Vol. 4, No. 1, 2019

 Recent Issues Volume 4, Issue 4 Volume 3, Issue 4 Volume 3, Issue 3 Volume 3, Issue 2 Volume 3, Issue 1 Volume 2, Issue 4 Volume 2, Issue 3 Volume 2, Issue 2 Volume 2, Issue 1
 The Journal About the Journal Editorial Board Subscriptions Ethics Statement Submission Guidelines Submission Form Editorial Login Ethics Statement ISSN: 2379-1691 (e-only) ISSN: 2379-1683 (print) Author Index To Appear Other MSP Journals
Segal operations in the algebraic $K\mkern-2mu$-theory of topological spaces

Vol. 4 (2019), No. 1, 1–56
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 ${\theta }^{n}:A\left(X\right)\to A\left(X×B{\Sigma }_{n}\right)$ in the algebraic $K$-theory of spaces. We show that our operations can be given the structure of ${E}_{\infty }$-maps. Let ${\varphi }_{n}:A\left(X×B{\Sigma }_{n}\right)\to A\left(X×E{\Sigma }_{n}\right)\simeq A\left(X\right)$ be the ${\Sigma }_{n}$-transfer. We also develop an inductive procedure to compute the compositions ${\varphi }_{n}\circ {\theta }^{n}$, and outline some applications.

Keywords
algebraic $K\mkern-2mu$-theory of topological spaces, Segal operations, operations
Primary: 19D10
Secondary: 19D23