Today, we will define relative versions of the homotopy groups, and show that they fit into an exact sequence. So let ${X}$ be a pointed space and ${A \subset X}$ a subspace containing the basepoint. Let ${n \geq 1}$. Then we define

$\displaystyle \pi_n(X, A)$

to be the pointed homotopy class of maps ${(D^n, S^{n-1}) \rightarrow (X, A)}$. Here ${D^n}$ has a basepoint, which is located on the boundary ${S^{n-1}}$.

Definition 1 ${\pi_n(X, A)}$ is called the ${n}$-th relative homotopy group of the pair ${(X, A)}$. (We have not yet shown that it is a group.)

Another perhaps more geometric way of thinking of the relative homotopy groups is as follows. Namely, it is the homotopy class of maps ${(I^n, I^{n-1}, J^n) \rightarrow (X, A, x_0)}$, where ${I^n}$ is the ${n}$-cube and ${J^n}$ is the complement of the front face ${I^{n-1}}$. So on the boundary, such a map is ${x_0}$ except possibly on the front face, where it is at least in ${A}$. The reason is that if we quotient by ${J^{n-1}}$, we get the pair ${(D^n, S^{n-1})}$.

And this is also how we can make ${\pi_n(X, A)}$ into a group if ${n \geq 2}$. Namely, given two maps ${f, g: I^n \rightarrow X}$, we join the two cubes together among a face that is not the front face or the back. This is possible if ${n \geq 2}$. Here is a picture.

We can also do this more formally. One way is to note that if we define ${P(X, A)}$ to be the family of paths ${I \rightarrow X}$ that start in ${A}$ and end in ${x_0}$, topologized with the compact-open topology and basepointed with the constant path at ${x_0}$, then ${\pi_n(X, A)}$ is just ${\pi_{n-1}(P(X, A))}$. The addition can be defined in this way, by the additive structure on ${\pi_{n-1}}$. In particular, we have seen that ${\pi_n(X, A)}$ is a group when ${n \geq 2}$. It is even abelian when ${n \geq 3}$.

The main result we want to prove today is:

Theorem 2 Let ${A \subset X}$ be a closed subspace. Then there is an exact sequence of homotopy groups:$\displaystyle \pi_n(A) \rightarrow \pi_n(X) \rightarrow \pi_n(X, A) \rightarrow \pi_{n-1}(A) \rightarrow \dots$

Before proving it, let’s at least describe what the boundary maps look like. (The maps ${\pi_n(A) \rightarrow \pi_n(X)}$ come by functoriality, and the maps ${\pi_n(X) \rightarrow \pi_n(X, A)}$ are obvious.) So say we have a class in ${\pi_n(X, A)}$ represented by ${f: (I^n, I^{n-1}, J^n) \rightarrow (X, A, x_0)}$. Then ${f|_{I^{n-1}}}$ takes values in ${A}$ and is ${x_0}$ on the boundary on ${\partial I^{n-1}}$. This is what we define to be the boundary of the homotopy class. Moreover, this way of defining the map is easily seen to be a group-homomorphism when ${n \geq 3}$.

Using the exact sequence for a pair and some purely algebraic nonsense, one can show:

Theorem 3 Let ${B \subset A \subset X}$ be a triple. Then there is a long exact sequence$\displaystyle \pi_n(A, B) \rightarrow \pi_n(X, B) \rightarrow \pi_n(X, A) \rightarrow \pi_{n-1}(A, B) \rightarrow \dots.$

This is a standard argument in axiomatic homology theory, where you go from the exact sequence of a pair to the exact sequence of a triple. (In singular homology theory, the exact sequence of a triple can of course be proved directly.) I won’t try to blog about the argument, because it’s really messy and completely un-topological.

The abstract approach

As in Hatcher, one can argue geometrically about the exact sequence. There is also the categorical approach. Since this is (to me, at least) harder to understand, I will blog about it.

