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
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
-connected (that means the homotopy groups
It is a consequence of Poincaré duality that if we can achieve this for
The word systematically above refers to the fact that one tries to do surgeries on
In fact it is more convenient to use homology of the universal covers to observe how connected the map
More precisely, one works with the surgery kernels
As a consequence of Poincaré duality on
, 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.
then the only nontrivial homology group is the kernel
It turns out that the cup-product pairings on
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 define elements in the L-groups
Such a thing defines an element in the odd-dimensional L-group
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
So this possibly destroys what has already been achieved.
In the simply connected case the following happens.
then the surgery obstruction can be calculated as the difference of the signatures of M and X.
then the surgery obstruction is the Arf-invariant of the associated kernel quadratic form over