Let textbfHoTop∗ be the homotopy category of pointed topological spaces. In the following, the word "isomorphism" shall always mean isomorphism in textbfHoTop∗, i.e. pointed homotopy equivalence. All constructions like cone or suspensions are pointed/reduced.
A triangle XtoYtoZtoSigmaX is called distinguished if it is isomorphic in textbfHoTop∗ to a triangle of the form XstackrelftoYhookrightarrowtextCftoSigmaX, where textCftoSigmaX is the map collapsing Y to a point.
Problem:
Let matrix{X & to & Y & to & Z & to & Sigma Xcrdownarrowalpha &&downarrowbeta&&downarrowgamma &&downarrow&Sigmaalphacr X^{prime} & to & Y^{prime} & to & Z^{prime} & to & Sigma X^{prime}} be a morphism of distinguished triangles such that alpha and beta are isomorphisms. Is it true that gamma is an isomorphism, too?
Suggestions:
For a morphism of triangles as above (where alpha and beta are not necessarily isomorphisms), the morphism gamma∗:[Zprime,−]to[Z,−] is equivariant with respect to [Sigmaalpha]∗:[SigmaXprime,−]to[SigmaX,−]. (edit: this is wrong -- see below) Therefore, I thought one could apply theorem 6.5.3 in Hoveys book on Model Categories. Unfortunately, there seems to be a gap at the end of the proof, as already pointed out here.
Therefore, I have the following
Questions:
(1) Am I misunderstanding something in Hovey's proof of 6.5.3(b), or is there really a gap in it? If it is a gap: Do you have any suggestions on how to fix the proof?
(2) If the proof can't be fixed in this generality: Do you have suggestions on how to prove the statement above only for textbfHoTop∗?
Edit:
(1) The usual proof of this fact for triangulated categories does not work here, because there one uses the fact that [X,−] is abelian-group valued for any X and uses the classical five lemma together with Yoneda to conclude that gamma is an isomorphism. This doesn't seem to work here.
(2) Since partial morphisms of distinguished triangles in textbfHoTop∗ can always be completed to morphisms of triangles, we can reduce to the case where alpha and beta both equal the identity. Therefore, we have a commutative diagram (in textbfHoTop∗, i.e. a homotopy commutative diagram in textbfTop∗)
matrix{X & to & Y & to & Z & to & Sigma Xcrdownarrow & text{id}_X &downarrow & text{id}_Y&downarrow&gamma&downarrow&text{id}_{Sigma X}cr X & to & Y & to & Z& to & Sigma X}
and we have to prove that gamma is a homotopy equivalence.
Hovey's proof
The way Hovey proceeds in his proof is as follows: We know the following things:
(1) gamma∗:[Z,−]to[Z,−] is [SigmaX,−]-equivariant
(2) Two maps c,din[Z,W] are equal in [Y,W] if and only if they lie in the same [SigmaX,W]-orbit.
From (2) and the commutativity of the middle square it follows that for any hin[Z,W] there is some rhoin[SigmaX,W] such that gamma∗(h)=h.rho; in other words gamma∗ doesn't change the [SigmaX,−]-orbit.
Now, suppose there are g,hin[Z,W] such that gamma∗(h)=gamma∗(g). Then, again by the commutativity of the middle square, there is some alphain[SigmaX,W] such that g=h.alpha. Thus, by (1), gamma∗(g)=gamma∗(h).alpha=gamma∗(g).alpha, and so alphaintextStab(gamma∗(g)).
The point is that Hovey now wants to show that textStab(gamma∗(g))=textStab(g); this would imply alphaintextStab(g), and thus h=g.alpha−1=g as required. The inclusion textStab(gamma∗(g))supsettextStab(g) is obvious. For the other inclusion, I have no idea how to prove it.
Do you see how one can fix the proof?
FINAL EDIT
I made a mistake in proving that for any morphism of triangles (alpha,beta,gamma) the morphism gamma∗ is equivariant with respect to (Sigmaalpha)∗. This is wrong.
So what remains is the question on how to fix the proof of theorem 6.5.3 in Hovey's book. Any suggestions?
Thank you.
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