What is it?
Definition:Given a topological space X, π1(X,x), its fundamental group at x ∈ X is the set of homotopy classes of closed loops starting at x composed by the operation of path joining.
If you're using the same crumby web browser that I'm using
, the symbol π is supposed to be a lower case pi
Those given to flights of fantasy might like to speculate on what π1, π2 etc are, but they will not be considered here.
I'm sorry, that makes no sense
The fundamental group is an object used in algebraic topology
to characterise properties which differ between spaces which are not homeomorpic
, eg. why a sphere
is not like a torus
. You might just say "one's got a hole in and the other dosen't
", but we need to make this idea precise.
I mentioned closed loops above, which are just continuous path
s which start at some point, and then return to the same place (to digress to a lower level of technicality, curves you can draw from a point with a pencil without lifting it from the page that return to where they started). It would now be nice if we could make a group
out of something like this, by joining paths together. First, it would be nice if there was some way in which we could think about paths that can be continously deformed into one another as the same thing (there would otherwise be many paths to consider!). The notion of homotopy
is precisely this; and if you're reading this section then hopefully a more formal definition is not necessary. As an example, the two paths on the left below are homptopic, whereas the ones on the right are not:
_ _ _ <end _ _ _ <end
/ | / |
__/ | __/ |
A / / C / /
| | | HOLE |
/ _ __/ B / _ __/ D
/ / / /
(they're not closed loops I know, but I drew them this way for clarity). We can consider all homotopic paths as the same thing by putting all paths homotopic to (say) β into a set denoted [ β ] . These sets will be the elements of the group
being we're constructing (formally, this is the set of equivalence classes arising from the equivalence relation
Joining two paths together is simple - just follow one then the other; if α and β are the two paths, call their join α+β . This then gives a group composition operation on the set of homotopy equivalences: [ α ] * [ β ] = [ α+β ] . Then the identity element is the set of all paths that can be contracted to a point, and the inverse of [ β ] is [-β ] , where -β is understood to mean traversing the path in the opposite direction. Closure and associativity of the operation follow easily. If all paths are based at x in X, then what you have now is precisely the group π1
What is the group is based at a point y instead? If so, and the space is path connected
, then there's some path γ going from y to x, a with this a group isomorphism can be defined as
π1(X,x) <-> π1(X,y) : [ α ]X <-> [(-γ)+α+γ ]Y
...in which case you can simply write the fundemental group as π1(X) for path connected spaces without ambiguity.
This is all a very nice framework to be using, but the actuall business about going and calculating the fundamental groups is a lot less pleasant. To illustrate: how in the above example could you show that there is no
way of continuously interpolating between paths C and D, given the presence of the hole? It's something that you can probably convince yourself of very easily (in that, at some point an intermeidate path must go through the hole); but given that you're reading the wishy-washy part of a mathematics node, that chances are that you're not very interested in the gritty details of such proofs, and so they will be gladly omitted.
So, having dropped the rigours tone of the proceedings, what are some of thse groups. Firstly, the easy ones: spaces like Rn
can be continously shrunk to a point (they are contractable
), so all paths can be shrunk to the identity - and therefore their fundamental groups are trivial. This is not true of the circle
(I'm thinking of it as just the rim), as you can have path β going all the way around which obviously is not homotopic to a constant path (ie. [ β ] is not the identity). This path and its inverse -β will then generate the group, giving elements [ β ]n
for n an integer, and [ β ]a
* [ β ]b
= [ β ]a+b
. Then it is clear that π1
(circle) is isomorphic
to the integers under addition.
Fundamental groups are not necessarily abelian: consider two circles joined at a point (making a figure of eight), and for simplicity's sake, base the group at the point of intersecton. Then if you go round one loop first then then other, the path is not homotopic to that formed by going round the second loop first.
Go and draw some pictures of wierd surfaces with holes in and try to see if you can work out what their fundamental groups are.
What's nice about them?
Well, other that being conceptually pleasant, they also crop up in an elegant way when dealing with Riemann surfaces
which are the quotients
of certain simply connected sets (for more info, have a look at that write up).
Also, fundamental groups extend naturally onto products of topological spaces, that is to say that:
π1(X×Y) = π1(X)×π1(Y)
which you can instantly see to be true by considering that a path in the product space is that same thing as a product of paths in the two individual spaces, with all the results about homotopy following quite easily. In short, fundamental groups behave as they ought to under various topological operations.
If you're wondering precisely what a path is in this context, it's just a continious function from [ 0,1 ] to the space X. To say that the path φ is closed in theis context means that φ(0) = φ(1).
A homotopy between two paths ψ0
is a continious function ψ from [ 0,1 ]x[ 0,1 ] to X, with ψ(0,t) = ψ0
(t) and ψ(1,t) = ψ1
(t) for all t in [ 0,1 ].