation on the E-cohomology of an extended power construction D_{ZZ / p} X for a (nonequivariant) ring spectrum E and space (or spectrum) X. And this norm operation is none other than the cyclic power operation on E. A goal of this project is to extend calculational tools for the cyclic power operation to other Galois extensions, allowing calculations of norm operations themselves and, via a Brauer-Galois exact sequence, Brauer group calculations.

Lifting homotopy T-algebra maps to strict maps

This project is joint with Justin Noel.
arXiv 1301.1511 (2013): [abs | pdf]

Abstract:

The settings for homotopical algebra---categories such as simplicial groups, simplicial rings, A_∞ spaces, E_∞ ring spectra, etc.---are oftentimes equivalent to categories of algebras over some monad or triple T. In such cases, T is acting on a nice simplicial model category in such a way that the T descends to a monad on the homotopy category and defines a category of homotopy T-algebras. In this setting there is a forgetful functor from the homotopy category of T-algebras to the category of homotopy T-algebras.

Under suitable hypotheses we provide an obstruction theory, in the form of a Bousfield-Kan spectral sequence, for lifting a homotopy T-algebra map to a strict map of T-algebras. Once we have a map of T-algebras to serve as a basepoint, the spectral sequence computes the homotopy groups of the space of T-algebra maps and the edge homomorphism on π_0 is the aforementioned forgetful functor. We discuss a variety of settings in which the required hypotheses are satisfied, including monads arising from algebraic theories and from operads.

We provide examples in G-spaces, G-spectra, rational E_∞-algebras, and A_∞-algebras under an Eilenberg-MacLane commutative ring spectrum. We give explicit calculations showing that the forgetful functor from the homotopy category of E_∞ ring spectra to the category of H_∞ ring spectra is generally neither full nor faithful. We also apply a result of the second named author and Nick Kuhn to compute the homotopy type of the space \(E_\infty(\Sigma^\infty_+ \text{Coker}\, J, L_{K(2)} R)\).

Homotopic Descent Reading Project

Homotopic Descent notes (UGA, 2011)

Abstract:

This is a collection of notes from a reading project on homotopic descent, following the preprint of Kathryn Hess. In these notes we deliberately suppress nearly all of the important technical details and follow an exposition motivated by intuition and key examples. It is very much a rough draft. You have been warned!

Azumaya Objects in Bicategories

arXiv 1005.4878 (2011): [abs | pdf]

Abstract:

We introduce the notion of Azumaya object in general homotopy-theoretic settings. We give a self-contained account of Azumaya objects and Brauer groups in bicategorical contexts, generalizing the Brauer group of a commutative ring. We go on to describe triangulated bicategories and prove a characterization theorem for Azumaya objects therein. This theory applies to give a homotopical Brauer group for derived categories of rings and ring spectra. We show that the homotopical Brauer group of an Eilenberg-Mac Lane spectrum is isomorphic to the homotopical Brauer group of its underlying commutative ring. We also discuss tilting theory as an application of invertibility in triangulated bicategories.

Galois Theory for Rings and Ring Spectra

This is work in progress.

Chase-Harrison-Rosenberg use Amitsur cohomology to give a 7-term exact sequence relating units, Picard, and Brauer groups to the Galois group in a Galois extension of commutative rings. This project aims to expand this relationship to include Galois extensions of commutative ring spectra, using the bicategorical description of Brauer groups developed in previous work.

Galois Theory Reading Project

Galois notes (UGA, 2010)

Abstract:

This is a collection of notes from our summer 2010 reading project on Galois theory for rings and ring spectra. An attempt is made to outline main ideas of Rognes and Chase-Harrison-Rosenberg, but only a minimal effort has been put into proofreading and checking details. The reader is encouraged not to read too carefully!

Many thanks are due to the various people who participated; without their interest, we certainly wouldn't have made it even this far.

Calculations for Complex Oriented Cohomology Theories

The formal group law structure on complex oriented cohomology theories enables explicit computation of power operations and related structure for these topological objects. The goal of this project is to expand existing computer code to support further research related to these objects, in terms of certain power series computations. Developing algorithms for an open-source computer algebra system makes them available to other researchers, both for scientific review and further applications.

The Universal Case

We used calculations of the universal p-typical formal group laws for p ≤ 13 to study power operations on the complex cobordism spectrum, MU.

Example: Consider the ZZ/13 cyclic power operation on MU_(13). We compute a summand of P_{ZZ/13} [CCP^{24}] obtained from Quillen's map MU_(p) —> BP. The appearance of non-zero coefficients is an obstruction to induced power operations on BP. For further details, see the project on Power Operations and p-typicality.

\begin{align} MC_{24}(\xi) \equiv \; & 11v_1^{40}\xi^{744} + (6v_1^{41} + 6v_1^{27}v_2)\xi^{756}\\\\ & + O(\xi^{768}) \quad \text{mod } \langle 13 \rangle \xi \end{align}

Related calculations have allowed us to formulate the following conjecture for general primes p: That the first non-zero coefficient of MC_{2(p-1)} occurs at

\[ v_1^{3p+1} \xi^n, \qquad n = 5p^2 − 8p + 3. \]

A summer undergraduate project with Eddie Beck extends our results and confirms the conjecture for higher primes by improving the algorithm for our computaitons.

Computational packages:

A first draft of the necessary algorithms was developed with Mathematica 7. Although the Mathematica software was easy to get started with, we were not able to develop a version which was practical for primes greater than 5. We have developed another package using the open-source software Sage. With this package, we were able to expand our results to the primes 7, 11, and 13. Currently, there is a rough working version available (2010). Further work will expand this package to support calculations for a range of complex oriented cohomology theories (see below).

Expanding Functionality and Usability

This is work in progress.

The plan for this project is as follows:

  • Implement multivariate power series for Sage.
    This step was completed during the summer of 2011, when the code was merged into Sage version 4.7.1.
  • Implement Hopf algebroids for Sage.
    Independent effort has already begun in this direction; the focus here will be incorporating Hopf algebroids into Sage's category-theoretic framework so that the functionality integrates seamlessly with the rest of Sage's object types.
  • Implement complex oriented cohomology theories as a subclass of Hopf algebroids.
    This is the stage at which existing code from previous projects will be merged into this project.
  • Improve the basic algorithms and functionality so that other topologists can make use of the calculator. This was one goal of the undergraduate summer project with Eddie Beck (2011).

Ecological Niche Topology

This is work in progress.

This project is joint with John Drake.

The concept of ecological niche is a way that ecologists abstract a species' environmental needs from its geographic location. It is described as the set of environmental conditions in which a species can survive, and is usually thought of as a convex subset of the space of relevant environmental parameters (temperature, precipitation, soil pH, etc.). The goals of this project are to provide mathematically precise models for niches, and to investigate their topological properties both theoretically and empirically.

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