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Quadratic K-Theory and Geometric Topology

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Handbook of K-Theory

Abstract

Suppose R is a ring with an (anti)-involution −: R → R, and with choice of central unit ε such that \( \bar{\epsilon}\epsilon = 1 \). Then one can ask for a computation of \( \mathbb{K} Quad (R,-,\epsilon) \), the K-theory of quadratic forms. Let H: \( \mathbb{K}R \rightarrow \mathbb{K} Quad (R,-,\epsilon) \) be the hyperbolic map, and let F : \( \mathbb{K} Quad (R,-,\epsilon) \rightarrow \mathbb{K}R \) be the forget map. Then the Witt groups

$$ W_{0}(R,-,\epsilon) = coker (K_{0}R \xrightarrow{H} K_{0}Quad(R,-,\epsilon)) $$
$$ W_{1}(R,-,\epsilon) = ker (F: KQuad_{1}(R,-,\epsilon) \xrightarrow{F} K_{1}R) $$

have been highly studied. See [6], [29, 32, 42, 46, 68], and [86]–[89]. However, the higher dimensional quadratic K-theory has received considerably less attention, than the higher K-theory of f.g. projective modules. (See however, [34, 35, 39], and [36].)

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  1. Department of Mathematics, University of Notre Dame, 46556-4618, Notre Dame, IN, USA

    Bruce Williams

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  1. Department of Mathematics, Northwestern University, 60208, Evanston, IL, USA

    Eric M. Friedlander

  2. Department of Mathematics, University of Illinois at Urbana-Champaign, 61801, Urbana, IL, USA

    Daniel R. Grayson

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Williams, B. (2005). Quadratic K-Theory and Geometric Topology. In: Friedlander, E., Grayson, D. (eds) Handbook of K-Theory. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27855-9_13

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