PhD Defense by Robin Janik Sroka
Title:
Patterns in the Homology of Algebras: Vanishing, Stability, and Higher Structures
Abstract:
This thesis studies patterns in the homology and cohomology of algebras. We investigate the vanishing of homology and cohomology groups, homological stability questions, and homology operations arising from $E_k$-structures. In Chapter 1, we introduce the notion of algebraic coset poset. This construction is inspired by work of Boyd, Hepworth and Patzt. It generalizes the notion of coset poset for groups considered in the literature and allows us to associate `geometrically flavored' semi-simplicial $A$-modules to certain algebras $A$. These `spaces with $A$-action' play an important role in the two subsequent chapters in which we use associated `isotropy' spectral sequences to prove theorems about the homology of $A$. In Chapter 2, we prove that the homology of any Temperley-Lieb algebra on an odd number of strands vanishes in all positive homological degrees. This improves a result of Boyd-Hepworth. In Chapter 3, we derive an explicit formula for the second homology of certain Iwahori-Hecke algebras. This generalizes a result of Boyd for the second homology of Coxeter groups and is the Iwahori-Hecke analogue of a theorem of Howlett. In Chapter 4, which is based on joint work with Richard Hepworth and Jeremy Miller, we specify conditions for the existence of an $E_k$-algebra structure on the `classifying space' of a family of abstract algebras, building on work of Berger, Fiedorowicz and Smith. We then describe an $E_2$-algebra structure on the `classifying space' of certain families of Iwahori-Hecke algebras and show that it does not extend to an $E_3$-structure in general. Chapter 5, which is based on joint work with Benjamin Brück and Peter Patzt, studies the top-dimensional rational cohomology of the integral symplectic groups. It follows from a theorem of Gunnells that this unstable cohomology group is trivial. We implement an idea of Putman for a new proof of Gunnells' theorem and explain how the vanishing result follows.
Link to Hybrid defense: https://ucph-ku.zoom.us/j/69356262600?pwd=NnhpMVNicEhySVBUUmFuTDIzdlFEZz09
Link to the Thesis:https://www.math.ku.dk/english/research/phd-theses/robin-sroka/
Advisor :
Nathalie Wahl (University of Copenhagen, Denmark)
Assessment committee:
S øren Galatius (University of Copenhagen, Denmark)
Alexander Kupers (University of Toronto, Canada)
Andrew Snowden (University of Michigan, USA)