Surgery theory

Builds on Smooth manifold n-sphere Cobordism

Definition

Let $M$ be a smooth manifold of dimension $n$ , and let $p, q \ge 0$ be integers with $p + q = n$ . Let $\varphi \colon S^p \times D^q \to M$ be a smooth embedding, where $S^p$ is the $p$ -sphere and $D^q$ the closed $q$ -disc. The surgery on $M$ along $\varphi$ is the following manifold: $M' = \bigl(M \setminus \varphi(S^p \times \operatorname{int} D^q)\bigr) \cup_{\varphi} \bigl(D^{p+1} \times S^{q-1}\bigr).$ Here the two pieces are glued along their common boundary $S^p \times S^{q-1}$ by $\varphi$ , and corners are smoothed. The number $p$ is the index of the surgery. Surgery theory is the study of manifolds through such operations. Its central problem is to turn a degree one normal map into a homotopy equivalence by a sequence of surgeries.

Remarks

The trace of the surgery is $W = (M \times [0, 1]) \cup_{\varphi \times \set{1}} (D^{p+1} \times D^q)$ , a $(p+1)$ -handle attached to $M \times [0,1]$ . It is a cobordism from $M$ to $M'$ , and every compact cobordism is obtained by a finite sequence of such traces. Example: let $M = S^1 \times S^1$ and $\varphi$ an embedding of $S^1 \times D^1$ as an annulus around the meridian circle. Removing the open annulus leaves a cylinder $S^1 \times [0,1]$ , and gluing in two discs $D^2 \times S^0$ caps it to $S^2$ . So surgery of index $1$ on a torus along a meridian gives the sphere. Non-example: let $C$ be the core circle of a Möbius band inside $\R P^2$ . Every tubular neighbourhood of $C$ is a Möbius band and not an annulus, so $C$ has no neighbourhood of the form $S^1 \times D^1$ . So there is no embedding $\varphi$ as in the definition with core $C$ , and no surgery along $C$ is defined.

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Cobordism concept
Smooth manifold concept
n-sphere object
Surgery theory definition

Tag	`NHY3HE7RCN`
URI	`m2://NHY3HE7RCN`
Kind	definition
Writing	W0 stub
MSC	57R65, 57R67
Defines	Surgery theory
Uses	Smooth manifold, n-sphere, Cobordism
Changed	2 November 2026

Surgery theory

Definition

Remarks

Reading path

Further reading

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