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Even-dimensional $l$–monoids and $L$–theory

Jörg Sixt

Algebraic & Geometric Topology 7 (2007) 479–515

Surgery theory provides a method to classify n–dimensional manifolds up to diffeomorphism given their homotopy types and n 5. In Kreck’s modified version, it suffices to know the normal homotopy type of their n 2 –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 θ l2k(Λ) for a ring Λ. Firstly, if θ l2k(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.

surgery theory, $L$-groups
Mathematical Subject Classification 2000
Primary: 57R67
Secondary: 57R65
Received: 14 November 2006
Revised: 22 January 2006
Accepted: 24 January 2006
Published: 25 April 2007
Jörg Sixt
Universität Heidelberg
Mathematisches Institut
Im Neuenheimer Feld 288
D-69120 Heidelberg