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December 21st, 2012, 03:38 AM   #1
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A question about topological invariant

Hi,

I'm a physicist and have a question. Is it possible to change the dimensionality of a manifold while its topology remains unchanged? If yes, what kind of topological invariant supports such a transformation?

Thank you all.
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December 21st, 2012, 06:56 AM   #2
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Re: A question about topological invariant

What do you mean by "change the dimensionality of a manifold"?
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December 21st, 2012, 07:35 AM   #3
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Re: A question about topological invariant

changing number of dimensions of manifold; I search for a topological invariant, if exists, which allows to transform a n-dimensional manifold to a n+1-dimensional one.
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December 21st, 2012, 10:28 AM   #4
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Re: A question about topological invariant

I think that I find the answer. Topological invariants remain unchanged under homeomorphism. however, it is impossible to define a homeomorphism between two manifolds of different dimensions since such a mapping is not one-to-one and therefore the mapping is not invertible (while a homeomorphism must be). If this argument is true, it then cannot be defined any topological invariant when the number of dimension of manifold changes.
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