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

Oliver Braunling; Michael Groechenig; Jesse Wolfson

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arXiv · 1806.08766

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

Oliver Braunling·Michael Groechenig·Jesse Wolfson

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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.

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BibTeX RIS

Oliver Braunling, Michael Groechenig, Jesse Wolfson. 2018-06-22. The $A_\infty$-structure of the index map. https://arxiv.org/abs/1806.08766

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