Since Stephen Hawking and Roger Penrose have successfully proven that the Universe began as a singularity (a single point at which density is infinite) and then expanded out to its present state, wouldn't this imply that the universe were simply connected? And, if it were simply connected, wouldn't that mean that the universe is homeomorphic to a disk? Also, if a 3 dimensional curve is drawn in space, then it can be contracted to a point - the very requirement for simply connectedness. It is difficult to imagine that the universe resides on the surface of a fourth dimensional torus (or any other multiply connected surface) because, if it did, then a 3 dimensional curve drawn on its surface would not be able to contract to a single point. Therefore, doesn't this rule out all multiply connected surfaces as being the geometry of the universe?

[Edited by unholynemesisx on 03/17/03 at 17:37]
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hello all ,

that goes more than a bit above my head, I must admit. I do not see why condensation is impossible with a fourth dimension.There might be other facets of conjointedness that are as yet unexplored.

It may well be that the definition of a point in geometry is in itself inadequate and not einstein's model of a fourth dimension. we have to accept that the concept of a point is pretty old, and that it is possible to have a complexly connected system even during condensation to a single point.

does that help one bit? i guess not

s'long,
alan

Basically, what I was getting at was this:

Imagine the surface of a donut (a torus.) If you put a circle on the donut, where the central point of the circle is not on the torus - its in the central hole of the donut. If you tried to contract that circle, you'd find that you couldn't do so. Correspondingly, on any multiply connected surface, of which the torus is an example, the you couldn't contract a circle to a point. Now, imagine that the universe, itself, were a multiply connected surface. That would mean, then, that a sphere, the 3 dimensional equivelent of a circle, if oriented in such a way, would not be able to contract to a point. My previous thinking was that since Stephen Hawking and Roger Penrose had proven that the universe had once been an infinitely dense point, this would be impossible. However, what I had not realized, is that although a circle on a multiply connected surface cannot be contracted through continous functions to a point, the torus, itself, has a limit at the point - thus, the problem is solved. However, if the universe were a multiply connected surface, then it would be possible to orient a sphere on the surface of the universe that could not be contracted to a point - that is, if the universe, itself, were not contracted to a point. The consequence of this is that in such a universe, if there was a spherical mass which was sufficiently large and oriented properly, then it could not be contracted to a point and remain in the universe. Thus, in such a universe, a sufficently large black hole could only form a singularity if it actually changed the topology of the universe. Thus, if it were assumed that the universe's topology were constant, then in a multiply connected universe black holes (or, more specifically, singularities) could not actually form. Thus, a great deal about the universe's topology could be discovered if the connectivity of the universe could be established - i.e. whether sufficiently large and properly oriented spheres can be contracted to a point.
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that Stephen Hawkings and Roger Penrose have not proven that the Universe began as a Singularity. It is unprovable. It may be a safe assumption based on observed evidence, and consistent with Big Bang theology, oops, I mean theory, however, it is just that theory. BTW, Dan, I stumbled on an article in a recent issue of Wired magazine that resonated of ideas I have tried to articulate myself in our past discussions. Check it out and tell me what you think. You to Alan! http://www.wired.com/wired/archive/10.12/convergence.html?pg=1&topic=&topic_set=
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