orientation

orientation

There are many definitions of an

orientation

of a

manifold

. The most
general, in the sense that it doesn’t require any extra
on the manifold, is based on
(co-)

homology

theory. For this article manifold means a

connected

,

topological

manifold possibly with

boundary

.

Theorem 1

.

Let

M

be a closed,

n

–dimensional

manifold

. Then

H

n

⁢

(

M

;

Z

)

the top dimensional
homology group of

M

, is either trivial (

{

0

}

) or

isomorphic

to

Z

.

Definition 2

.

A closed

n

–manifold is called

orientable

if its top
homology group is isomorphic to the integers.
An

orientation

of

M

is a choice of a particular

isomorphism

𝔬

:

ℤ

→

H

n

⁢

(

M

;

ℤ

)

.

An oriented manifold is a (necessarily orientable) manifold

M

endowed with
an orientation.
If

(

M

,

𝔬

)

is an oriented manifold then

𝔬

⁢

(

1

)

is called
the

fundamental class

of

M

, or the

orientation class

of

M

, and is denoted
by

[

M

]

.

Remark 3

.

Notice that since

ℤ

has exactly two

automorphisms

an

orientable manifold

admits two possible
orientations.

Remark 4

.

The above definition could be given using

cohomology

instead of homology.

The top dimensional homology of a non-closed manifold is always
trivial, so it is trickier to define orientation for those
beasts. One approach (which we will not follow) is to use special
kind of homology (for example relative to the boundary for

compact
manifolds

with boundary). The approach we follow defines (global)
orientation as

compatible

fitting together of local orientations. We
start with manifolds without boundary.

Theorem 5

.

Let

M

be an

n

-manifold without boundary and

x

∈

M

. Then the

relative homology group

H

n

⁢

(

M

,

M

∖

x

;

ℤ

)

≅

ℤ

Definition 6

.

Let

M

be an

n

-manifold and

x

∈

M

. An

orientation
of

M

at

x

is a choice of an isomorphism

𝔬

x

:

ℤ

→

H

n

⁢

(

M

,

M

∖

x

;

ℤ

)

.

to make precise the notion of nicely
fitting together of orientations at points, is to require that for
nearby points the orientations are defined in a
way.

Theorem 7

.

Let

U

be an

open subset

of

M

that is

homeomorphic

to

R

n

(e.g. the domain of a chart). Then,

H

n

⁢

(

M

,

M

∖

U

;

ℤ

)

≅

ℤ

.

Definition 8

.

Let

U

be an open subset of

M

that is homeomorphic
to

ℝ

n

. A

local orientation

of

M

on

U

is a choice
of an isomorphism

𝔬

U

:

H

n

⁢

(

M

,

M

∖

U

;

ℤ

)

→

ℤ

.

Now notice that with

U

as above and

x

∈

U

the inclusion

ı

x

U

:

M

∖

U

↪

M

∖

x

a map (actually isomorphism)

ı

x

⁣

*

U

:

H

n

⁢

(

M

,

M

∖

U

;

ℤ

)

→

H

n

⁢

(

M

,

M

∖

x

;

ℤ

)

and therefore a local orientation at

U

(by composing with the above
isomorphism) an orientation at each point

x

∈

U

. It is
to declare that all these
orientations fit nicely together.

Definition 9

.

Let

M

be a manifold with non-empty boundary,

∂

⁡

M

≠

∅

.

M

is called

orientable

if its double

M

^

:=

M

⁢

⋃

∂

⁡

M

M

is orientable, where

⋃

∂

⁡

M

denotes gluing along the boundary.

An orientation of

M

is determined by an orientation of

M

^

.

Title

orientation

Canonical name

Orientation1

Date of creation

2013-03-22 12:56:24

Last modified on

2013-03-22 12:56:24

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

15

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 57N99

Related topic

ThomClass

Defines

orientable

Defines

oriented

Defines

orientable manifold

Defines

oriented manifold

Defines

fundamental class

Defines

orientation class

Defines

local orientation

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