Surgery obstruction explained

In mathematics, specifically in surgery theory, the surgery obstructions define a map

\theta \colon l{N} (X) \to L _{n} (\pi _{1} (X)) 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

(f,b) \colon M \to X is normally cobordant to a homotopy equivalence if and only if the image \theta (f,b) =0 in L _{n} ( Z [\pi _{1} (X)]) .

Sketch of the definition

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

Consider a degree-one normal map

(f,b) \colon M \to X . 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 m > \lfloor n/ 2 \rfloor 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 K _{i} (\tilde M) : = ker \{f _{*} \colon H _{i} (\tilde M) → H _{i} (\tilde X)\} which one views as Z [\pi _{1} (X)] -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 Z [\pi _{1} (X)] -modules Poincaré duality K ^{n-i} (\tilde M) \cong K _{i} (\tilde M) , so one only has to watch half of them, that means those for which i \leq \lfloor n/ 2 \rfloor .

Any degree-one normal map can be made

\lfloor n/ 2 \rfloor -connected by the process called surgery below the middle dimension. This is the process of killing elements of K _{i} (\tilde M) for i < \lfloor n/ 2 \rfloor described here when we have p + q = n such that i = p < \lfloor n/ 2 \rfloor . After this is done there are two cases.

1. If

n =2 k then the only nontrivial homology group is the kernel K _{k} (\tilde M) : = ker \{f _{*} \colon H _{k} (\tilde M) → H _{k} (\tilde X)\} . It turns out that the cup-product pairings on M and X induce a cup-product pairing on K _{k(\tilde} M) . This defines a symmetric bilinear form in case k =2 l and a skew-symmetric bilinear form in case k =2 l +1 . It turns out that these forms can be refined to \varepsilon -quadratic forms, where \varepsilon = ( -1 ) ^{k} . These \varepsilon -quadratic forms define elements in the L-groups L _{n} (\pi _{1} (X)) .

2. If

n =2 k +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 L _{n} (\pi _{1} (X)) .

If the element

\theta (f,b) 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

K _{k} (\tilde M) possibly creates an element in K _{k-1} (\tilde M) when n = 2 k or in K _{k} (\tilde M) when n =2 k +1 . So this possibly destroys what has already been achieved. However, if \theta (f,b) 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 = 2 k +1 there is no obstruction.

If

n = 4 l then the surgery obstruction can be calculated as the difference of the signatures of M and X.

If

n = 4 l +2 then the surgery obstruction is the Arf-invariant of the associated kernel quadratic form over Z _{2}