---
tag: Q7C480HQ9J
kind: statement
role: theorem
name: Surgery exact sequence
msc: [57R67, 57R65, 57N65]
uses: [MHHN8386TG, QQQ9B1DK4P, AR8SP73SXX, S1XK3HXHFX]
about: [MHHN8386TG]
sources:
  - url: https://en.wikipedia.org/wiki/Surgery_exact_sequence
same_as:
  wikidata: Q7646195
  wikipedia: Surgery exact sequence
---

## Statement

Let $X$ be a closed connected [[topological manifold|QQQ9B1DK4P]] of dimension $n \ge 5$, with [[fundamental group|S1XK3HXHFX]] $\pi$ and orientation character $w \colon \pi \to \set{\pm 1}$.
Consider [[simple homotopy equivalences|AR8SP73SXX]] $f \colon M \to X$ 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|E1E5ABP89W]] 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.

## Explanations

### L1 human

Suppose a space looks like a manifold to homotopy theory.
How many genuinely different manifolds have this homotopy type?
The surgery exact sequence answers this by fitting the set of such manifolds between two computable pieces.
One piece is homotopy theory of a classifying space, the other is algebra of quadratic forms over the group ring of the fundamental group.

### L2 human

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|X057MN9YY9]], which says that for aspherical $X$ the structure set is a single point.
The algebraic theory of these groups is L-theory.