Re: Moebius Band is not homeomorphic with a Torus

David Marcus <DavidMarcus@xxxxxxxxxxxxxx> writes:

Narasimham wrote:
....
David Marcus wrote:
....
However, two surfaces can be homotopic without being homeomorphic.

You have to give some more examples (of course other than the present
one) and its logic, if there is something that can be brought in here.

Do you mean you want an example of surfaces that are homotopic, but not
homeomorphic?

A nice example, which ought to (but probably won't) clear up N.'s confusion,
is the following. Take that inner tube he had a while ago, and rip a
patch off it. This physical object, which many people would be content
to abstract to S, a torus with an open disc removed--i.e., an orientable
surface of genus 1 with a single boundary component--*does* have thickness,
and so it would appear that N. would like to abstract it differently,
as the "square tube" (if I understand his term correctly) of that
surface, in other words, as a trivial I-bundle over S, which is a
3-dimensional solid with piecewise smooth boundary comprising three
parts--a (very narrow) annulus (which is the restriction of the I-bundle
to the boundary of S) and two copies of S (the "inside" and "outside"
surfaces of the inner tube). Of course homeomorphisms don't care
about corners, so topologically this solid is a "handlebody of genus 2",
bounded by a surface F of genus 2 (with empty boundary); the residuum
of the corners is a decomposition (what Gabai calls a "suturing") of
F into an annulus and the two (as it happens, homeomorphic) components
of its complement.

Now start again, this time from a flat disk from the interior of which
two smaller open disks with disjoint closures have been removed. A
physical counterpart of this might be a mask with two eye-holes (and
no mouth or nostrils), or a pair of underpants (with no fly). Again,
many of us would be content to abstract the mask or pants to S',
the original orientable surface of genus 0 with three boundary components;
but N., presumably, would once again take the flat tube. This time
we *again* get a 3-dimensinal handlebody of genus 2, but now the
sutures are different: there are 3 of them.

The process of replacing S or S' with a tribial I-bundle over it does not
change the homotopy type, and indeed creates two homeomorphic total spaces;
yet (of course!) S is not homeomorphic to S'.

Put differently, cancellation doesn't work for homeomorphism types:
you can have A x X = B x X with X non-empty (even, like I, contractible)
and yet not have A = B.

Lee Rudolph
.