Annals of K-Theory

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

Oliver Bräunling, Michael Groechenig, and Jesse Wolfson

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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.

Article information

Ann. K-Theory, Volume 3, Number 4 (2018), 581-614.

Received: 24 May 2016
Revised: 7 June 2018
Accepted: 21 June 2018
First available in Project Euclid: 5 January 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 19D55: $K$-theory and homology; cyclic homology and cohomology [See also 18G60]
Secondary: 19K56: Index theory [See also 58J20, 58J22]

Waldhausen construction boundary map in $K$-theory Tate space


Bräunling, Oliver; Groechenig, Michael; Wolfson, Jesse. The $A_\infty$-structure of the index map. Ann. K-Theory 3 (2018), no. 4, 581--614. doi:10.2140/akt.2018.3.581.

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