Surgery obstruction

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In mathematics, specifically in surgery theory, the surgery obstructions define a map from the normal invariants to the L-groups which is in the first instance a set-theoretic map (that means not necessarily a homomorphism) with the following property when :

A degree-one normal map is normally cobordant to a homotopy equivalence if and only if the image in .

Sketch of the definition[edit]

The surgery obstruction of a degree-one normal map has a relatively complicated definition.

Consider a degree-one normal map . The idea in deciding the question whether it is normally cobordant to a homotopy equivalence is to try to systematically improve so that the map becomes -connected (that means the homotopy groups for ) for high . It is a consequence of Poincaré duality that if we can achieve this for then the map already is a homotopy equivalence. The word systematically above refers to the fact that one tries to do surgeries on to kill elements of . In fact it is more convenient to use homology of the universal covers to observe how connected the map is. More precisely, one works with the surgery kernels which one views as -modules. If all these vanish, then the map is a homotopy equivalence. As a consequence of Poincaré duality on and there is a -modules Poincaré duality , so one only has to watch half of them, that means those for which .

Any degree-one normal map can be made -connected by the process called surgery below the middle dimension. This is the process of killing elements of for described here when we have such that . After this is done there are two cases.

1. If then the only nontrivial homology group is the kernel . It turns out that the cup-product pairings on and induce a cup-product pairing on . This defines a symmetric bilinear form in case and a skew-symmetric bilinear form in case . It turns out that these forms can be refined to -quadratic forms, where . These -quadratic forms define elements in the L-groups .

2. If the definition is more complicated. Instead of a quadratic form one obtains from the geometry a quadratic formation, which is a kind of automorphism of quadratic forms. Such a thing defines an element in the odd-dimensional L-group .

If the element is zero in the L-group surgery can be done on to modify to a homotopy equivalence.

Geometrically the reason why this is not always possible is that performing surgery in the middle dimension to kill an element in possibly creates an element in when or in when . So this possibly destroys what has already been achieved. However, if is zero, surgeries can be arranged in such a way that this does not happen.


In the simply connected case the following happens.

If there is no obstruction.

If then the surgery obstruction can be calculated as the difference of the signatures of M and X.

If then the surgery obstruction is the Arf-invariant of the associated kernel quadratic form over .