Vtyjkyuk

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 $$X ast Y=Xtimes Ytimes I/(x_1,y,1)sim(x_2,y,1), (x,y_1,0)sim(x,y_2,0)$$ is associative for CW-complexes.

I know that there is a natural bijection between $Xast Yast Z$ as a quotient of $Xtimes Ytimes Z times Delta$ and $(Xast Y)ast Z$, 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 $$X hatast Y= xi+eta=1 $$ 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 $f: Xtimes Ytimes Itimes Ito X times Ytimes I, f(x,y,xi,eta)=(x,y,xi(1-eta))$. $f$ is continuous, as it is basically a map from $Itimes I to I$. 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 $CXtimes CY$ $(x_1,y,1,eta)sim(x_2,y,1,eta),$ 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 $Xtimes Ytimes I/(x_1,y,1)sim(x_2,y,1)$. Proceeding similarly for the other pair of equivalence relations we obtain a homeomorphism betweem $Xast Y$ and $X hatast Y$.

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 $Asubset X$, then $X/A simeq Y/f(A)$? I think it is.

2) Something breaks down due to the order in which I go from $f: Xtimes Ytimes Itimes I$ 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 $(Xast Y)ast Z$ and $Xast(Yast Z)$ 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 $Xtimes Y$ 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 $Xtimes Y$ 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.