Surgery obstruction

In mathematics, specifically in surgery theory, the surgery obstructions define a map formula_1 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 formula_2:
A degree-one normal map formula_3 is normally cobordant to a homotopy equivalence if and only if the image formula_4 in formula_5.
Sketch of the definition.
The surgery obstruction of a degree-one normal map has a relatively complicated definition.
Consider a degree-one normal map formula_3. The idea in deciding the question whether it is normally cobordant to a homotopy equivalence is to try to systematically improve formula_7 so that the map formula_8 becomes formula_9-connected (that means the homotopy groups formula_10 for formula_11) for high formula_9. It is a consequence of Poincaré duality that if we can achieve this for formula_13 then the map formula_8 already is a homotopy equivalence. The word "systematically" above refers to the fact that one tries to do surgeries on formula_15 to kill elements of formula_16. In fact it is more convenient to use homology of the universal covers to observe how connected the map formula_8 is. More precisely, one works with the surgery kernels formula_18 which one views as formula_19-modules. If all these vanish, then the map formula_8 is a homotopy equivalence. As a consequence of Poincaré duality on formula_15 and formula_22 there is a formula_19-modules Poincaré duality formula_24, so one only has to watch half of them, that means those for which formula_25.
Any degree-one normal map can be made formula_26-connected by the process called surgery below the middle dimension. This is the process of killing elements of formula_27 for formula_28 described here when we have formula_29 such that formula_30. After this is done there are two cases.
1. If formula_31 then the only nontrivial homology group is the kernel formula_32. It turns out that the cup-product pairings on formula_15 and formula_22 induce a cup-product pairing on formula_35. This defines a symmetric bilinear form in case formula_36 and a skew-symmetric bilinear form in case formula_37. It turns out that these forms can be refined to formula_38-quadratic forms, where formula_39. These formula_38-quadratic forms define elements in the L-groups formula_41.
2. If formula_42 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 formula_41.
If the element formula_44 is zero in the L-group surgery can be done on formula_15 to modify formula_8 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 formula_47 possibly creates an element in formula_48 when formula_49 or in formula_50 when formula_42. So this possibly destroys what has already been achieved. However, if formula_44 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 formula_53 there is no obstruction.
If formula_54 then the surgery obstruction can be calculated as the difference of the signatures of M and X.
If formula_55 then the surgery obstruction is the Arf-invariant of the associated kernel quadratic form over formula_56.