Vol. 3, No. 4, 2018

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The $A_\infty$-structure of the index map

Oliver Braunling, Michael Groechenig and Jesse Wolfson

Vol. 3 (2018), No. 4, 581–614
Abstract

Let F be a local field with residue field k. The classifying space of GLn(F) comes canonically equipped with a map to the delooping of the K-theory space of k. Passing to loop spaces, such a map abstractly encodes a homotopy coherently associative map of A-spaces GLn(F) Kk. Using a generalized Waldhausen construction, we construct an explicit model built for the A-structure of this map, built from nested systems of lattices in Fn. More generally, we construct this model in the framework of Tate objects in exact categories, with finite dimensional vector spaces over local fields as a motivating example.

Keywords
Waldhausen construction, boundary map in $K“$-theory, Tate space
Mathematical Subject Classification 2010
Primary: 19D55
Secondary: 19K56
Milestones
Received: 24 May 2016
Revised: 7 June 2018
Accepted: 21 June 2018
Published: 18 December 2018
Authors
Oliver Braunling
Freiburg Institute for Advanced Studies
University of Freiburg
Freiburg
Germany
Mathematical Institute
University of Bonn
Bonn
Germany
Michael Groechenig
Department of Mathematics
University of Toronto
Toronto, ON
Canada
Jesse Wolfson
Department of Mathematics
University of California
Irvine, CA
United States