|Homological Algebra, Its Non- Abelian and Categorical Topics: Applications to Homotopy Theory, K-Theory, Algebraic Geometry and Galois Theory |
Project #GM1-115 The project was devoted to the study of homological and homotopical properties of algebraic structures using methods and techniques of simplicial , combinatorial and categorical algebra with applications to important fields of mathematics. The following results were obtained. Quillen's algebraic K-theory was extended to the category of normed algebras over commutative Banach rings and its relationship with topological K-theory was established. Sufficient conditions were given for the isomorphism of algebraic and topological K-groups. New descriptions of non- abelian homology groups in terms of derived functors were given. Sufficient conditions were established for non- abelian homology of groups to be finite, finitely generated, p-groups and torsion groups. Some calculation formulas for lower homology groups were obtained. Graded automorphisms of polytopal semigroup rings were described in terms of toric and elementary automorphisms and symmetries.