Algebraic Topology Functors

This will be old news to anyone who does algebraic topology, but oddly enough I can’t seem to find it all written in one place anywhere, and in particular I can’t find the bit about ${\mathsf{hPairTop}}$ at all.

In algebraic topology you (for example) associate every topological space ${X}$ with a group, like ${\pi_1(X, x_0)}$ or ${H_5(X)}$ . All of these operations turn out to be functors. This isn’t surprising, because as far as I’m concerned the definition of a functor is “any time you take one type of object and naturally make another object”.

The surprise is that these objects also respect homotopy in a nice way; proving this is a fair amount of the “setup” work in algebraic topology.

1. Homology, ${H_n : \mathsf{hTop} \rightarrow \mathsf{Grp}}$

Note that ${H_5}$ is a functor

$\displaystyle H_5 : \mathsf{Top} \rightarrow \mathsf{Grp}$

i.e. to every space ${X}$ we can associate a group ${H_5(X)}$ . (Of course, replace ${5}$ by integer of your choice.) Recall that:

Definition 1

Two maps ${f, g : X \rightarrow Y}$ are homotopy equivalent if there exists a homotopy between them.

Thus for a map we can take its homotopy class ${[f]}$ (the equivalence class under this relationship). This has the nice property that ${[f \circ g] = [f] \circ [g]}$ and so on.

Definition 2

Two spaces ${X}$ and ${Y}$ are homotopic if there exists a pair of maps ${f : X \rightarrow Y}$ and ${g : Y \rightarrow X}$ such that ${[f \circ g] = [\mathrm{id}_X]}$ and ${[g \circ f] = [\mathrm{id}_Y]}$ .

In light of this, we can define

Definition 3

The category ${\mathsf{hTop}}$ is defined as follows:

The objects are topological spaces ${X}$ .
The morphisms ${X \rightarrow Y}$ are homotopy classes of continuous maps ${X \rightarrow Y}$ .

Remark 4

Composition is well-defined since ${[f \circ g] = [f] \circ [g]}$ . Two spaces are isomorphic in ${\mathsf{hTop}}$ if they are homotopic.

Remark 5

As you might guess this “quotient” construction is called a quotient category.

Then the big result is that:

Theorem 6

The induced map ${f_\sharp = H_n(f)}$ of a map ${f: X \rightarrow Y}$ depends only on the homotopy class of ${f}$ . Thus ${H_n}$ is a functor

$\displaystyle H_n : \mathsf{hTop} \rightarrow \mathsf{Grp}.$

The proof of this is geometric, using the so-called prism operators. In any case, as with all functors we deduce

Corollary 7

${H_n(X) \cong H_n(Y)}$ if ${X}$ and ${Y}$ are homotopic.

In particular, the contractible spaces are those spaces ${X}$ which are homotopy equivalent to a point. In which case, ${H_n(X) = 0}$ for all ${n \ge 1}$ .

2. Relative Homology, ${H_n : \mathsf{hPairTop} \rightarrow \mathsf{Grp}}$

In fact, we also defined homology groups

$\displaystyle H_n(X,A)$

for ${A \subseteq X}$ . We will now show this is functorial too.

Definition 8

Let ${\varnothing \neq A \subset X}$ and ${\varnothing \neq B \subset X}$ be subspaces, and consider a map ${f : X \rightarrow Y}$ . If ${f(A) \subseteq B}$ we write

$\displaystyle f : (X,A) \rightarrow (Y,B).$

We say ${f}$ is a map of pairs, between the pairs ${(X,A)}$ and ${(Y,B)}$ .

Definition 9

We say that ${f,g : (X,A) \rightarrow (Y,B)}$ are pair-homotopic if they are “homotopic through maps of pairs”.

More formally, a pair-homotopy ${f, g : (X,A) \rightarrow (Y,B)}$ is a map ${F : [0,1] \times X \rightarrow Y}$ , which we’ll write as ${F_t(X)}$ , such that ${F}$ is a homotopy of the maps ${f,g : X \rightarrow Y}$ and each ${F_t}$ is itself a map of pairs.

Thus, we naturally arrive at two categories:

${\mathsf{PairTop}}$ , the category of pairs of toplogical spaces, and
${\mathsf{hPairTop}}$ , the same category except with maps only equivalent up to homotopy.

Definition 10

As before, we say pairs ${(X,A)}$ and ${(Y,B)}$ are pair-homotopy equivalent if they are isomorphic in ${\mathsf{hPairTop}}$ . An isomorphism of ${\mathsf{hPairTop}}$ is a pair-homotopy equivalence.

Then, the prism operators now let us derive

Theorem 11

We have a functor

$\displaystyle H_n : \mathsf{hPairTop} \rightarrow \mathsf{Grp}.$

The usual corollaries apply.

Now, we want an analog of contractible spaces for our pairs: i.e. pairs of spaces ${(X,A)}$ such that ${H_n(X,A) = 0}$ for ${n \ge 1}$ . The correct definition is:

Definition 12

Let ${A \subset X}$ . We say that ${A}$ is a deformation retract of ${X}$ if there is a map of pairs ${r : (X, A) \rightarrow (A, A)}$ which is a pair homotopy equivalence.

Example 13 (Examples of Deformation Retracts)

If a single point ${p}$ is a deformation retract of a space ${X}$ , then ${X}$ is contractible, since the retraction ${r : X \rightarrow \{\ast\}}$ (when viewed as a map ${X \rightarrow X}$ ) is homotopic to the identity map ${\mathrm{id}_X : X \rightarrow X}$ .
The punctured disk ${D^2 \setminus \{0\}}$ deformation retracts onto its boundary ${S^1}$ .
More generally, ${D^{n} \setminus \{0\}}$ deformation retracts onto its boundary ${S^{n-1}}$ .
Similarly, ${\mathbb R^n \setminus \{0\}}$ deformation retracts onto a sphere ${S^{n-1}}$ .

Of course in this situation we have that

$\displaystyle H_n(X,A) \cong H_n(A,A) = 0.$

3. Homotopy, ${\pi_1 : \mathsf{hTop}_\ast \rightarrow \mathsf{Grp}}$

As a special case of the above, we define

Definition 14

The category ${\mathsf{Top}_\ast}$ is defined as follows:

The objects are pairs ${(X, x_0)}$ of spaces ${X}$ with a distinguished basepoint ${x_0}$ . We call these pointed spaces.
The morphisms are maps ${f : (X, x_0) \rightarrow (Y, y_0)}$ , meaning ${f}$ is continuous and ${f(x_0) = y_0}$ .

Now again we mod out:

Definition 15

Two maps ${f , g : (X, x_0) \rightarrow (Y, y_0)}$ of pointed spaces are homotopic if there is a homotopy between them which also fixes the basepoints. We can then, in the same way as before, define the quotient category ${\mathsf{hTop}_\ast}$ .