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 ??? 2. Call two simple closed curves C_0, C_1 on the Klein bottle "equivalent" if there is a continuous family {C_t ⊂ K | 0 ≤ t ≤ 1} of simple closed curves on K. How many inequivalent simple closed curves are there on K, and what is an example of each one ??? —Dan Appendix -------- A surface is called "nonorientable" if it has a subset that is a Möbius band. Otherwise it is called "orientable". The (compact) orientable and nonorientable surfaces (without boundary) were first classified up to topological equivalence by August Ferdinand Möbius in 1861, but this was not proved rigorously until Henry Roy Brahana did so in 1921. Any two surfaces can be combined topologically by the operation called "connected sum", which means to remove the interior of a disk from each surface and then connect the resulting two circular boundaries by a cylinder. Each orientable surface M is the connected sum of g tori (where the connected sum of 0 tori is defined to be the sphere). Each nonorientable surface M is the connected sum of g projective planes for some integer g ≥ 1, also called the "genus" of M. If S denotes the sphere, T denotes the torus and P denotes the projective plane, then every surface (compact and without boundary) can be expressed as the connected sum of S with finitely many copies of T and finitely many copies of P. This operation, denoted by #, has S as the identity element and is commutative with just one relation: P # P # P = P + T. It turns out that each compact surface M can be "triangulated" — expressed as the union of finitely many 2D triangles any two of which intersect in a common edge or vertex. Then if V, E, F denote the number of vertices, edges, and faces (triangles) of the surface, the computation 𝜒(M) = V - E + F is a topological invariant called the Euler characteristic of M. If M is orientable its genus satisfies g = 2 - 2 𝜒(M), and if M is nonorientable its genus satisfies g = 2 - 𝜒(M). Thus Euler characteristic and orientability determine the surface up to topological equivalence. The Klein bottle K shows up in this classification as K = P # P, the connected sum of two projective planes. It can also be viewed as the result of starting with a square and identifying the top and bottom sides in the same direction, but identifying the left and right sides in the reverse direction: |—————➞————| | | ↑ // ↓ | | |—————➞———— |