- #1

dkotschessaa

- 1,060

- 783

Anyway,

I am given a space defined by three polygons with directed edges as follows (a description should suffice).

1) A triangle with edges ##bca^{-1}##.

2) A triangle with edges ##ad^{-1}e##

3) A square with edges ##bdc^{-1}e^{-1}##.

I am to find a presentation for ## \pi_1(S) ## which I suppose is just what I have given above as labels, so ## < a,b,c,d,e | bca^{-1} = ad^{-1}e = bdc^{-1}e^{-1} = 1 > ##.

I mean, first of all, is this correct? In most examples I have seen, we are given a single polygonal region, not multiple ones.

I'm to find ## H_1(S) ## which is equal to ## \pi_1(S)/ [\pi_1(S), \pi_1(S)] ## i.e. the mod of the fundamental group with its commutator subgroup.

I don't know of the most efficient procedure. I've got 9 commutators ## [a,b], [a,c]...## etc. So that's 27 operations? If I was clever I might know which ones aren't worth doing. For the first element against the first commutator ##[a,b] = aba^{-1}b^{-1} ## I'm getting stuff like:

## bcba^{-1}b^{-1} ##

## bcca^{-1}c^{-1}##

## bcda^{-1}d^{-1}##

Where I begin to feel like:

http://i1.kym-cdn.com/photos/images/original/000/234/739/fa5.jpg

Lastly I'm asked to classify this space from the classification theorem. All I can think to do is glue all the pieces together along their similarly labeled edges, respecting the orientations, etc.

e.g. ## bca^{-1}ad^{-1}ee^{-1}bdc^{-1} ##

which reduces to ##bcd^{-1}bdc^{-1} ##

Which I can relabel ## abcac^{-1}b^{-1}##

which I see is ## abca(bc)^{-1}## and I can relabel ##bc## (I think) so that I end up with ##abab^{-1}##.

And darnit, the first time I did this, I swear I ended up with ##aba^{-1}b^{-1}## which is a torus, but I do not know what I have here.

'elp?

-Dave K