Surgery obstruction

Short description: Map from the normal invariants to the L-groups

In mathematics, specifically in surgery theory, the surgery obstructions define a map θ:𝒩(X)->Ln(π1(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≥5:

A degree-one normal map (f,b):M->X is normally cobordant to a homotopy equivalence if and only if the image θ(f,b)=0 in Ln(ℤ[π1(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):M->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 π*(f)=0 for *≤m) for high m. It is a consequence of Poincaré duality that if we can achieve this for m>⌊n/2⌋ 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 πi(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 Ki(M~):=ker{f*:Hi(M~)->Hi(X~)} which one views as ℤ[π1(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[π1(X)]-modules Poincaré duality Kn−i(M~)≅Ki(M~), so one only has to watch half of them, that means those for which i≤⌊n/2⌋.

Any degree-one normal map can be made ⌊n/2⌋-connected by the process called surgery below the middle dimension. This is the process of killing elements of Ki(M~) for i<⌊n/2⌋ described here when we have p+q=n such that i=p<⌊n/2⌋. After this is done there are two cases.

1\. If n=2k then the only nontrivial homology group is the kernel Kk(M~):=ker{f*:Hk(M~)->Hk(X~)}. It turns out that the cup-product pairings on M and X induce a cup-product pairing on Kk(M~). 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 ε-quadratic forms, where ε=(−1)k. These ε-quadratic forms define elements in the L-groups Ln(π1(X)).

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 Ln(π1(X)).

If the element θ(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 Kk(M~) possibly creates an element in Kk−1(M~) when n=2k or in Kk(M~) when n=2k+1. So this possibly destroys what has already been achieved. However, if θ(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=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 ℤ2.

## References

* Browder, William (1972), Surgery on simply-connected manifolds, Berlin, New York: Springer-Verlag
* Lück, Wolfgang (2002), A basic introduction to surgery theory, ICTP Lecture Notes Series 9, Band 1, of the school "High-dimensional manifold theory" in Trieste, May/June 2001, Abdus Salam International Centre for Theoretical Physics, Trieste 1-224, http://wwwmath.uni-muenster.de/u/lueck/publ/lueck/ictp.pdf
* Ranicki, Andrew (2002), Algebraic and Geometric Surgery, Oxford Mathematical Monographs, Clarendon Press, ISBN 978-0-19-850924-0, http://www.maths.ed.ac.uk/~aar/books/index.htm
* Wall, C. T. C. (1999), Surgery on compact manifolds, Mathematical Surveys and Monographs, 69 (2nd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0942-6

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