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

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.

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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\textit{A. S. Anan'evskii}, Dokl. Math. 87, No. 1, 76--78 (2013; Zbl 1291.55008); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 448, No. 5, 503--505 (2013)

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### References:

[1] | Topological Library. Cobordisms in Soviet Union. 1967-1979, Ed. by S. P. Novikov and I. A. Taimanov (Inst. Komp’yut. Issled., Moscow, 2011) [in Russian]. |

[2] | E. Friedlander and D. Grayson, Handbook of K-Theory (Springer-Verlag, Berlin, 2005). · Zbl 1070.19002 |

[3] | P. E. Conner and E. E. Floyd, The Relation of Cobordism to K-Theories (Springer-Verlag, Berlin, 1966). · Zbl 0161.42802 |

[4] | Jardine, J F, No article title, Doc. Math., 5, 445-552, (2000) · Zbl 0969.19004 |

[5] | Morel, F; Voevodsky, V, No article title, Publ. Math. IHES, 90, 45-143, (1999) · Zbl 0983.14007 |

[6] | I. Panin and C. Walter, On the Algebraic Cobordism Spectra MSL and MSp, arXiv:1011.0651 [math.AG], Nov. 2, 2010. |

[7] | I. Panin and C. Walter, On the Relation of the Symplectic Algebraic Cobordism to Hermitian K-Theory, arXiv:1011.0652 [math.AG], Nov. 2, 2010. |

[8] | Schlichting, M, No article title, J. K-Theory, 5, 105-165, (2010) · Zbl 1328.19009 |

[9] | Voevodsky, V, No article title, Doc. Math. Extra Vol., 1, 579-604, (1998) |

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