# Surgery obstruction

In mathematics, specifically in surgery theory, the surgery obstructions define a map $\theta \colon {\mathcal {N}}(X)\to L_{n}(\pi _{1}(X))$ 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 $n\geq 5$ :

## Sketch of the definition

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

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 $L_{n}(\pi _{1}(X))$ .

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 $n=2k$ or in $K_{k}({\tilde {M}})$ when $n=2k+1$ . 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 $n=4l$ 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 $\mathbb {Z} _{2}$ .