Algebraic and homological aspects of Hermitian \(K\)-theory (Q6633158)

The paper under review surveys contributions made by Sergei Novikov with regard to algebraic and homological aspects of Hermitian \(K\)-theory. In the 1970s he proposed a systematization of algebraic results of the surgery theory based on the Hamiltonian formalism over rings with involution. His results have had a significant impact on the development of Hermitian analogs of algebraic \(K\)-theory. Recently new questions about the current state of Hermitian \(K\)-theory arose, namely how one should construct a Hermitian algebraic \(K\)-theory of rings with involution so that it is a kind of ``homology theory'' on the category of rings with involution, that for group rings of (finitely presented) discrete groups, it coincides with the surgery obstruction groups, that for the rings of (complex-valued) functions on a smooth closed manifold, it coincides with the topological \(K\)-theory of the manifold, and that for group rings and rings of functions, Bott elements are defined that guarantee the 4-periodicity of the obstruction groups and 2-periodicity of the topological \(K\)-theory? These questions constitute a whole research program, which, as an analysis of the literature has shown, on the whole can be considered fairly complete. The aim of the paper under the review is to describe the results of this program; many results are presented in sufficient detail, while for some other results only literature references are given.