I am currently self-studying a course in algebraic topology and one of the problems I encountered is to prove that the join operation defined as $$X \ast Y=X\times Y\times I/(x_1,y,1)\sim(x_2,y,1), (x,y_1,0)\sim(x,y_2,0)$$ is associative for CW-complexes.
I know that there is a natural bijection between $X\ast Y\ast Z$ as a quotient of $X\times Y\times Z \times \Delta$ and $(X\ast Y)\ast Z$, but I don't understand how to prove that it's a homeomorphism in the case of CW-complexes.
I found one related proposition in Fomenko's "Homotopical Topology" saying that the join operation is associative for compact spaces. However, it was left as an exercise for the reader. I think I understand how to prove something of the sort for compact Hausdorff spaces, but I am at a loss how compactness could be sufficient. And even then I have no idea how this could be generalized to CW complexes because the weak topology on a join of CW-complexes does not always coincide with the topology of the join.
Update: In the exercise mentioned in the comments it is recommended to consider a new join operation $$X\ \hat{\ast}\ Y=\{(x,\xi,y,\eta)\in CX \times CY\ |\ \xi+\eta=1\ \}$$ and show that for "good" spaces the two operations are equivalent. The problem is that my reasoning leads to the two operations being equivalent in all cases and I can't find the error. I suppose I don't understand quotient maps that well.
Consider $f: X\times Y\times I\times I\to X \times Y\times I, f(x,y,\xi,\eta)=(x,y,\xi(1-\eta))$. $f$ is continuous, as it is basically a map from $I\times I \to I$. Moreover, its restriction to the subset $A$ where $\xi+\eta=1$ is a homeomorphism. Now note that it behaves well with respect to quotients: if weconsider two equivalence relations in the join $(x_1,y,1)\sim(x_2,y,1)$ and in $CX\times CY$ $(x_1,y,1,\eta)\sim(x_2,y,1,\eta),$ then the preimages of equivalent points are equivalent and vice-versa. Hence, $f$ induces a homeomorphism between $A/(x_1,y,1,\eta)\sim(x_2,y,1,\eta)$ and $X\times Y\times I/(x_1,y,1)\sim(x_2,y,1)$. Proceeding similarly for the other pair of equivalence relations we obtain a homeomorphism betweem $X\ast Y$ and $X\ \hat{\ast}\ Y$.
This argument is clearly wrong. Right now I see two possible issues:
1) Is it true that if $f$ is a homeomorphism between $X$ and $Y$, and $A\subset X$, then $X/A \simeq Y/f(A)$? I think it is.
2) Something breaks down due to the order in which I go from $f: X\times Y\times I\times I$ to its subset, take the quotients and products. But again, I can't pinpoint the point at which these operations do not commute.
Any help would be greatly appreciated.
One other thing I thought of: is the original statement about the join operation being associative for CW-complexes even true? Maybe I misunderstood the problem and the aim is not to show that $(X\ast Y)\ast Z$ and $X\ast(Y\ast Z)$ are homeomorphic, but that they are homeomorphic as CW-complexes with the join structure?
Update 2: Thanks to another helpful comment by Steve D. I realised that the above argument (if written down more rigorously) works in the case when $X\times Y$ is Hausdorff and locally compact because of Whitehead's thorem.
Now it is true that if $X$ and $Y$ are CW-complexes and at least one of them is locally compact, then their join $X\times Y$ is also a CW-complex. Overall, for locally compact (equivalently, locally finite) CW-complexes there is no ambiguity in the question and it is indeed true that the join operation is associative. However, the case of general CW-complexes still remains out of reach for me.
