Introduction
My area of research is algebraic topology. Loosely speaking algebraic topology is the study of topological spaces via functors from topological spaces to algebraic objects (groups, rings, modules, etc). Within algebraic topology I am working in homotopy. My current research concerns convergence conditions for the Eilenberg-Moore spectral sequence for Morava K-Theory.
Homotopy Theory
The original problems motivating homotopy theory are easy to state. Given two topological spaces X and Y and two continuous functions between them f,g:X \rightarrow Y these maps are said to be homotopic if there exists a continuous function F:[0,1]\times X \rightarrow Y with F(0,x) = f(x) and F(1,x) = g(x) . That is, if one function can be continuously deformed into the other, the functions are homotopic. Similarly, two topological spaces X and Y are said to be homotopy equivalent if there are continuous functions f:X \rightarrow Y and g:Y \rightarrow X such that gf and fg are homotopic to the identity.
One goal of homotopy theory is to classify all topological spaces up to homotopy equivalence. Also, given two topological spaces, it would be nice to understand [X,Y] , the homotopy classes of maps from X to Y. Both of these goals turn out to be overly ambitious. Even for very well known and easily defined spaces such as spheres, these are hard problems.
Among the tools that have been developed to classify homotopy types of spaces and homotopy types of maps between spaces are homology theories. Homology theories are functors from topological spaces to groups (or modules or rings). This allows us to compare algebraic objects to discriminate between topological spaces (or between maps). The best known examples are ordinary homology, mod-p homology, and K-theory. But, as it turns out, there are many other homology theories. In fact there are as many homology theories as topological spaces. Unfortunately most of these are not computable in any reasonable sense.
I have particular interest in a set of generalized homology theories called the Morava K-theories. These are closely related to complex cobordism. For a set prime, p, there is a Morava K-theory for each m \in \mathbb{N}, denoted K(m)_\ast(X). A property that makes Morava K-theory particularly attractive is that each Morava K-theory has a perfect Kunneth theorem. In part because of this the Morava K-theories are among the few homology theories available that have proven computable. Furthermore, because of their close relationship to complex cobordism, information about Morava K-theories may be enough for information on bordism (i.e. for geometric problems).
The bulk of my research has involved developing tools for calculating Morava K-theory via the Morava K-theory Eilenberg-Moore spectral sequence.
My Research
The object of my research comes from applying Morava K-theory to a tower of cofibrations of spectra associated to the path loop fibration. This tower comes from the cosimplicial space construction of the Eilenberg-Moore spectral sequence as done by Rector [Rec70].
\matrix{{P(1)\simeq\Sigma^{-1}X \leftarrow}&{P(2)\leftarrow}&{\cdots}\leftarrow &{P(n)\leftarrow}&{\cdots\leftarrow}&{P(X)\simeq\Omega X}\\ & \uparrow & & \uparrow \\ { }&{\Sigma^{-2}X^{\wedge2}}&{ }& {\Sigma^{-n}X^{\wedge n}}& }
If you apply ordinary homology with \mathbb{Z}/p coefficients to this tower you get a spectral sequence with E^2_{\ast,\ast}=\text{Cotor}^{H_{\ast}(X)}(\mathbb{Z}/p,\mathbb{Z}/p). When this spectral sequence converges (i.e., actually calculates what you expect) it converges to the homology of the loop space of X denoted \Omega X. This occurs under fairly modest conditions on X, for example if X is 1-connected. In part because looping a space shifts its homotopy groups, understanding loop spaces is of special interest to topologists.
A related spectral sequence can be defined for any generalized homology theory since a cofibration always leads to a long exact sequence in homology. So applying Morava K-theory (or any homology theory) to this tower leads to a spectral sequence. Because Morava K-theory has a perfect Kunneth theorem, this gives us E^2_{\ast,\ast}=\text{Cotor}^{K(m)_{\ast}(X)}(K(m)_\ast,K(m)_\ast). This is an E_2 page we might expect to be able to calculate. The hope then is to find explicit conditions on X such that this spectral sequence will converge to the Morava K-theory of the loop space of X, i.e. to K(m)_\ast(\Omega X).
To see that this is not a trivial problem, consider the following well known example.
Example
The Morava K-theory of the t^{th} Eilenberg-MacLane space is K(m)_\ast(K(t,\mathbb{Z}/p)) = \begin{cases} \neq 0 & t\leq m\\0 & t>m\end{cases}
Recalling that \Omega K(t,\mathbb{Z}/p)=K(t-1,\mathbb{Z}/p), this implies that if X = K(t+1,\mathbb{Z}/p) and we apply K(t)_\ast(-) to the above tower, all of the groups in the spectral sequence are trivial. But what we hope for it to converge to, \Omega K(t,\mathbb{Z}/p)=K(t+1,\mathbb{Z}/p), is non-trivial. In this case the spectral sequence doesn't calculate what we want.
Example
In a paper by Lansetmo and Stanley [LS], the authors identified certain K-theory equivalences. M(t+k(2p^n-2),\mathbb{Z}/p)\rightarrow_{f} M(t,\mathbb{Z}/p) Such that K(m)_\ast(C(f))=0 \text{ but }K(m)_\ast(\Omega C(f))\neq 0
Here the same thing happens as with the Eilenberg-MacLane spaces. These are very small CW complexes, only 4 cells, and can be given with arbitrarily large connectivity.
These examples suggest we should expect somewhat restrictive convergence conditions. The convergence conditions I have so far are:
Theorem
Let B be a 1-connected space defined by a CW complex with finitely many cells in each degree such that the Eilenberg-Moore spectral sequence for ordinary homology collapses at the E^2 page and \text{Rank}(\bigoplus_i H_{j-i}(\Omega B,K(m)_i)) is not infinite for consecutive values of j. Then the Morava K-theory Eilenberg-Moore spectral sequence for B converges to K(m)_*(\Omega B).
Recall that \bigoplus_i H_{j-i}(\Omega B,K(m)_i) consists of all elements of total degree j on the E^2 page of the Atiyah-Hirzebruch spectral sequence for \Omega B.
The same method produces the following theorem, but it is not clear that it can be applied in practice.
Theorem
Let X be a space defined by a finite CW complex such that the Eilenberg-Moore spectral sequence for ordinary homology collapses at the E_2 page and theAtiyah-Hirzebruch spectral sequence for Morava K-theory for P(n)collapses at the E_k page for all n and some fixed k.Then the Morava K-theory Eilenberg-Moore spectral sequence for X converges to K(m)_*(\Omega X).
Research Goals
There are several directions to go with my research but my first order of business will be to understand more precisely where the Eilenberg-MacLane space example goes wrong to see if I can pose more general convergence results. Also I can more or less immediately begin to calculate the Morava K-theory of the non-trivial class of spaces that satisfy my convergence criteria. Furthermore in the process of exploring this spectral sequence I have proven a conjecture in [LS, 4.9] and have some hope for solving other conjectures I have stumbled across.
Bibliography
- [R] Rector D.L., Steenrod Operations in the Eilenberg-Moore Spectral Sequence, Comentarii Mathematici Helvetici, 45:540-552, 1970.
- [B] Boardman, Homotopy Invarient Algebraic Structures, American Mathematical Society Contemporary Mathematics Series 242
- [LS] Lisa Langsetmo and Don Stanley, Nondurable K-Theory Equivalences and Bousfield Localization, K-theory, 24:397-410, 2001.
- [Rav] D.C.Ravenel, W.S.Wilson, The Morava K-theories of Eilenberg-MacLane spaces and the Connor-Floyd conjecture, Am. J. Math, 691-748, 1980.