Quadratic K-Theory and Geometric Topology

Bruce Williams ()
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Bruce Williams: University of Notre Dame, Department of Mathematics

Chapter III.3 in Handbook of K-Theory, 2005, pp 611-651 from Springer

Abstract: 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].)

Keywords: Chain Complex; Loop Space; Hermitian Form; Surgery Theory; Homotopy Equivalent (search for similar items in EconPapers)
Date: 2005
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DOI: 10.1007/978-3-540-27855-9_13

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