## Annals of K-Theory

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

#### 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.

#### Article information

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

Dates
Revised: 7 June 2018
Accepted: 21 June 2018
First available in Project Euclid: 5 January 2019

https://projecteuclid.org/euclid.akt/1546657257

Digital Object Identifier
doi:10.2140/akt.2018.3.581

Mathematical Reviews number (MathSciNet)
MR3892960

Zentralblatt MATH identifier
07000853

#### Citation

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. https://projecteuclid.org/euclid.akt/1546657257

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