Orientation of a manifold

Builds on Topological manifold Relative homology Excision theorem Orientation of a real vector space Smooth manifold Möbius strip n-sphere

Definition

Let $M$ be a topological manifold of dimension $n \ge 1$ , and use integral relative homology. For each $x \in M$ , the excision theorem gives $H_n(M, M \setminus \set{x}) \cong H_n(\R^n, \R^n \setminus \set{0}) \cong \Z$ . A local orientation of $M$ at $x$ is a choice of generator $\mu_x$ of $H_n(M, M \setminus \set{x})$ . An orientation of $M$ is a function $x \mapsto \mu_x$ assigning a local orientation to each point, with the following local consistency. Each $x \in M$ has a chart neighbourhood $U \cong \R^n$ containing an open ball $B$ of finite radius about $x$ , and a class $\mu_B \in H_n(M, M \setminus B)$ . For every $y \in B$ , the image of $\mu_B$ in $H_n(M, M \setminus \set{y})$ is $\mu_y$ . $M$ is orientable if it has an orientation.

Remarks

A connected orientable manifold has exactly two orientations. For a smooth manifold, an orientation in this sense is the same as a choice of orientation of each tangent space $T_x M$ that is locally given by a chart. It is also the same as an atlas whose transition maps have positive Jacobian determinant. Replacing $\Z$ by a commutative ring $R$ gives $R$ -orientations; every manifold has a unique $\Z/2$ -orientation.

Example: the sphere $S^n$ is orientable. Fix a generator $\mu \in H_n(S^n) \cong \Z$ ; its image in each $H_n(S^n, S^n \setminus \set{y})$ is a generator, and one takes this image as $\mu_y$ . The condition holds with $\mu_B$ the image of $\mu$ in $H_n(S^n, S^n \setminus B)$ for every chart ball $B$ .

Non-example: the open Möbius strip is not orientable. Following a local orientation once around the central circle returns the opposite generator, so no continuous choice exists.

Reading path

To understand this page from the ground up, read these 43 pages in order. The foundations come first.

Barycentric subdivision concept
Closure concept
Element definition
Interior concept
Orientation of a real vector space concept
Proposition concept
Reduced homology concept
Relative homology concept
Singular homology concept
Smooth manifold concept
Topological manifold concept
n-sphere object
Set definition
Excision theorem statement
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Empty set object
Family of sets definition
Ordered pair definition
Intersection definition
Union definition
Set-builder notation concept
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Subset definition
Union of a family of sets definition
Uniqueness quantifier definition
Binary relation definition
Topological space definition
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Function definition
Equivalence relation definition
Preimage of a set definition
Equivalence class definition
Quotient set definition
Quotient topology definition
Möbius strip example
Orientation of a manifold definition

Tag	`MKBSS15J3Z`
URI	`m2://MKBSS15J3Z`
Kind	definition
Writing	W0 stub
MSC	57N65, 55M05
Defines	Orientation of a manifold
Uses	Topological manifold, Relative homology, Excision theorem, Orientation of a real vector space, Smooth manifold, Möbius strip, n-sphere
Changed	2 November 2026

Orientation of a manifold

Definition

Remarks

Reading path

Further reading

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