The first thing to do is to understand the homotopy groups and relative homotopy groups in such a manner as to apply the Barratt-Puppe sequence. So start with the inclusion of pairs ${(S^0, S^0) \rightarrow (D^1, S^0)}$. The mapping cone (i.e., pairs of mapping cones) is ${(S^1, X)}$, where ${X}$ is a proper (and consequently) subarc of ${S^1}$. So we can replace the last term by ${(S^1, x_0)}$ for ${x_0}$ the basepoint.

But wait, you say. We never talked about pairs before, right? OK, perhaps we should explain what’s going on in outline.

Let us consider the category ${\mathbf{PT_2}}$ of pairs of pointed topological spaces ${(X, A)}$ and basepoint homotopy classes of maps of pairs. So two maps ${f, g: (X, A) \rightarrow (Y, B)}$ are homotopic if there exists ${H: X \times I \rightarrow Y}$ such that ${H(\cdot, 0) = f, H(\cdot, 1) = g}$, ${H(x_0 \times I) = y_0}$, and ${H(A \times I) \subset B}$. This is the pair analog of the category ${\mathbf{PT}}$. It too has an initial object (namely, ${(\ast, \ast)}$), so it makes sense to talk about coexact sequences in this category.

Next, we note that there is a version of the Barratt-Puppe sequence for pairs.

Theorem 4 (Barratt-Puppe) Let ${(X, A) \stackrel{f}{\rightarrow} (Y, B)}$ be a morphism. There is then a coexact sequence in ${\mathbf{PT)_2}}$ $(X, A) \rightarrow (Y, B) \rightarrow (Y \cup CX, B \cup CA) \rightarrow (\Sigma X, \Sigma A) \rightarrow (\Sigma Y, \Sigma B) \rightarrow (\Sigma(Y \cup CX), \Sigma(B \cup CA)) \rightarrow \dots$

I can’t see how this theorem is a corollary of the theorem I stated last time. Actually, perhaps I should have been more thoughtful and discussed it the first time in the context of pairs instead of the context of one pointed space. But anyway, it turns out that the proof is just the same. So I will not write it out here.

This implies that, as above, we have a coexact sequence beginning ${(S^0, S^0) \rightarrow (D^1, S^0) \rightarrow (S^1, x_0)}$. Let us continue this. It goes

$\displaystyle \Sigma^n(S^0, S^0) \rightarrow \Sigma^n(D^1, S^0) \rightarrow \Sigma^{n}(S^1, x_0) \rightarrow \Sigma^{n+1}(S^0, S^0) \rightarrow \dots$

But these elements are very nice. In fact, we know that the reduced suspension of an ${n}$-sphere is just an ${n+1}$ sphere. Actually, this is not completely trivial because of the reduction, but I will omit the proof. (You can look it up in Bredon’s book or Switzer’s book, for instance.) The suspension of a point, by contrast, is just a point. Finally, what is ${\Sigma^n(D^1)}$? It turns out to be, up to homotopy, ${D^{n+1}}$. This is clear if ${\Sigma}$ is replaced with the “unreduced” suspension; it is a general fact that for reasonable (well-pointed, to be precise) spaces, the unreduced suspension is homotopy equivalent to the regular reduced suspension.

Now we know that this sequence is coexact. So let’s take a pair ${(X, A)}$. When we hom out in ${\mathbf{PT_2}}$ from the long sequence into the pair ${(X, A)}$, we get an exact sequence of pointed sets. The associated sequence of pointed sets is, however, just

$\displaystyle \pi_n(A) \rightarrow \pi_n(X) \rightarrow \pi_n(X, A) \rightarrow \pi_{n-1}(X, A) \rightarrow \dots.$

This is why we have an exact sequence for a pair.

Next time, we’ll discuss a compression criterion for deciding when a map represents zero in the relative homotopy groups. After that, over the next several posts, I would like to talk about some of the homotopy theory of CW complexes: the Whitehead theorem, CW approximation, Eilenberg-Maclane spaces, homotopy excision, and the Brown representability theorem.  I will also need to go back to general topology to discuss fibrations (the dual idea to cofibrations) and how this long exact sequence of homotopy groups turns into something special when you have a fibration.

About these ads