Surgery obstruction groups

Remarks

Wall's theorem: let $X$ be a finite Poincaré complex of dimension $n \ge 5$ with fundamental group $\pi$ and orientation character $w$. A degree one normal map $f \colon M \to X$ from a closed manifold has an obstruction $\sigma(f) \in L^s_n(\Z[\pi], w)$. It vanishes if and only if $f$ is normally bordant to a simple homotopy equivalence. Example: for $\pi$ trivial, $L_n(\Z) \cong \Z, 0, \Z/2, 0$ for $n \equiv 0, 1, 2, 3 \pmod 4$. For $n = 4k$ the obstruction is $(\operatorname{sign} M - \operatorname{sign} X)/8$, and for $n = 4k + 2$ it is an Arf invariant. The groups depend on $\pi$: by Shaneson's splitting, $L^s_n(\Z[\pi \times \Z]) \cong L^s_n(\Z[\pi]) \oplus L^h_{n-1}(\Z[\pi])$ for trivial $w$, so $L_1(\Z[\Z]) \cong \Z$ while $L_1(\Z) = 0$. They are the algebraic terms of the surgery exact sequence.