Re: [math-fun] Klein bottle puzzles

On Wed, Nov 25, 2020 at 4:40 PM Dan Asimov <asimov@msri.org> wrote:

Let K denote a Klein bottle.

1. The cartesian product

S^1 x S^2

of a circle (S^1) and a sphere (S^2) is a certain 3-dimensional manifold.

Puzzle: Does this manifold contain a Klein bottle

K ⊂ S^1 x S^2

as a subset ???

My intuition is that an orientable 3-manifold can't contain a nonorientable compact surface, but I can't come up with a proof.

g = 2 - 2 𝜒(M),

This is backwards: it should be 𝜒(M) = 2 - 2 g

and if M is nonorientable its genus satisfies

g = 2 - 𝜒(M).

And I spent an embarrassingly long time trying to figure out whether this one was also backwards. Andy

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Andy.Latto@pobox.com