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 n \geq 5: A degree-one normal map is normally cobordant to a homotopy equivalence if and only if the image in.

Sketch of the definition

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 (f,b) so that the map f becomes m-connected (that means the homotopy groups \pi_* (f)=0 for * \leq m) for high m. It is a consequence of Poincaré duality that if we can achieve this for then the map f already is a homotopy equivalence. The word systematically above refers to the fact that one tries to do surgeries on M to kill elements of \pi_i (f). In fact it is more convenient to use homology of the universal covers to observe how connected the map f is. More precisely, one works with the surgery kernels which one views as -modules. If all these vanish, then the map f is a homotopy equivalence. As a consequence of Poincaré duality on M and X 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 p+q = n such that. After this is done there are two cases.

1. If n=2k then the only nontrivial homology group is the kernel. It turns out that the cup-product pairings on M and X induce a cup-product pairing on. This defines a symmetric bilinear form in case k=2l and a skew-symmetric bilinear form in case k=2l+1. It turns out that these forms can be refined to \varepsilon-quadratic forms, where. These \varepsilon-quadratic forms define elements in the L-groups.
2. If n=2k+1 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 M to modify f 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 n = 2k or in when n=2k+1. 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.

Example

In the simply connected case the following happens. If n = 2k+1 there is no obstruction. If n = 4l then the surgery obstruction can be calculated as the difference of the signatures of M and X. If n = 4l+2 then the surgery obstruction is the Arf-invariant of the associated kernel quadratic form over.