---
tag: MHHN8386TG
kind: concept
name: Surgery theory
msc: [57R65, 57R67]
sources:
  - url: https://en.wikipedia.org/wiki/Surgery_theory
same_as:
  wikidata: Q2963975
  wikipedia: Surgery theory
---

## Summary

Surgery is an operation that cuts out a thickened sphere $S^p \times D^q$ from a manifold and glues in $D^{p+1} \times S^{q-1}$ along the common boundary.
The result is a new manifold, [[cobordant|T8BVGPPSZP]] to the old one.
Surgery theory uses this operation to decide whether a space is homotopy equivalent to a manifold.
It also classifies the manifolds within a homotopy type.
In dimensions at least $5$ it reduces these questions to homotopy theory and to algebra in the surgery obstruction groups.

## Explanations

### L1 human

Surgery is a cut-and-paste move on a manifold.
In a surface, cut out two small discs and glue in a tube between the two holes: the surface gains a handle.
This is surgery on a $0$-sphere, the pair of centres of the discs.
Repeating such moves, one can kill unwanted loops and holes and steer a manifold towards a desired shape.
Surgery theory is the systematic study of when this steering succeeds.