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

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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.

Example

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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  .

References

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  • Browder, William (1972), Surgery on simply-connected manifolds, Berlin, New York: Springer-Verlag, MR 0358813
  • Lück, Wolfgang (2002), A basic introduction to surgery theory (PDF), 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
  • Ranicki, Andrew (2002), Algebraic and Geometric Surgery, Oxford Mathematical Monographs, Clarendon Press, ISBN 978-0-19-850924-0, MR 2061749
  • Wall, C. T. C. (1999), Surgery on compact manifolds, Mathematical Surveys and Monographs, vol. 69 (2nd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0942-6, MR 1687388