The Index Map in Algebraic K-Theory
Oliver Braunling, Michael Groechenig, Jesse Wolfson

TL;DR
This paper constructs a universal torsor in algebraic K-theory for Tate modules over a ring, linking higher loop groups with K-theory and generalizing classical index maps.
Contribution
It introduces a universal K_R-torsor on the K-theory space of Tate R-modules, connecting higher loop groups with algebraic K-theory in a novel way.
Findings
Constructed a universal K_R-torsor on K-theory space of Tate R-modules.
Related the torsor to classical central extensions of loop groups.
Analyzed the classifying index map and its analogy with Fredholm operators.
Abstract
For a ring , we construct a universal -torsor on the -theory space of Tate -modules. This torsor is closely related to canonical central extensions of loop groups. Just like classical loop group theory has features of -theory (e.g. determinant bundles, tame symbol cocycle for Kac-Moody extension), the -theory torsor relates higher loop groups with higher -theory. We study the classifying "index" map of this torsor in detail. We explain how it arises in analogy with the classical index map of Fredholm operators, and we relate the -theory torsor to previously studied dimension and determinant torsors.
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