Fast Computation of Fundamental Groups: Practical Methods and Tricks

[mich0144]

So I've read through beginning alg topology really fast and there are a lot of theorems and methods for computing fundamental groups but what are the most useful tools? When asked to compute the fundamental group what should one do? try to find a deformation retract and compute the fund group of the retraction instead? use seifert van kampen immediately? what other practical methods are there for fast computation.

 

[Tinyboss]

Seifert-Van Kampen is probably the best, although homology groups are more used than homotopy groups, mainly because homotopy groups beyond the first are nasty to compute.

 

[mich0144]

I haven't gotten to homology yet but thanks

 

[Tinyboss]

If you can see that the space is homotopic to something familiar, that's the easiest. If not, then try to decompose it with S-VK. For this it also helps to be familiar with the classification of compact connected 2-manifolds as quotients of polygons (a 4n-gon for the genus-n surface, and a 2n-gon for RP^n).

 

[mich0144]

yea that's a good idea i know the 2manifolds classifications I think, is there a good book or any kind of document with lots of examples for using seifert van kampen I'm reading hatcher but it's a little too terse and my algebra background isn't too good

 

[Tinyboss]

Massey's Introduction to Algebraic Topology does S-VK, but either way you'll want to know some algebra.

 

[mich0144]

so if I were using seifert van kampen on a torus as an illustration. decompose it as U = torus with a hole, and V as a patch larger than the whole. now fundamental group of U is the figure eight which is Z*Z i think if I unroll it into a square and fundamental group of patch is trivial. fundamental group of the intersection which is an annulus is that of a retracted circle so it's the infinite cyclic Z. This is the part i don't get. seifert van kampen says pi(X) = free group of pi(U) and pi(V) / smallest normal subgroup (in other text it says its the kernel)

so I looked up amalgamated free product
http://en.wikipedia.org/wiki/Free_product

phi:F -> G and psi:F -> H
where F is some random group
start with G*H free product and adjoin as relations :
phi(f)psi(f)^-1 = 1 for every f in F. In other words take the smallest normal subgroup N of G ∗ H containing all elements on the left-hand side of the above equation, which are tacitly being considered in G ∗ H by means of the inclusions of G and H in their free product. The free product with amalgamation of G and H, with respect to φ and ψ, is the quotient group

G*H / N

how exactly would I compute this for the torus for example. Most of the examples in hatcher have the trivial fundamental group for the intersection of U and V so pi(x) just reduces to the free group of the 2 individual fundamental groups. I think I understand free groups now but a little fuzzy on quotient group.

 

[mich0144]

ok so I figured out some tricks you basically just mod it by the relation generated by identifying the edges of the polygon representation and set it equal to 1, this seems to work for most case.