Title: Properties of Brown-Gitler spectra with application to constructing type 2 complexes.
Abstract: The (dual) Brown-Gitler spectra T(n) are a family of finite complexes with remarkable properties. Historically, they played an important role in work on immersions of manifolds, Mahowald's construction of the eta_j family in the stable homotopy groups of spheres, and in the proof of the Sullivan conjecture.
Here we revisit some old results of Paul Goerss and apply them in new ways.
Working at the prime 2, we construct maps T(n)-->T(m) of Adams filtration 1 that make the family of Adams spectral sequences converging to [T(*),X] into a single spectral sequence of modules over the Dyer-Lashof algebra.
Then, in joint work with William Balderrama, Justin Barhite, and Don Larson, and overlapping with the 2019 Ph.D. thesis work of my student Brian Thomas, we have determined exactly when the fibers of these Dyer-Lashof maps are complexes of type 2. There are lots of infinite families of these, and indeed, they all have mod 2 cohomology free over A(1).
In my talk, I will outline how this all goes.
Noah Riggenbach
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