proof of invariance of dimension

An application of the invariance of dimension theorem shows that n is homeomorphicMathworldPlanetmath to m if and only if m=n. Already this is a difficult question. (We will assume nm throughout this article.)

Simple arguments suffice for small dimensionsMathworldPlanetmath.

  • If n=0 cardinality is sufficient: there can be no bijection between 0={0} and n, m>0, as the latter is uncountable.

  • If n=1, then suppose f:m is a homeomorphism with m>1. Then certainly the following restrictionPlanetmathPlanetmathPlanetmath of f


    is also a homeomorphism. Yet, as m>1, m-{f(0)} is (path) connectedPlanetmathPlanetmath but -{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 homologyMathworldPlanetmathPlanetmathPlanetmath, 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 n and m respectively. Assume that f:UV is a homeomorphism.

Choose a point xU (akin to the point we removed when n=1.) Then consider the relative homology groups Hi(U,U-{x}), i. As U is open we may apply the Excision Theorem (axiom) to claim Hi(U,U-{x})Hi(n,n-{x}) – basically, to look at a punctured open disk it to look at a punctured n. Now we look at the induced long exact sequence from the relative pair (n,n-{x}) and find Hi(n,n-{x}) is isomorphicPlanetmathPlanetmath to the reduced homology H~i(n-{x}). But n-{x} contracts to the sphere Sn-1 – and homologoy preserves homotopy typeMathworldPlanetmath – so we now have Hi(U,U-{x})Hi(Sn-1). (Puncture a disk, and it deflates to a sphere of lower dimension.)

Now it is an exercise in homology to prove that Hi(Sn-1)=0 if i0,n-1 and 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. In particular,


Thus either m=1 which implies n=0,1 as nm, 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: hatcher/AT/ATpage.html 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