Morse-witten cohomology for non-orientable target spaces

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In summary, the conversation discusses the computation of cohomology groups for different spaces, specifically the sphere, torus, Klein bottle, and projective plane. The question of whether Morse homology is defined for non-orientable spaces is raised and the possibility of extending it is considered. The use of distance and height functions as Morse functions is also discussed, along with the idea of using immersions to define Morse functions. The conversation concludes with a potential Morse function for the projective plane.
  • #1
electroweak
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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?
 
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  • #2
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.
 
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  • #3
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?
 
  • #4
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.
 
  • #5
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.
 
  • #6
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.
 

What is Morse-Witten cohomology?

Morse-Witten cohomology is a mathematical tool used to study the topology and geometry of a space. It combines elements of Morse theory, which studies the critical points of a function on a manifold, and de Rham cohomology, which studies the global structure of a manifold.

How is Morse-Witten cohomology different for non-orientable target spaces?

In Morse-Witten cohomology, the target space is usually assumed to be orientable, meaning that it has a consistent notion of "clockwise" and "counterclockwise" rotations. For non-orientable target spaces, this assumption does not hold, and the theory must be modified to account for this lack of orientation.

What are some applications of Morse-Witten cohomology for non-orientable target spaces?

Morse-Witten cohomology has been used to study various mathematical objects, such as moduli spaces, mapping class groups, and symplectic manifolds. It has also found applications in physics, particularly in the study of gauge theories and string theory.

What are the main challenges in studying Morse-Witten cohomology for non-orientable target spaces?

One challenge is the lack of a consistent orientation, which can complicate the definition and computations of the cohomology groups. Additionally, the non-orientability can lead to more complicated topological structures, making it difficult to generalize results from the orientable case.

Are there any open questions or conjectures related to Morse-Witten cohomology for non-orientable target spaces?

Yes, there are still many open questions and conjectures related to Morse-Witten cohomology for non-orientable target spaces. Some of these include generalizing the theory to higher dimensions and finding alternative approaches to computing the cohomology groups.

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