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?