The Witt group of real algebraic varieties. (3rd November 2016)

• Record Type: Journal Article
• Authors: Karoubi, Max; Schlichting, Marco; Weibel, Charles
• Is Part Of: Journal of topology. Volume 9:Part 4(2016)
• Page Start: 1257
• Page End: 1302
• Publication Date: 2016-11-03
• Subjects: Topology -- Periodicals; 514.05
• DOI: 10.1112/jtopol/jtw024
• Languages: English
• ISSNs: 1753-8416

Abstract

Abstract : The purpose of this paper is to compare the algebraic Witt groupW ( V ) of quadratic forms for an algebraic varietyV overR with a new topological invariant, WR ( V C ), based on symmetric forms on Real vector bundles (in the sense of Atiyah) on the space of complex points ofV . This invariant lies betweenW ( V ) and the groupK O ( V R ) ofR ‐linear topological vector bundles on the spaceV R of real points ofV . We show that the comparison mapsW ( V ) → WR ( V C ) andWR ( V C ) → K O ( V R ) are isomorphisms modulo bounded 2‐primary torsion. We give precise bounds for the exponent of the kernel and cokernel, depending upon the dimension ofV . These results improve theorems of Knebusch, Mahé and Brumfiel. Along the way, we prove the comparison theorem between algebraic and topological HermitianK ‐theory, and homotopy fixed point theorems for the latter. We also give a new proof (and a generalization) of a theorem of Brumfiel.