Homology, Homotopy and Applications

Volume 18 (2016)

Number 1

On the relation of special linear algebraic cobordism to Witt groups

Pages: 205 – 230

DOI: http://dx.doi.org/10.4310/HHA.2016.v18.n1.a11

Author

Alexey Ananyevskiy (Chebyshev Laboratory, St. Petersburg State University, Saint Petersburg, Russia)

Abstract

We reconstruct derived Witt groups via special linear algebraic cobordism. There is a morphism of ring cohomology theories that sends the canonical Thom class in special linear cobordism to the Thom class in the derived Witt groups. We show that for every smooth variety $X$, this morphism induces an isomorphism\[{\mathrm{MSL}}_{\eta *}^{[\star]}(X)\otimes_{{\mathrm{MSL}}^{[2\star]}_{\hphantom{[}0}({\rm pt})}\mathrm{W}^{2\star}({\rm pt}) \to \mathrm{W}^\star(X)[\eta,\eta^{-1}],\]where $\eta$ is the stable Hopf map. This result is an analogue of the result by Panin and Walter reconstructing hermitian $K$-theory using symplectic algebraic cobordism.

Keywords

Witt groups, algebraic cobordism, SL-oriented cohomology, Hopf map

2010 Mathematics Subject Classification

14F42, 19E20, 19G12, 19G38

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