Recent content by pp31

Hey guys, There was a definition of linking number that used the fact that H_1(S3\knot, Z) = Z. But to do that I was trying to compute the homology of S^3\knot and had no idea how to do it. Any help would be appreciated. Thanks

Determine the homeomorphism classes of compact 3-manifolds obtained from D^3 by identifying finitely many pairs of disjoint disks in the boundary? I just started reading some low dimensional topology on my own and I came across this question. I have realized that based on how the...

In local coordinates what does it mean for a critical point of a function f:M\rightarrowR to be non degenerate? In addition how can you show that the definition is independent of the choice of local coordinates? I know that being a critical point is independent of the choice of local...

THanks a lot for the help

I think we could use the fact that the boundary operators vanish when we construct the chain complex of the projective plane over Z2 which in turn implies that Hn(RP^{n};Z2 ) =Z2 to show that we need atleast one critical point of each index and since f(x) = f(-x) we have at least two critical...

However I think there are different decomposition of a sphere as a CW complex for instance one way is attaching a 0 cell and a n cell and the other one is 2 0 cells,..., 2 n cells. So we have different morse funcitions based on two different decompositions so I think we have to be careful when...

So what you are saying is that not only can we find the CW complex structure of a space from a Morse function but we can use the CW structure to determine the critical points and the index of a Morse function. Correct me if I am wrong

Suppose a Morse function f:S^n\rightarrow R satisfies f(x) = f(-x). Show that f must have atleast 2 critical points of index j for all j = 0,...,n. Show that any MOrse function on a compact set of genus g has at least 2g+2 critical points. These are the questions and I have no idea how to get...

Thanks quasar ... that was really helpful

Employment areas with PhD in Pure Maths?? I am currently filling out my graduate school applications for PhD in Algebraic topology and there are a lot of questions asking about my career goals. So I was wondering what are the employment opportunities for a person with PhD in Algebraic...