The center of $\mathbf{hTop}$

Question

What is the center of the homotopy category $\mathbf{hTop}$? I strongly believe that it is trivial, but it is hard to prove since $\mathbf{hTop}$ is not concretizable and hence has no small separator. This also means that currently I have no proof why the center is even a set ...

The center of $\mathbf{hTop}$ consists of homotopy classes of continuous maps $\alpha_X : X \to X$ for every space $X$. For every continuous map $f : X \to Y$ there should be a homotopy between $\alpha_Y \circ f$ and $f \circ \alpha_X$. These homotopies are not subject to any compatibility conditions when $f$ changes.

Clearly, $\alpha_X$ is (homotopic to) the identity when $X$ is contractible. Also, $\alpha_{\coprod_{i \in I} X_i}$ identifies with $\coprod_{i \in I} \alpha_{X_i}$, so the same holds when $X$ is a coproduct of contractible spaces. The easiest space which is not of this form is the circle $S^1$. I have no idea how to approach $\alpha_{S^1}$.

For $x \in X$, naturality with respect to $x : \star \to X$ shows that there is a path from $x$ to $\alpha_X(x)$.

Notice that for any functor $K : \mathbf{hTop} \to \mathcal{C}$ (for example $H_*,H^*,\pi_*$) we have a map $Z(\mathbf{hTop}) \to \mathrm{End}(K)$, so we get lots of operations.

I am open for $1$-categorical variations of the spaces, such as CW complexes, CGWH spaces or pointed spaces. But in this question I am not asking about higher categorical versions of the center.

Answer 1

Here's a partial answer constraining $\alpha_{S^1}$. First of all, I think my comment shows that (at least if we take $\operatorname{hTop}$ to be the homotopy category of CW complexes), it is possible for $\alpha_{S^1}$ to be constant. Let's think about what other degrees besides $0$ and $1$ are possible!

Assume $\alpha_{S^1}$ has degree $n$. By looking at the commutative diagrams for the quotient maps $(S^1)^{\times k}\to S^k$, we see that $\alpha_{S^k}$ has to be the degree $n^k$-map. Now look at the commutative diagram for $\eta: S^3\to S^2$. Precomposing $\eta$ by the degree $n^3$-map on $S^3$ yields the element $n^3\cdot \eta\in \pi_3(S^2)$. On the other hand, postcomposing $\eta$ with the degree $n^2$ map yields $n^4\cdot\eta\in\pi_3(S^2)$. So we learn $n^4=n^3$, and thus $n=0$ or $1$.

(That postcomposition on $\eta$ behaves in this quadratic way can be seen by viewing $\eta$ as the map $\mathbb{C^2}\setminus\{0\}\to \mathbb{C}P^1$ taking $(z,w)\mapsto \frac{z}{w}$, and realising that this commutes with taking $d$-th powers on either side. But on the right, $d$-th power is of degree $d$, on the left, of degree $d^2$.)

I'm not sure what to expect about the full description of the center. If $\alpha_{S^1}$ is the identity, we certainly see that all $\alpha_{S^k}$ are identities as above, but $\alpha$ could still do funny things on more complicated cell complexes. I think it does follow though that $\alpha$ induces identities on homology. Similarly, if $\alpha_{S^1}$ is the constant map, it follows that this holds on spheres and generally on homology, but maybe not that $\alpha_X$ is constant on each connected space (as in the example I gave in the comment).