Geodesic
Splitting obstruction groups and quadratic extensions of anti-structures
Izvestiya. Mathematics , Tome 59 (1995) no. 6, pp. 1207-1232.

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A natural generalization is given of the LS groups of obstructions to splitting in the case of one-sided manifolds. These LS groups depend functorially on four anti-structures and maps between them, generating a square of a special form. The connection between the LS groups and the Wall groups of anti-structures involved in the diagram is studied. “Intermediate” LS groups (LS with decorations) are also defined, by analogy with the corresponding Wall groups, and the connection between LS groups with different decorations is studied. 
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Yu. V. Muranov. Splitting obstruction groups and quadratic extensions of anti-structures. Izvestiya. Mathematics , Tome 59 (1995) no. 6, pp. 1207-1232. http://geodesic.mathdoc.fr/item/IM2_1995_59_6_a5/

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