It seems to me that commuting up to homotopy certainly implies commuting in $\pi_1$, but when is the converse also true?
I am reading something on surfaces, and it says in that case commuting in $\pi_1$ implies commuting up to homotopy, by some "standard facts from algebraic topology", but I am unable to find it by searching the internet. Does it have anything to do with the higher homotopy groups (which I am also not very familiar with)? How about in general in higher dimensions?
Thanks in advance!!
Edit: I probably didn't articulate very well… I am looking into this case. If I have the following commutative diagram.
\begin{array}{ccc} \pi_1(M)&\longrightarrow & \pi_1(M)\\
\downarrow & & \downarrow{\alpha_*}\\
\pi_1(N)&\longrightarrow & \pi_1(N)
\end{array}
I would like to know when I can tell that I have the following diagram
\begin{array}{ccc} M &\longrightarrow & M \\
\downarrow & & \downarrow{\beta}\\
N &\longrightarrow & N
\end{array}
that also commutes but up to homotopy?
Additionally, suppose as in the above diagrams the corresponding maps are $\alpha_*$ which is induced by some map $\alpha$ and $\beta$ induces the same homomorphism on $\pi_1$, when can I tell $\alpha$ is actually homotopic to $\beta$? I guess this is essentially the same as asking, when are the two maps that induce the same homomorphism on $\pi_1$ really homotopic to each other?
I am in the case where the above maps are surjective, but I would love to know if it is true in a more general case.
Best Answer
This is about the fact that the universal cover of a closed surface of positive genus is contractible. In particular, all of its "higher homotopy groups" are zero. Such spaces are called "aspherical" and more generally, if precisely one homotopy group is nonzero, "Eilenberg MacLane spaces". In the aspherical case, these are denoted $K(G,1)$.
The general theorem is this. Pointed homotopy classes of pointed maps $K(G,1) \to K(H,1)$ are in bijection with the set of homomorphisms $G \to H$. (There is a proof of this in Hatcher's 1.B.) If you want unbased homotopy classes, you quotient by the action of $G$ and $H$ acting on this set by conjugacy.
In particular, two based maps that induce the same maps on fundamental groups of surfaces of positive genus are homotopic.
Such a property is essentially never true if the domain or codomain are not both Eilenberg MacLane spaces $K(G,n)$ with the same $n$, unless your result is of the form "All maps are null-homotopic".