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

TL;DR

This paper constructs an explicit $A_ abla$-structure model for the index map from the classifying space of $GL_n(F)$ to the $K$-theory space of the residue field, using nested lattices and Tate objects.

Contribution

It introduces a new explicit model for the $A_ abla$-structure of the index map in $K$-theory, applicable to local fields and Tate objects.

Findings

Explicit $A_ abla$-structure model constructed

Model built from nested lattices in local fields

Framework extended to Tate objects in exact categories

Abstract

Let $F$ be a local field with residue field $k$. The classifying space of $GL_n(F)$ 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-infinity-spaces $GL_n(F)\to K_k$. 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.