Morse-witten cohomology for non-orientable target spaces

[electroweak]

I'm reading "Mirror Symmetry" by Hori et al. In it, they compute the cohomology groups for the sphere and the torus. I tried to do the same for the Klein bottle, and it isn't working out.

Is Morse homology defined for non-orientable spaces? If not, why not? Can it be extended?

 

[lavinia]

There is a theorem that any compact manifold possesses a Morse function. The proof in Milnor's Morse Theory relies on the ability to embed any smooth manifold in Euclidean space and then proving - using Sard's Theorem - that the distance function from the points in the manifold to some fixed point in Euclidean space has no degenerate critical points.

Before I thought I had an explicit example of Morse function for the Klein bottle but on further thought the function seemed to have an entire circle of critical points. I would like to see a Morse function for the Klein bottle and also for the projective plane.

 

[electroweak]

Consider the height function on the image of some nice immersion of the Klein Bottle in R3. Pullback the height function by the immersion to obtain a potential on the actual Klein Bottle.

Is this a Morse function? If so, I can't get it to work. If not, why isn't it?

 

[lavinia]

I think the same arguments that work for embeddings also work for immersions. For an immersed Klein bottle in 3 space the distance function to most points - except for a set of measure zero - will be a Morse function. I am not sure about the height function.

Here is another thought.

Take a rectangle - not a square - in the plane and make a flat Klein bottle by pasting the opposite sides to each other. It seems that the function that measures the Euclidean distance to the center of the rectangle is a Morse function. This also works on a flat torus. I was confused since there is one maximum and minimum and two saddle point in each but I guess the cell attaching maps are different.

 

[lavinia]

I should have said but I think you already knew this that the distance function for an immersion has to follow the surface and not switch at intersections.

 

[lavinia]

Here is a thought for the projective plane. Each point in the plane has two inverse images in the 2 sphere each either lying on the equator or one in the northern and the other its antipode in the southern hemisphere. The function will be the distance of the point in the northern hemisphere to the north pole. For points along the equator, it is also the distance to the north pole. This seems to be a Morse function with one critical point - which is correct since its Euler characteristic is 1.