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

Abstract

Surgery theory provides a method to classify $n$–dimensional manifolds up to diffeomorphism given their homotopy types and $n\ge 5$. 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 $l_n\left(Z\left[\pi \right]\right)$. Unlike the $L$–groups, the Kreck monoids are not well-understood.

We present three obstructions to help analyze $\theta \in l_{2k}\left(\Lambda \right)$ for a ring $\Lambda$. Firstly, if $\theta \in l_{2k}\left(\Lambda \right)$ 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 $\theta$ is elementary. Alternatively, there is an obstruction in $L_{2k}\left(\Lambda \right)$ for certain flip-isomorphisms which is trivial if and only if $\theta$ is elementary.