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**Introduction to algebra over “brave new rings”.**
*(English)*
Zbl 0969.55003

Slovák, Jan (ed.) et al., Proceedings of the 18th winter school “Geometry and physics”, Srní, Czech Republic, January 10-17, 1998. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 59, 49-82 (1999).

This nice survey paper provides an introduction to the algebra over so-called “brave new rings”. The term “brave new rings” has been introduced by F. Waldhausen to denote ring structures on topological spaces up to coherent homotopies.

In chapter I the author describes \(A_\infty\)-spaces and loop spaces. Basic notions and properties are given. More specifically, the author reviews the definitions of non-\(\Sigma\)-operad, \(A_\infty\)-operad and describes the composition in detail. Applications of these techniques to loop spaces are described in section 3 of chapter I.

Chapter II is devoted to \(E_\infty\)-spaces and infinite loop spaces. After the definitions of an operad and \(E_\infty\)-structure the author introduces the little cube operad and shows its connection with infinite loop spaces. These results are by now classical in homology theory. As the author is one of the main contributors to the subject the results are stated very clearly and appropriate references are given. In chapter III brave new rings and Waldhausen’s proposal, how to extend algebraic structures to homotopy theory are discussed. Detailed historic account concerning evolution of these ideas is given. In chapter IV algebraic applications to topological Hochschild homology, topological cyclic homology and \(K\)-theory of integers are discussed. The paper is written in an elegant and clear style and gives an overview of the subject.

For the entire collection see [Zbl 0913.00039].

In chapter I the author describes \(A_\infty\)-spaces and loop spaces. Basic notions and properties are given. More specifically, the author reviews the definitions of non-\(\Sigma\)-operad, \(A_\infty\)-operad and describes the composition in detail. Applications of these techniques to loop spaces are described in section 3 of chapter I.

Chapter II is devoted to \(E_\infty\)-spaces and infinite loop spaces. After the definitions of an operad and \(E_\infty\)-structure the author introduces the little cube operad and shows its connection with infinite loop spaces. These results are by now classical in homology theory. As the author is one of the main contributors to the subject the results are stated very clearly and appropriate references are given. In chapter III brave new rings and Waldhausen’s proposal, how to extend algebraic structures to homotopy theory are discussed. Detailed historic account concerning evolution of these ideas is given. In chapter IV algebraic applications to topological Hochschild homology, topological cyclic homology and \(K\)-theory of integers are discussed. The paper is written in an elegant and clear style and gives an overview of the subject.

For the entire collection see [Zbl 0913.00039].

Reviewer: Piotr Krasoń (Szczecin)