Determining Homology Groups of S^2 with a Morse Function

[seydunas]

Hi,

i want to determine the homology groups of S^2 using a Morse function with at least 3 critical points. Is there anyone to helpm me in this way. I know how i can describe the homology of sphere in usual way. That is by using a Morse function with 2 critical points( index 0 and 2)

Also, can you say me a heigt function from connected,compact,oriented surface of genus g.

 

[quasar987]

Are you asking how to deduce the homology of S^2 from the existence of a Morse function with 1 max, 1 saddle and 1 min by using Morse homology?

 

[seydunas]

Yes, but i ask not only the Morse function has three critical points but only more than 3. As you know there is no Morse function that has three criticak points because the characteristic number of S^2 is 2, and the number of the critical points of index k determine the characteristic of S^2,i.e the characteristic number number is equal the sum of index 0 and index2 minus index1, so any Morse function on S^2 must have even number critical points. I actually want to ask that the homology of S^2 by using the Morse function with at least 3 critical points. Do you know the Morse function preciely.

 

[lavinia]

Yes, but i ask not only the Morse function has three critical points but only more than 3. As you know there is no Morse function that has three criticak points because the characteristic number of S^2 is 2, and the number of the critical points of index k determine the characteristic of S^2,i.e the characteristic number number is equal the sum of index 0 and index2 minus index1, so any Morse function on S^2 must have even number critical points. I actually want to ask that the homology of S^2 by using the Morse function with at least 3 critical points. Do you know the Morse function preciely.

Warp the sphere so that its height function has two maxima - it will look like a rounded out letter U. The height function will have two maxima, one minimum, and one saddle point.

 

[quasar987]

This is what I thought too, but there is something bugging me. The space that you are describing is symmetric with respect to reflection across the.. zy plane say.

There is obviously a gradient line from the saddle point to each of the max, so if M1, M2 are the maxes, S the saddle and m the min, then dM_i=±S, so in order to have d² = 0, it must be that dS=0. That is, there are no gradient flow line from m to S. This is saying that all gradient flow lines that start at m end up at one of the M_i. But this just seem very weird to me for consider the flow line starting at m and pointing in the direction of S. Then by symmetry, it would be absurd that it ends at one of the Mi and not the other. So it must end up at S!

 

[lavinia]

This is what I thought too, but there is something bugging me. The space that you are describing is symmetric with respect to reflection across the.. zy plane say.

There is obviously a gradient line from the saddle point to each of the max, so if M1, M2 are the maxes, S the saddle and m the min, then dM_i=±S, so in order to have d² = 0, it must be that dS=0. That is, there are no gradient flow line from m to S. This is saying that all gradient flow lines that start at m end up at one of the M_i. But this just seem very weird to me for consider the flow line starting at m and pointing in the direction of S. Then by symmetry, it would be absurd that it ends at one of the Mi and not the other. So it must end up at S!

I don't know much about Morse theory but it seems that this picture must be right because starting with a torus and its height function, one can tear the torus apart along the circle connecting the max and the top saddle point then separate the two ends slightly and cap each one off with a disk.The height function will be virtually unchanged on the torus and should extend smoothly to the two disks if you round them at the edges.

There is still one gradient line connecting the saddle point to the min. Is there only one?

But can you explain what you mean by d^2 = 0 and your other notation?

 

[quasar987]

Oh, I'm silly! dS does not imply that there are no gradient flow lines btw S and m, it just means that there are 0 of them counting algebraically.

Since there are obviously two flow lines, they must have opposite signs so that d²=0.

Here, "d" is the boundary operator for the Morse chain complex. The ith chain group is just the free abelian group on the set of critical points of index i. Basically, if c is a critical point of index i, then dc counts the gradient flow lines btw c and the critical points of index i-1. The Morse homology is the homology of this complex and it is naturally isomorphic to singular theory.

See the excellent online note of M Hutchings.

 

[seydunas]

Warp the sphere so that its height function has two maxima - it will look like a rounded out letter U. The height function will have two maxima, one minimum, and one saddle point.

This warping is a goodidea but wonder that i can write function precisely.That s think about a function from sphere to R, i want to get it is Morse function and it has critical points more than 3. Precisely i want to see a function. But as quasar' s answer, i think i is the best way to tend the homology of sphere by using gradient flows. Again i want to see a function as i desired.

 

[quasar987]

An explicit function like this should not be too hard to write i would think. Use the stereographic on S²-{S} and define on R^2 a positive map with two max at (x,y)= (±1,0) and a saddle at (0,0) and which goes to 0 at infinity. Then patch it to 0 at S. (Or something a little more refined using bump function if this doesn't quite work.)

 

[lavinia]

Oh, I'm silly! dS does not imply that there are no gradient flow lines btw S and m, it just means that there are 0 of them counting algebraically.

Since there are obviously two flow lines, they must have opposite signs so that d²=0.

Here, "d" is the boundary operator for the Morse chain complex. The ith chain group is just the free abelian group on the set of critical points of index i. Basically, if c is a critical point of index i, then dc counts the gradient flow lines btw c and the critical points of index i-1. The Morse homology is the homology of this complex and it is naturally isomorphic to singular theory.

See the excellent online note of M Hutchings.

thanks! this sounds like it is very cool. It will be interesting to see how torsion is obtained from this theory.

 

[lavinia]

This warping is a goodidea but wonder that i can write function precisely.That s think about a function from sphere to R, i want to get it is Morse function and it has critical points more than 3. Precisely i want to see a function. But as quasar' s answer, i think i is the best way to tend the homology of sphere by using gradient flows. Again i want to see a function as i desired.

Take a standard torus - any equations you like. Replace the top half with two hemispherical caps. These will attach smoothly to the bottom half of the torus.

 

[seydunas]

"Take a standard torus - any equations you like. Replace the top half with two hemispherical caps. These will attach smoothly to the bottom half of the torus."

What do you mean when you say it? This is the answer for S^2 or another question? Then i can not find the function as quasar' s advising. But i will try to find it.

 

[quasar987]

I just stumbled upon this. Looks interestings to read. http://math.berkeley.edu/~alanw/240papers03/chen.pdf