Segal operations in the algebraic K-theory of topological spaces

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 $Σ_{n}$ , we define operations $θ^{n} : A (X) \to A (X \times B Σ_{n})$ in the algebraic $K$ -theory of spaces. We show that our operations can be given the structure of $E_{\infty}$ -maps. Let $ϕ_{n} : A (X \times B Σ_{n}) \to A (X \times E Σ_{n}) ≃ A (X)$ be the $Σ_{n}$ -transfer. We also develop an inductive procedure to compute the compositions $ϕ_{n} \circ θ^{n}$ , and outline some applications.

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Keywords

algebraic $K\mkern-2mu$-theory of topological spaces, Segal operations, operations

Mathematical Subject Classification 2010

Primary: 19D10

Secondary: 19D23

Milestones

Received: 11 July 2017

Revised: 9 July 2018

Accepted: 31 October 2018

Published: 24 March 2019

Authors

Thomas Gunnarsson
Department of Mathematics Luleå University of Technology Luleå Sweden

Ross Staffeldt
Department of Mathematical Sciences New Mexico State University Las Cruces, NM United States

Vol. 4, No. 1, 2019

Thomas Gunnarsson and Ross Staffeldt

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors