Here we talk about the Hurewicz theorem. Let be a path connected space with for . Then is determined up to homotopy by .
What does “determined up to homotopy” mean? It means that all the spaces that satisfy the condition above are homotopic to each other. When two spaces are homotopic, what does that mean? Say and are homotopic. This means that there exist maps and such that and . Can we think of any examples of spaces that are not homotopic? Yes. Just map a disconnected space to a connected space. Like mapping two non-intersecting discs to a connected disc. A homotopy between spaces tends to preserve the same “kind” of connectivity between spaces; i.e. two disconnected discs would be homotopic to two discs, and not three disconnected discs.
Why do we need the Hurewicz theorem? Because it is often difficult to calculate the fundamental group of a space, but much easier to calculate the homology groups. Hence, knowing that we can map homology groups to homotopy groups, we can discover properties of the homotopy groups that we couldn’t earlier.
We state and prove the Hurewicz Theorem for the case. Let be defined such that a path goes to . How is a member of the homology group? Because it is a map from , which is a -simplex, to ! Now for the map to be well-defined, if , then . Why is this true? We shall find out later.
Anyway, we have a homomorphism , and is an abelian group. Hence, we have a homomorphism , where is just the group modulo its commutator subgroup. We have abelianized the fundamental homotopy group. Hurewicz Theorem says that the map is an isomorphism. How do we see this? Moreover, we have not even proven that is a homomorphism, or even a well-defined map.
We need to prove three things: that is well defined, a homomorphism, and an isomorphism.
!. Well-defined- Consider two homotopic paths . We need to prove that and belong to the same homology group. There exists a map such that and . The solid square can be thought of as the union of two -simplices and . We’ll orient the boundary edges in a compatible fashion, and consider the restriction of to and .
On a completely different but related topic, note that the constant map from is just the boundary of the constant map from . Coming back to the above argument, we note that . Here just denotes the mapping of a -simplex to the point .
So what exactly is happening here? How do we know that ? Because , and all three of and belong to . Hence, we’ve proved that is well-defined.
2. Homomorphism: Let and be elements of the fundamental group. Consider . Moreover, let be and let . On the edge, let the restriction of be . On the other two edges, the restriction of should be and .
First we need to prove that reparametrization does not affect homotopy. Why’s that? The homotopy lies in continuously deforming one interval to another, and then making the path (say ) act on it. This homotopy is clear. Now the proof says that is a reparametrization of . How is that?
Anyway, we have . Hence, we get a homomorphism.
3. Surjectivity: Let us take a -cycle . Here it is possible that for . We can re-write this sum as a sum of loops, but putting together non-loops in the obvious way. Hence, we know that all cycles can be mapped to. These cycles may be disjoint too. It’s just that we’ve covered all cycles.
Now let us move to more general -simplices. Let be the path from to the base point of . Then is homologous to . What does this mean? It means that their difference is a boundary. Why are they homologous? This is because , as is a homomorphism. Now is homologous to . Hence, we get is homologous to . By doing this, we can center all our loops at . Now any sum of loops centred at can be mapped to from the fundamental group, as that is what the fundamental group is, the collection of all loops centred at any point (remember that is path-connected). Hence, we’ve proven that is surjective.
4. Kernel: We need to prove that the kernel is the commutator subgroup. We can see why belongs to the kernel. This is because is homologous to . Now we need to prove that the kernel is a subset of the commutator subgroup.
I’m not typing up the inclusion of the kernel in the commutator subgroup, but it can be found here.