Segal operations in the algebraic $K$-theory of topological spaces

Abstract

We extend earlier work of Waldhausen which defines operations on the algebraic $K$-theory of the one-point space. For a connected simplicial abelian group $X$ and symmetric groups ${\Sigma }_n$, we define operations ${\theta }^n:A(X)\to A(X\times B{\Sigma }_n)$ in the algebraic $K$-theory of spaces. We show that our operations can be given the structure of $E_{\infty }$-maps. Let ${\varphi }_n:A(X\times B{\Sigma }_n)\to A(X\times E{\Sigma }_n)\simeq A(X)$ be the ${\Sigma }_n$-transfer. We also develop an inductive procedure to compute the compositions ${\varphi }_n\circ {\theta }^n$, and outline some applications.

Revision corrects typographical errors, corrects some omissions, replaces quoted (long) definitions with specific references to literature, and reorganizes material appropriately

References in corpus (1)

• Operations on A-theoretic nil-terms