proof of invariance of dimension

proof of invariance of dimension

An application of the

invariance of dimension

theorem shows that

ℝ

n

is

homeomorphic

to

ℝ

m

if and only if

m

=

n

. Already this is a difficult question. (We will assume

n

≤

m

throughout this article.)

Simple arguments suffice for small

dimensions

.

•

If

n

=

0

cardinality is sufficient: there can be no bijection between

ℝ

0

=

{

0

}

and

ℝ

n

,

m

>

0

, as the latter is uncountable.

•

If

n

=

1

, then suppose

f

:

ℝ

→

ℝ

m

is a

homeomorphism

with

m

>

1

. Then certainly the following

restriction

of

f

f

:

ℝ

-

{

0

}

→

ℝ

m

-

{

f

⁢

(

0

)

}

is also a homeomorphism. Yet, as

m

>

1

,

ℝ

m

-

{

f

⁢

(

0

)

}

is (path)

connected

but

ℝ

-

{

0

}

is not connected. Thus this restriction of

f

cannot be a homeomorphism so indeed the original

f

could not be a homeomorphism.

Unfortunately neither of these two arguments extends well to the cases where

n

,

m

>

1

. Indeed even the case for

n

=

1

requires a reasonable amount of work to fill in the details. However, the latter approach does provide the necessary hint for a full solution.

To solve the problem outright depends on algebraic invariants from

homology

, a
surprisingly big hammer for such a basic topological question. But the conceptual steps are still basic, and we will attempt to highlight them in our exposition of the proof.

Let

U

and

V

be non-empty

open subsets

of

ℝ

n

and

ℝ

m

respectively. Assume that

f

:

U

→

V

is a homeomorphism.

Choose a point

x

∈

U

(akin to the point we removed when

n

=

1

.) Then consider the

relative homology groups

H

i

⁢

(

U

,

U

-

{

x

}

)

,

i

∈

ℕ

. As

U

is open we may apply the Excision Theorem (axiom) to claim

H

i

⁢

(

U

,

U

-

{

x

}

)

≅

H

i

⁢

(

ℝ

n

,

ℝ

n

-

{

x

}

)

– basically, to look at a punctured

open disk

it to look at a punctured

ℝ

n

. Now we look at the

induced

long exact sequence

from the relative pair

(

ℝ

n

,

ℝ

n

-

{

x

}

)

and find

H

i

⁢

(

ℝ

n

,

ℝ

n

-

{

x

}

)

is

isomorphic

to the reduced homology

H

~

i

⁢

(

ℝ

n

-

{

x

}

)

. But

ℝ

n

-

{

x

}

contracts to the sphere

S

n

-

1

– and homologoy preserves

homotopy type

– so we now have

H

i

⁢

(

U

,

U

-

{

x

}

)

≅

H

i

⁢

(

S

n

-

1

)

. (Puncture a disk, and it deflates to a sphere of lower dimension.)

Now it is an exercise in homology to prove that

H

i

⁢

(

S

n

-

1

)

=

0

if

i

≠

0

,

n

-

1

and

ℤ

otherwise. In particular we are using the fact that the invariance of dimension of spheres is (more) easily established by the homology groups.

We now repeat the process with

V

. If

U

and

V

are indeed homeomorphic, then this process will result in isomorphic homology groups for every

i

∈

ℕ

. In particular,

ℤ

≅

H

m

-

1

⁢

(

S

m

-

1

)

≅

H

m

-

1

⁢

(

V

,

V

-

{

f

⁢

(

x

)

}

)

≅

H

m

-

1

⁢

(

U

,

U

-

{

x

}

)

≅

H

m

-

1

⁢

(

S

n

-

1

)

.

Thus either

m

=

1

which implies

n

=

0

,

1

as

n

≤

m

, or

m

=

n

. If

n

=

0

we have already seen

m

=

n

. So the result stands for all

m

,

n

.

For a detailed accounting of this theorem together with the necessary lemmas refer to:

Allen Hatcher,

Algebraic Topology

, Cambridge University Press, Cambridge, 2002. Available on-line at:

http://www.math.cornell.edu/ hatcher/AT/ATpage.html

http://www.math.cornell.edu/ hatcher/AT/ATpage.html

Title

proof of invariance of dimension

Canonical name

ProofOfInvarianceOfDimension

Date of creation

2013-03-22 15:56:39

Last modified on

2013-03-22 15:56:39

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

8

Author

Algeboy (12884)

Entry type

Proof

Classification

msc 55-00

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