July 14th, 2026
What I worked on:
- Finished reading up on Topological cyclic homology
- Worked up to a sketch of Topological Hochschild Homology
- in the Nikolaus-Scholze definition of cylotomic spectra, no equivariant homotopy theory is needed which differs from previous definitions
- the definition of THH is given as follows \begin{align*} THH(\mathcal{C})_n = \text{colim}\left(\bigotimes_{0 \leq i \leq n} \text{map}_{\mathcal{C}}(x_i, x_{i+1})\right) \end{align*}
- to understand this definition of THH we need to understand the colimit, to do this we need the following steps:
- we need to construct a functor: \begin{align*} (\mathcal{C}^{\backsimeq})^{n+1} \to \text{Sp} \end{align*}
- then we need to lift the map $n \mapsto THH(\mathcal{C})_n$ to a functor $\Lambda^{op} \to \text{Sp}$ from Conne's cyclic category such that the face and degeneracy maps are given by composing adjacent morphisms and by inserting identities.
- Lastly, you want to construct the cycolotmic Frobenius maps, which are defined in the paper I read yesterday
- we need to construct a functor:
- Also read through a computation of $THH(\mathbb{Z}/p^n)$ for all $n$. This can be found in Krause-Nikolaus.
- Worked up to a sketch of Topological Hochschild Homology
- Continued reading about limit sets and its properties
- classified the elementary subgroups of $Isom(\mathbb{H}^n)$
- learned about minimal actions in the context of $\Gamma$ acting on its limit set $\Lambda(\Gamma)$. A minimal action means that $\Lambda(\Gamma)$ has no invariant non-empty proper closed subset
- showed that if $\Gamma$ is not elementary then its action on $\Lambda(\Gamma)$ is minimal.
- a corollary of this result is that if $\Gamma$ is not elementary and $\Gamma'$ is an infinite normal subgroup of $\Gamma$ then their two limit sets are eaqual.
- the proof of this theorem relied on the fact that a convex hull $C(S)$ always exists for any closed subset $S \subset \overline{\mathbb{H}^n}$.
- showed that if $\Gamma$ is not elementary then its action on $\Lambda(\Gamma)$ is minimal.
- Read Folland
- finished section 4.1 on Topological spaces. Learned about the Kuratowski closure operator $(-^*)$ and showed that for any subset $A \subset X$, $A^*$ is its closure.
- started on continuity, not many interesting results but tomorrow I get to prove my favorite result, which is the Urysohn lemma