Manifolds and cell complexes Cobordism and surgery Theorem

Surgery exact sequence

Builds on Surgery theory Topological manifold Simple homotopy equivalence Fundamental group

Statement

Let $X$ be a closed connected topological manifold of dimension $n \ge 5$ , with fundamental group $\pi$ and orientation character $w \colon \pi \to \set{\pm 1}$ . Consider simple homotopy equivalences $f \colon M \to X$ from closed topological manifolds $M$ . Identify $f$ with $f' \colon M' \to X$ when there is a homeomorphism $h \colon M \to M'$ with $f' \circ h$ homotopic to $f$ . The structure set $\mathcal{S}(X)$ is the set of classes. Let $\mathcal{N}(X)$ be the set of normal bordism classes of degree one normal maps to $X$ . Let $\mathcal{N}(X \times [0,1], \partial)$ be the same set for $X \times [0,1]$ relative to the boundary. Let $L_m(\Z[\pi], w)$ be the simple surgery obstruction groups of Wall. Then there is an exact sequence of pointed sets: $\mathcal{N}(X \times [0,1], \partial) \xrightarrow{\ \theta\ } L_{n+1}(\Z[\pi], w) \xrightarrow{\ \partial\ } \mathcal{S}(X) \xrightarrow{\ \eta\ } \mathcal{N}(X) \xrightarrow{\ \theta\ } L_n(\Z[\pi], w).$ Here $\eta$ sends a homotopy equivalence to its normal invariant and $\theta$ is the surgery obstruction. The map $\partial$ comes from an action of $L_{n+1}(\Z[\pi], w)$ on $\mathcal{S}(X)$ . Exactness at $\mathcal{S}(X)$ means the following. Two elements have the same image under $\eta$ if and only if they lie in the same orbit of this action.

Remarks

Idea of the proof: surgeries below the middle dimension make a degree one normal map to $X$ highly connected. By Wall's theorem, the remaining obstruction to reach a homotopy equivalence is $\theta$ . The action of $L_{n+1}$ realises an element as the obstruction of a normal bordism from a given homotopy equivalence; the s-cobordism theorem turns trivial obstructions into homeomorphisms. In the topological category $\mathcal{N}(X)$ is in bijection with the set $[X, G/\mathrm{TOP}]$ of homotopy classes of maps from $X$ to the space $G/\mathrm{TOP}$ .

The sequence at work: for $X = S^n$ with $n \ge 5$ , the structure set is a single point, which is the topological generalized Poincaré conjecture. For simply connected $X$ the groups $L_n(\Z)$ are $\Z, 0, \Z/2, 0$ for $n \equiv 0, 1, 2, 3 \pmod 4$ , and the obstruction in $L_{4k}(\Z) = \Z$ is a multiple of a difference of signatures.

Going deeper

For a student

The sequence is the main tool of the Browder–Novikov–Sullivan–Wall surgery theory. To classify manifolds homotopy equivalent to $X$ , one computes the normal invariants, the surgery obstruction map and the action of $L_{n+1}$ . It is the setting of the Borel conjecture, which says that for aspherical $X$ the structure set is a single point. The algebraic theory of these groups is L-theory.

Reading path

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Tag	`Q7C480HQ9J`
URI	`m2://Q7C480HQ9J`
Kind	statement, theorem
Writing	W0 stub
MSC	57R67, 57R65, 57N65
Status	This copy of the site has no build data. Run `m2 status Q7C480HQ9J`.
About	Surgery theory
Uses	Surgery theory, Topological manifold, Simple homotopy equivalence, Fundamental group
Same as	Wikidata Q7646195 Wikipedia Surgery exact sequence
Changed	2 November 2026