I will straight off assume that continuity is something everyone’s comfortable with, and build on top of that.

Homotopies and homotopy equivalences

We say that two continuous maps, f,g:X→Y between topological spaces are homotopical, and write , if there is a continuous map such that H(x,0)=f(x) and H(x,1)=g(x). This captures the intuitive idea of step by step nudging one map into the other in formal terms.

Two spaces X,Y are homeomorphic if there are maps , such that and .

Two spaces X,Y are homotopy equivalent if there are maps , such that and .

Now, if f,g are maps X→Y and f=g, then , since we can just set H(x,t)=f(x)=g(x) for all t, and get a continuous map out of it. Thus homeomorphic spaces are homotopy equivalent, since the relevant maps are equal, and thus homotopic.

There are a couple of more properties for homotopic maps we’ll want. It respects composition – so if and h:Y→Z and e:W→X then and . This can be seen by considering h(H(x,t)) and H(e(x),t) respectively.

Denote by D² the unit disc in , and by {*} the subset {(0,0)} in . Then . In one direction, the relevant map is just the embedding, and in the other direction, it collapses all of D² onto {*}. One of the two relevant compositions is trivially equal the identity map, and in the other direction, the linear homotopy H(x,t)=tx will do well. Thus the disc and the one point space are homotopy equivalent.

The fundamental groupoid

Let X be a topological space (most probably with a number of neat properties – I will not list just what properties are needed though), and consider for each pair x,y of points in X, the set [x,y] of homotopy classes of paths from the point x to the point y. A path, here, is a continuous map [0,1]→X. We can compose classes – if and , then we can consider the map
. This is a path from x to z, and so belongs to a class in [x,z]. This class is well defined from the choices of γ, γ' since homotopies and composition of maps work well together.

This gives us a composition. It is associative - on homotopy classes. What happens if we look at maps instead of homotopy classes is part of the subject of my own research. It has an identity at each point x - the constant path γ(t)=x, and for each class in [x,y] there is a class in [y,x] such that their composition is homotopic to the constant path in [x,x].

Thus, we get a groupoid. This is called the fundamental groupoid, and denoted by . If we fix a point, and consider [x,x], then this is a group, called the fundamental group with basepoint x, and denoted by .

For , a linear homotopy will make any two paths in [x,y] homotopic – and so |[x,y]|=1 in for any x,y.

For S¹ – the circle – we can choose to view it as [0,1]/(0=1). Then we can consider the paths f_m(t)=a(1-t)+bt+nt. This is a path from a to b, and it winds n times around the circle. Each path in [a,b] is homotopic to a f_m, by a linear homotopy, which just rescales the speeds through various bits and pieces, and possibly straightens out when you double back. Thus, . Furthermore, if you compose f_mf_n, you’ll get f_n+m.

If we pick out the fundamental group out of this groupoid, we’ll get the well known fundamental group .

Now, suppose we have two homotopy equivalent spaces X and Y, with the homotopy equivalence given by f:X→Y and g:Y→X. Then consider the map f_*:[x,y]_X→[f(x),f(y)]_Y given by f_*γ(t)=f(γ(t)). I claim
1) f_* is bijective.
2) f_* works well with composition of classes.

For bijectivity we start with injectivity in one direction. Consider two paths in [x,y]. We need to show . If , then . However, then

which contradicts . Thus , and so also .

The proof is symmetric in the choice of direction, and so we can just repeat the same argument to get that g_* is also an injection. Thus we can conclude that f_* is in fact a bijection.

Now, for the second part, we consider and . We need to show that . But is the path that first runs through in half the time, then runs through in the rest of the time, and just transports this path point by point to Y. And transports point by point to Y and transports point by point to Y, and just runs through the first of these in half the time, then the rest in the rest of the time.

Thus, homotopy equivalent spaces have the same fundamental groupoid.

They got, today, Armstrong’s Basic Topology, and an extract from the German topology book by Jänich, and on monday, we shall go over the formal definition of topological spaces, and of continuous functions together.

I plan to keep updates on our progress here on the blog – with the questions I send them off with each meeting as well as some sort of discussion about how this setup is working out, if at all.

For the first trip, the questions I dumped in their laps were:

• What topologies are possible on the set {0,1}?
• What topologies are possible on the set {0,1,2}?
• Which are the continuous functions between the topologies above?
• Give an example of a continuous and a discontinuous function each for the following cases
  1. with the standard topology on
  2. with the standard topology on
  3. with the discrete topology
  4. with the finite-complement topology