In mathematics, specifically in surgery theory, the surgery obstructions from the normal invariants to the L-groupss which is in the first instance a set-theoretic map (that means not necessarily a homomorphism:

is normally cobordant to a homotopy equivalence.

Sketch of the definition

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

. It is a consequence of Poincaré duality already is a homotopy equivalence. The word systematically. In fact it is more convenient to use homology of the universal covers is. More precisely, one works with the surgery kernels.

described here. After this is done there are two cases.

. This defines a symmetric bilinear form.

the definition is more complicated. Instead of a quadratic form.

to a homotopy equivalence.

is zero, surgeries can be arranged in such a way that this does not happen.

Example

In the simply connected case the following happens.

there is no obstruction.

then the surgery obstruction can be calculated as the difference of the signatures of M and X.

.

References