Supervisors info:
Εμμανουήλ Ιωάννης, καθηγητής, Τμήμα Μαθηματικών, Εθνικό Καποδιστριακό Πανεπιστήμιο Αθηνών
Summary:
The thesis presents the concept of periodic cohomology for infinite groups (periodic cohomology after k-steps). Periodic cohomology after k-steps generalizes the concept of periodic (Tate) cohomology of finite groups in the case of infinite groups.
The first chapter introduces some basic concepts of Homological Algebra that will be needed, with emphasis on the pushout and the pullback, induced and coinduced modules.
The second chapter covers the diagonal action of a group G on the abelian group of Z-homomorphisms and the abelian group of the tensor product over Z fro two ZG-modules A and B. We prove that with the diagonal action, these abelian groups become ZG modules and we study the relation between the diagonal action and induced and coinduced modules.
The third chapter covers the Ext groups as groups of module extensions. We introduce the Baer sum of two extensions, the Yoneda product on the level of extensions and on the level of projective resolutions, and the cup product. Lastly, we prove that every natural transformation between Ext groups is induced by a homomorphism and we give necessary and sufficient conditions in order for such a transformation to be monomorphism, epimorphism and isomorphism.
The fourth chapter covers the algebraic invariants silpZG, spliZG and findimZG of a group G. We prove that silpZG is less than or equal to spliZG, whereas if spliZG is finite, then silpZG, spliZG and findimZG are all equal.
The fifth chapter is the core of the thesis. We define the concept of periodic cohomology with period q after k-steps, the concept of periodic projective resolution with period q after k-steps and we prove that these two notions are equivalent. We introduce the concept of a complete resolution and we prove that every group with periodic cohomology with period q after k-steps has a complete resolution with coincidence index k. Lastly, we give necessary and sufficient conditions in order for the periodicicity isomorphisms for a group G to be induced by cup product.
The sixth chapter presents an application of the results of the fifth chapter for a class of groups. Specifically, we prove that if a group G has finite cohomological dimension m and has an infinite cyclic subgroup, then the quotient group has periodic cohomology with period 2 or 4 after m-1 steps and the periodicity isomorphism is induced by cup product. Namely, if G is a finitely generated, torsion-free nilpotent group with Hirsch number m, then for every central element x, the group G/ has periodic cohomology with period 2 after m-1 steps which is induced by cup product.
