Even-dimensional l–monoids and L–theory

Sixt, Jörg

doi:10.2140/agt.2007.7.479

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

Jörg Sixt

Algebraic & Geometric Topology 7 (2007) 479–515

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 $n \geq 5$ . In Kreck’s modified version, it suffices to know the normal homotopy type of their $\frac{n}{2}$ –skeletons. While the obstructions in the original theory live in Wall’s $L$ –groups, the modified obstructions are elements in certain monoids $l_{n} (Z [π])$ . Unlike the $L$ –groups, the Kreck monoids are not well-understood.

We present three obstructions to help analyze $θ \in l_{2 k} (Λ)$ for a ring $Λ$ . Firstly, if $θ \in l_{2 k} (Λ)$ 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 $L_{2 k} (Λ)$ for certain flip-isomorphisms which is trivial if and only if $θ$ is elementary.

Keywords

surgery theory, $L$-groups

Mathematical Subject Classification 2000

Primary: 57R67

Secondary: 57R65

References

Publication

Received: 14 November 2006

Revised: 22 January 2006

Accepted: 24 January 2006

Published: 25 April 2007

Authors

Jörg Sixt
Universität Heidelberg Mathematisches Institut Im Neuenheimer Feld 288 D-69120 Heidelberg Germany

Volume 7, issue 1 (2007)