3-manifold

Builds on Topological manifold Hausdorff space Homeomorphism

Definition

A $3$ -manifold is a topological space $M$ such that:

$M$ is Hausdorff;
$M$ is second countable;
every point of $M$ has an open neighbourhood homeomorphic to an open subset of $\R^3$ .

Equivalently, it is a topological manifold of dimension $3$ . A $3$ -manifold with boundary allows open subsets of the closed half-space $\set{x \in \R^3 : x_3 \ge 0}$ in clause 3, and a closed $3$ -manifold is a compact one without boundary.

Remarks

Example: the $3$ -torus $T^3 = \R^3 / \Z^3$ is a closed $3$ -manifold. The quotient map $\R^3 \to T^3$ restricted to any open cube of side $1/2$ is a homeomorphism onto an open subset, and $T^3$ is Hausdorff, second countable and compact.

Non-example: the cone $C = (T^2 \times [0, 1]) / (T^2 \times \set{0})$ on the torus fails clause 3 at the cone point $v$ . The local homology group $H_3(C, C \setminus \set{v})$ is $\tilde H_2(T^2) \cong \Z$ , which matches $\R^3$ , but $H_2(C, C \setminus \set{v}) \cong \tilde H_1(T^2) \cong \Z^2$ , while at a point of $\R^3$ this group is $0$ .

Other standard examples are the $3$ -sphere $S^3$ , the lens spaces and the Poincaré homology sphere.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Hausdorff space concept
Homeomorphism concept
Topological manifold concept
3-manifold definition

Tag	`Z5M5ESDBGK`
URI	`m2://Z5M5ESDBGK`
Kind	definition
Writing	W0 stub
MSC	57K30
Defines	3-manifold
Uses	Topological manifold, Hausdorff space, Homeomorphism
Changed	2 November 2026

3-manifold

Definition

Remarks

Reading path

Further reading

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