For #1, take a family of great circles on S^2 that all intersect in the same two poles, like the meridional lines on a globe. Have the circle rotate 180 degrees, while its partner, a single point, goes all the way around S^1. For #2, I have a memory of having done this exercise as an undergraduate, and I think the answer was 4, but I don't remember the proof. It was something like showing that the curve that wraps twice around the "neck" could be deformed to a point by pulling one of the loops all the way around the long way. On Fri, Nov 27, 2020 at 10:56 AM Dan Asimov <asimov@msri.org> wrote:
I should at this point give some hints:
1. S^1 x S^2 in fact does have a Klein bottle as a subset. Can you find one?
2. The number of inequivalent simple closed curves on a Klein bottle is in fact finite (and less than 10). Can you find one example of each one?
—Dan ————— PS The book "Topology and Geometry" mentioned by Jean is written at an advanced graduate-student level, but is excellent. William Goldman's review of it is around the middle of this page: < http://www.math.umd.edu/~wmg/reviews.html>.
On Friday/27November/2020, at 7:42 AM, Jean Gallier <jean@seas.upenn.edu> wrote:
As a corollary of Alexander duality, if M is a connected orientable, compact manifold of dimension n, and if H_1(M,Z) = (0) (the first homology group with coeffs in Z vanishes), then no nonorientable compact manifold N of dimension n -1 can be embebded in M. (for example, see Bredon, Topology and Geometry, Corollary 8.9, Page 353).
This implies that K can’t be embedded in S^3.
But this is not NOT Dan’s question!
Unfortunately, by the Kunneth formula H_1(S^1xS^2, Z) = Z. (H_0(S^1,Z) = H_1(S^1,Z) = Z, H_0(S^2,Z) = Z, H_1(S^2,Z) = (0), H_2(S^2,Z) = Z).
It appears that S^1 x S2 is a bit “bigger” than S^3, so I don’t know whether K can be embedded in S^1 x S^2. I would lean for no.
Best, — Jean
On Nov 25, 2020, at 10:35 PM, Dan Asimov <asimov@msri.org> wrote:
I'll just say that, so far, nothing correct has been mentioned about the answer to either puzzle.
—Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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