Morse homology of a tilted torus.

[MathematicalPhysicist]

How do I compute Morse homology of a tilted torus?

Thanks.

 

[Tinyboss]

First of all, the torus is tilted because you've got it embedded in R^3 and you're using the height function (projection onto the vector (0,0,1)) as the Morse function. The tilt is necessary in this case because without it, the ascending and descending manifolds of the index-1 critical points would not have transverse intersection. Typically you'd just start out by assuming all the ascending and descending manifolds have transverse intersection (which is a generic condition) and then just talk about the "Morse homology of the torus".

I'll assume you're familiar with other notions of homology (e.g. cellular, simplicial). The generators of the i-th chain group are the index-i critical points of the Morse function, and the boundary map $\partial_i:C_i\to C_{i-1}$ is defined on generators by by assigning to an index-i critical point a linear combination of index-(i-1) critical points, where the coefficient for each index-(i-1) point is the number of flow lines connecting it to the index-i point.

If you're doing the homology over Z/2Z, then you just count flow lines (mod 2), but if you're doing it over Z then you have to take orientation into account as well. The flow lines correspond to intersections between the descending and ascending manifolds of the critical points, and these intersections can be positive or negative, and are counted accordingly.

So now you have chain groups and a boundary map defined on the generators. Extend it by linearity and compute homology groups in the usual way (kernel mod image).

 

[MathematicalPhysicist]

Hi, TinyBoss, do you have some reference where such a calculation or similar is being computed?

Thanks.

 

[Tinyboss]

Hi, TinyBoss, do you have some reference where such a calculation or similar is being computed?

Thanks.

I don't know of a specific textbook or set of notes. Have you checked out the references on Wikipedia's page on the topic?

 

[blue_raver22]

To compute the Morse homology of a tilted torus, you will first need to understand the concept of Morse homology and its application to manifolds. Morse homology is a mathematical tool used to study the topology of a manifold, which is a geometric object that can be described by a set of coordinates. In simple terms, it helps us understand the shape and structure of a manifold by analyzing the critical points of a function defined on it.

In the case of a tilted torus, we can consider it as a two-dimensional manifold with a specific shape and structure. To compute its Morse homology, you will need to follow these steps:

1. Define a Morse function: The first step is to define a Morse function on the tilted torus. This function should have non-degenerate critical points, which means that its Hessian matrix (second derivatives) is non-singular at each critical point.

2. Compute the critical points: Once you have a Morse function, you can compute its critical points by finding the points where its derivative is equal to zero. In the case of a tilted torus, there will be four critical points, two maxima and two minima.

3. Label the critical points: Each critical point will be assigned a label based on its index, which is determined by the number of negative eigenvalues of the Hessian matrix. For example, a maximum will have an index of zero, while a minimum will have an index of two.

4. Construct the Morse complex: The Morse complex is a chain complex that captures the topology of a manifold. It is constructed by taking the critical points as vertices and connecting them with edges based on their indices. For a tilted torus, the Morse complex will have four vertices and four edges.

5. Compute the Morse homology: The Morse homology of a manifold is obtained by taking the homology groups of its Morse complex. In the case of a tilted torus, the homology groups will be isomorphic to the homology groups of a circle, which are Z (integers) in dimensions 0 and 1, and zero in all other dimensions.

In summary, to compute the Morse homology of a tilted torus, you will need to define a Morse function, compute its critical points, label them, construct the Morse complex, and finally compute its homology. This process can be extended to higher-dimensional manifolds as well. I hope this helps in your understanding of Morse hom