3. Postdoctoral Program - MSRI

Annual Report 2010
Vigleik Angeltveit
June 10, 2010
This is the 2010 annual report as a postdoctoral fellow with the MSRI.
The paper “Uniqueness of Morava K-theory” has been accepted for publication,
pending revisions, in Compositio Mathematica. We submitted the paper
“RO(S 1 )-graded TR-groups”, which is joint with Teena Gerhardt, to the Journal
of Pure and Applied Algebra. We received a referee report indicating that
we needed to make the paper more accessible, after which we submitted a revised
version. We also submitted our paper “On the algebraic K-theory of the
coordinate axes over the integers” to Mathematical Research Letters.
I have spent much of my time the last year trying to understand the algebraic
K-theory of some very simple rings, like Z/p n . While Quillen computed
the algebraic K-theory of finite fields almost 40 years ago, and Bökstedt and
Madsen computed the algebraic K-theory of the p-adic integers in the early
1990s, K(Z/p n ) remains mysterious.
The plan is to use the cyclotomic trace map from K(R) to the topological
cyclic homology spectrum TC(R), which is built from the fixed points of the
topological Hochschild homology spectrum T HH(R) under the action of finite
subgroups of the circle. If we have a multiplicative filtration of a ring R, we get
a filtration of T HH(R) and hence a spectral sequence computing π∗T HH(R).
This filtration is S 1 -equivariant, so it gives a filtration and a spectral sequence
for computing the fixed points of T HH(R).
The case R = Z/p n , filtered by powers of p, is well suited for this approach.
While a complete understanding of K(Z/p n ) is still out of reach, I believe I
can at least extend the range where K∗(Z/p n ) is understood. In particular, I
believe I can prove that for any n ≥ 2, the first nontrivial p-torsion element
α1 ∈ π2p−3S in the stable homotopy groups of spheres maps nontrivially to
K2p−3(Z/p n ). This was only known at the primes p = 2 and p = 3.
Charles Weibel suggested that Teena Gerhard and I try to understand the
algebraic K-theory of the group ring Z[Z/2]. This would tell us about things
like diffeomorphisms of manifolds with fundamental group Z/2. Again using
the cyclotomic trace map seems like a good approach, and we are able to set
up spectral sequences for computing the homotopy groups of fixed points of
T HH(Z[Z/2]). This is still work in progress.
In March, Mike Hill, Teena Gerhardt, Andrew Blumberg and I participated
in a SQuaREs workshow at the American Institute of Mathematics in Palo Alto,
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