Surgery obstruction

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

A degree-one normal map ${\displaystyle (f,b)\colon M\to X}$ is normally cobordant to a homotopy equivalence if and only if the image ${\displaystyle \theta (f,b)=0}$ in ${\displaystyle L_{n}(\mathbb {Z} [\pi _{1}(X)])}$.