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2007 Even-dimensional $l$–monoids and $L$–theory
Jörg Sixt
Algebr. Geom. Topol. 7(1): 479-515 (2007). DOI: 10.2140/agt.2007.7.479

Abstract

Surgery theory provides a method to classify n–dimensional manifolds up to diffeomorphism given their homotopy types and n5. In Kreck’s modified version, it suffices to know the normal homotopy type of their n2–skeletons. While the obstructions in the original theory live in Wall’s L–groups, the modified obstructions are elements in certain monoids ln(Z[π]). Unlike the L–groups, the Kreck monoids are not well-understood.

We present three obstructions to help analyze θl2k(Λ) for a ring Λ. Firstly, if θl2k(Λ) is elementary (ie trivial), flip-isomorphisms must exist. In certain cases flip-isomorphisms are isometries of the linking forms of the manifolds one wishes to classify. Secondly, a further obstruction in the asymmetric Witt-group vanishes if θ is elementary. Alternatively, there is an obstruction in L2k(Λ) for certain flip-isomorphisms which is trivial if and only if θ is elementary.

Citation

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Jörg Sixt. "Even-dimensional $l$–monoids and $L$–theory." Algebr. Geom. Topol. 7 (1) 479 - 515, 2007. https://doi.org/10.2140/agt.2007.7.479

Information

Received: 14 November 2006; Revised: 22 January 2006; Accepted: 24 January 2006; Published: 2007
First available in Project Euclid: 20 December 2017

zbMATH: 1132.57029
MathSciNet: MR2308954
Digital Object Identifier: 10.2140/agt.2007.7.479

Subjects:
Primary: 57R67
Secondary: 57R65

Rights: Copyright © 2007 Mathematical Sciences Publishers

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