Relationship between algebraic \(MSL\)-cobordisms and derived Witt groups. (English. Russian original) Zbl 1291.55008

Anan’evskii, A. S.

Dokl. Math. 87, No. 1, 76-78 (2013); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 448, No. 5, 503-505 (2013).

From the text: We prove a theorem which expresses derived Witt groups [E. M. Friedlander (ed.) and D. R. Grayson (ed.) Handbook of \(K\)-theory. Vol. 1 and 2. Berlin: Springer. xiv, 1163 p. (2005; Zbl 1070.19002)] in terms of cobordism groups.
As the first step, we prove the existence of an isomorphism \(BO^{*,*}(X)/\langle\eta- 1\rangle\cong W^*(X)\), where \(\eta\in \pi^{-1,-1}(pt)\) is the stable Hopf map. After this, using a theorem from Friedlander and Grayson [loc. cit., arXiv 1011.0656], it is not difficult to obtain a formula expressing derived Witt groups in terms of symplectic cobordisms. Hower, it turns out that, in this formula, unlike in the case of Hermitian \(K\)-theory groups of symplectic cobordisms can be replaced by \(MSL\)-cobordisms; this is shown in the second half of the paper.

MSC:

55R40	Homology of classifying spaces and characteristic classes in algebraic topology
19G38	Hermitian \(K\)-theory, relations with \(K\)-theory of rings
14F42	Motivic cohomology; motivic homotopy theory

Citations:

Zbl 1070.19002

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