This is an old qual question, and I want to see if I have it right. I had virtually no instruction in homology despite this being about 1/4 of our qualifying exam, so I am feeling a bit stupid and frustrated.(adsbygoogle = window.adsbygoogle || []).push({});

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

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# A Fundamental and Homology groups of Polygons

Have something to add?

Draft saved
Draft deleted

Loading...

Similar Threads - Fundamental Homology groups | Date |
---|---|

A Homology calculation using Mayer-Vietoris sequence | Feb 17, 2018 |

A Fundamental group of a sphere with 6 points removed | Oct 15, 2017 |

A Fundamental group of n connect tori with one point removed | Oct 13, 2017 |

A Fundamental group of Project Plane with 2 points missing | Oct 12, 2017 |

I Homotopy Class vs Fundamental Group. | Sep 21, 2017 |

**Physics Forums - The Fusion of Science and Community**