TY - BOOK

T1 - Patterns in the homology of algebras

T2 - Vanishing, stability, and higher structures

AU - Sroka, Robin Janik

PY - 2021

Y1 - 2021

N2 - 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 Ek-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 certainalgebras 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 Ek-algebra structure on the “classifying space” of a family of abstract algebras, building on work of Berger, Fiedorowicz and Smith. We then describe an E2-algebra structure on the “classifying space” of certain families of Iwahori–Hecke algebras and show that it does not extend to an E3-structure in general. Chapter 5, which is based on joint work with Benjamin Br ̈uck and Peter Patzt, studies the top-dimensional rational cohomology of the integral symplectic groups. It follows from at heorem 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.

AB - 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 Ek-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 certainalgebras 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 Ek-algebra structure on the “classifying space” of a family of abstract algebras, building on work of Berger, Fiedorowicz and Smith. We then describe an E2-algebra structure on the “classifying space” of certain families of Iwahori–Hecke algebras and show that it does not extend to an E3-structure in general. Chapter 5, which is based on joint work with Benjamin Br ̈uck and Peter Patzt, studies the top-dimensional rational cohomology of the integral symplectic groups. It follows from at heorem 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.

M3 - Ph.D. thesis

SN - 978-87-7125-043-5

BT - Patterns in the homology of algebras

PB - Department of Mathematical Sciences, Faculty of Science, University of Copenhagen

ER -