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

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.