Nordic Topology Meeting
Thursday, August 27, Short Presentation
Anders Husebo: Homotopy fixed points in iterated topological Hochschild homology.
In 2013 Veen calculated in his thesis that the self map $k(n-1)_*(\Sigma^{2p^{n-1}-2 (\Lambda_{T^{n}}H\mathbb{F}_p)^{hT^{n}})\rightarrow k(n-1)_*((\Lambda_{T^{n}}H\mathbb{F}_p)^{hT^{n}})$ induced by multiplication by $v_{n-1}$ maps $1$ to something nonzero, for some range of $n$ depending on which prime $p$ we are interested in, supporting the conjecture that $(\Lambda_{T^{n}}H\mathbb{F}_p)^{hT^{n}}$ has telescopic complexity $n-1$. We would like to also be able to say something about the periodic behavior of $v_{n-1}$ by using the same techniques, but for this we need a spectral sequence in which we can calculate more terms. One way of doing this is taking into account that $GL_{n}\mathbb{Z}$ acts on $T^n$ and hence on $(\Lambda_{T^n}H\mathbb{F}_p)$ so we could look at the homotopy fixed points $(\Lambda_{T^{n}}H\mathbb{F}_p)^{h(GL_n\mathbb{Z}\ltimes T^{n})}$. Some preliminary results in the case $n=2$ will be presented.