5-local Computation of the Homotopy of $eo_4$
Massey Products in the Rational Cohomology of $Sp(5)/SU(5)$
A Fast Computation of the Cohomology of $\mathbb Z/3[\xi_1,\xi_2]/(\xi_1^9,\xi_2^3)$
The tmf homology of $B\Sigma_3$ and the homotopy of the $\Sigma_3$ Tate spectrum of tmf.
I wrote these notes up a few years back to help cement my understanding of the role Massey products play to compute extensions in Adams style spectral sequences. Since then, I have been working to expand on this theme, staring with the statement of my Massey product lemma which relates differentials on classes in a spectral sequence to Massey products on lower differentials of related elements. This is still very much a work in progress, and so it is available by request only.
A while back, I decided to use Mike Hopkins' Hopf algebroid to compute the homotopy algebra of the conjectural spectrum $eo_4$. After setting this aside of a while, I've returned to it. This is the most recent draft, updated on January 25th, 2006.
In an old paper of Stacheff, I found a comment concerning the existence of Massey products in the cohomology of this homogeneous space. These allow us to distinguish between it and a connect-sum of a product of spheres. Using my lemma, I was able to verify the existence of these secondary products, and I wrote it up for the benefit of my fellow grad students at MIT.
Haynes Miller suggested that I compute the cohomology of the dual of $A(1)$ at primes bigger than 2. I wrote up the computation for the prime 3, since it has the most readily apparent multiplicative extensions. At primes bigger than 3, similar extensions occur; they live only as Massey products. This note provides another example of the usefulness of the Massey product lemma found in the first set of notes.
In this paper, I prove the structure of the Hopf algebra conjectured by Andre Henriques and me to be the homotopy of $HZ/3^_{tmf} HZ/3$. Using the Adams spectral sequence in the world of tmf modules, I was able to compute the tmf homology of $B\Sigma_3$ and the homotopy of the Tate spectrum. I have also included results about the tmf homology of the truncated classifying spaces and more results about the cases of larger primes. This was updated January 26th, 2006.
Last
changed: