Surgery theory provides a method to classify –dimensional manifolds up to diffeomorphism given their homotopy types and . In Kreck’s modified version, it suffices to know the normal homotopy type of their –skeletons. While the obstructions in the original theory live in Wall’s –groups, the modified obstructions are elements in certain monoids . Unlike the –groups, the Kreck monoids are not well-understood.
We present three obstructions to help analyze for a ring . Firstly, if 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 is elementary. Alternatively, there is an obstruction in for certain flip-isomorphisms which is trivial if and only if is elementary.
"Even-dimensional $l$–monoids and $L$–theory." Algebr. Geom. Topol. 7 (1) 479 - 515, 2007. https://doi.org/10.2140/agt.2007.7.479