---
tag: FZH5T5340Q
kind: statement
role: theorem
name: Prime decomposition of 3-manifolds
msc: [57K30]
uses: [R20TB2KEQQ, Y0K0K5NGWW, T62369RFH6, 702SNFQG0W]
about: [R20TB2KEQQ, Y0K0K5NGWW]
sources:
  - url: https://en.wikipedia.org/wiki/Prime_decomposition_of_3-manifolds
same_as:
  wikidata: Q3527192
  wikipedia: Prime decomposition of 3-manifolds
---

## Statement

A closed connected $3$-manifold $P$ is *prime* if $P$ is not homeomorphic to $S^3$ and the following holds.
Whenever $P$ is homeomorphic to a [[connected sum|Y0K0K5NGWW]] $A \# B$ of closed $3$-manifolds, one of $A$ and $B$ is homeomorphic to $S^3$.

Let $M$ be a closed connected [[orientable|702SNFQG0W]] [[3-manifold|R20TB2KEQQ]] not homeomorphic to the [[3-sphere|T62369RFH6]] $S^3$.
Then:

1. $M$ is homeomorphic to a connected sum $P_1 \# \cdots \# P_k$ of finitely many prime $3$-manifolds.
2. The summands are unique.
   If $M$ is also homeomorphic to $Q_1 \# \cdots \# Q_l$ with each $Q_j$ prime, then $k = l$, and after reordering $P_i$ is homeomorphic to $Q_i$ for each $i$.

## Remarks

Existence is due to Kneser and uniqueness to Milnor.
For existence, one cuts $M$ along embedded $2$-spheres that do not bound balls and caps the pieces with balls; Kneser's finiteness argument shows that this process stops.
Uniqueness comes from an innermost-disk argument that makes two systems of splitting spheres disjoint.
A closed orientable prime $3$-manifold is either $S^2 \times S^1$ or irreducible, meaning that every embedded $2$-sphere bounds an embedded $3$-ball.
For example, $\mathbb{RP}^3 \# \mathbb{RP}^3$ has the two prime summands $\mathbb{RP}^3$, and no other decomposition.
Without orientability, uniqueness fails in this form: for a non-orientable $P$, the sums $P \# (S^2 \times S^1)$ and $P \# (S^2 \tilde{\times} S^1)$ are homeomorphic, where $S^2 \tilde{\times} S^1$ is the non-orientable $S^2$-bundle over the circle.

## Explanations

### L1 human

Every closed orientable $3$-manifold breaks up, in exactly one way, into basic pieces glued together by connected sums.
It is the analogue of factoring an integer into primes.
To understand all $3$-manifolds, it is enough to understand the prime ones.

### L2 human

The theorem reduces the classification of closed orientable $3$-manifolds to irreducible ones and $S^2 \times S^1$.
For irreducible manifolds, the next step is the JSJ decomposition along incompressible tori.
These two decompositions are the first stages of the geometrization theorem.
On the algebra side, the fundamental group of a connected sum is the free product of the fundamental groups of the summands.