The Witt group of real algebraic varieties

Max Karoubi, Marco Schlichting, Charles Weibel

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

The purpose of this paper is to compare the algebraic Witt group W(V ) of quadratic forms for an algebraic variety V over ℝ with a new topological invariant, WR(VCℂ), based on symmetric forms on Real vector bundles (in the sense of Atiyah) on the space of complex points of V . This invariant lies between W(V ) and the group KO(Vℝ) of ℝ-linear topological vector bundles on the space Vℝ of real points of V . We show that the comparison maps W(V ) → WR(Vℂ) and WR(Vℂ) → KO(Vℝ) are isomorphisms modulo bounded 2-primary torsion.We give precise bounds for the exponent of the kernel and cokernel, depending upon the dimension of V. These results improve theorems of Knebusch, Mahé and Brumfiel. Along the way, we prove the comparison theorem between algebraic and topological Hermitian K-theory, and homotopy fixed point theorems for the latter. We also give a new proof (and a generalization) of a theorem of Brumfiel.

Original languageEnglish (US)
Pages (from-to)1257-1302
Number of pages46
JournalJournal of Topology
Volume9
Issue number4
DOIs
StatePublished - Jan 1 2016

Fingerprint

Witt Group
Algebraic Variety
Vector Bundle
Michael Francis Atiyah
Topological Invariants
Comparison Theorem
K-theory
Algebraic Groups
Theorem
Quadratic form
Homotopy
Torsion
Fixed point theorem
Modulo
Isomorphism
Exponent
kernel
Invariant
Form
Generalization

All Science Journal Classification (ASJC) codes

  • Geometry and Topology

Cite this

Karoubi, Max ; Schlichting, Marco ; Weibel, Charles. / The Witt group of real algebraic varieties. In: Journal of Topology. 2016 ; Vol. 9, No. 4. pp. 1257-1302.
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The Witt group of real algebraic varieties. / Karoubi, Max; Schlichting, Marco; Weibel, Charles.

In: Journal of Topology, Vol. 9, No. 4, 01.01.2016, p. 1257-1302.

Research output: Contribution to journalArticle

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