Theorem 1B.9 in Hatcher's Algebraic Topology says that for a (pointed) connected CW complex $X$ and group $G$, there is a bijection $\text{Hom}(\pi_1(X), G) \cong [X,K(G,1)]$, where $\pi_1(X)$ is the first fundamental group of $X$, and $K(G,1)$ is the first Eilenberg-MacLane space of $G$. I guess he is describing an adjunction of functors here, between the category of homotopy classes of maps between connected pointed CW complexes and groups.
This surprised me. If $\pi_1$ is a left-adjoint functor, then we should conclude that it is cocontinuous, i.e. takes pushouts to pushouts. But I had understood the van Kampen theorem to say something like "$\pi_1$ takes certain pushouts in $\text{hTop}_*$ to pushouts in groups". For example, van Kampen requires the morphisms to be inclusions, among other things. Presumably then not all pushouts are preserved under $\pi_1$, for example if the maps are not inclusions.
I tried to come up with a pushout of non-injective pointed topological spaces which would give a counterexample to van Kampen, but I could not. Is there one? Can you give one?
And if there is one, why doesn't that contradict the status of $\pi_1$ as a left-adjoint? And if there isn't one, then why can't the hypotheses of the van Kampen theorem be weakened?
Best Answer
The problem is that there are not a lot of actual colimits in the homotopy category of (connected) CW complexes, so knowing that $\pi_1$ preserves them (which is true) is pretty much useless. The pushouts appearing in the van Kampen theorem are pushouts in $Top$ but not in the homotopy category, so the van Kampen theorem does not follow from this adjunction. On the other hand, the functor $\pi_1$ preserves all homotopy colimits, and the hypotheses in the van Kampen theorem guarantee that the pushout in Top is a homotopy pushout.