proof of invariance of dimension
Simple arguments suffice for small dimensions.
If cardinality is sufficient: there can be no bijection between and , , as the latter is uncountable.
Unfortunately neither of these two arguments extends well to the cases where . Indeed even the case for 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 and be non-empty open subsets of and respectively. Assume that is a homeomorphism.
Choose a point (akin to the point we removed when .) Then consider the relative homology groups , . As is open we may apply the Excision Theorem (axiom) to claim – basically, to look at a punctured open disk it to look at a punctured . Now we look at the induced long exact sequence from the relative pair and find is isomorphic to the reduced homology . But contracts to the sphere – and homologoy preserves homotopy type – so we now have . (Puncture a disk, and it deflates to a sphere of lower dimension.)
Now it is an exercise in homology to prove that if 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 . If and are indeed homeomorphic, then this process will result in isomorphic homology groups for every . In particular,
Thus either which implies as , or . If we have already seen . So the result stands for all .
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.htmlhttp://www.math.cornell.edu/ hatcher/AT/ATpage.html
|Title||proof of invariance of dimension|
|Date of creation||2013-03-22 15:56:39|
|Last modified on||2013-03-22 15:56:39|
|Last modified by||Algeboy (12884)|