#### 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 ${GL}_{n}\left(F\right)$ 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 ${A}_{\infty }$-spaces ${GL}_{n}\left(F\right)\to {K}_{k}$. Using a generalized Waldhausen construction, we construct an explicit model built for the ${A}_{\infty }$-structure of this map, built from nested systems of lattices in ${F}^{n}$. 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
Primary: 19D55
Secondary: 19K56