Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $ f:X\to Y $ be a map in the pointed category of topological spaces $ Top_* $. And let $ U:Top_*\to Top $ be the "forgetful" functor (which "forgets" the basepoint). We can look at the reduced mapping cylinder $ M_f $ and at the unreduced mapping cylinder $ M_{U(f)} $ in the category $ Top $. Until yesterday, I thought $ M_f $ has the same homotopy type of $ M_{U(f)} $. But, in May's Concise Course, he says "If $X, Y$ are well-pointed, then $ M_f $ has the same homotopy type of $ M_{U(f)} $."

Is this hypothesis necessary?

If it is truly necessary, I want to know where I'm wrong. I thought this: Since $ M_{U(f)} $ is a pushout of a trivial cofibration, we have that $ Y\to M_{U(f)} $ is a trivial cofibration. The same way (or only using a explicit homotopy), we have that $ Y\to M_f $ is a homotopy equivalence. So we have that $ M_{U(f)}\equiv Y\equiv M_f $.

I know that the first statement is right. If there is something wrong, it is in the second statement. I believed that we can factor any function in $ Top _* $ in the same way as in $ Top $, id est, $ f= R\circ j $, where $ j: X\to M_f $ is a cofibration and $ R: M_f\to Y $ is a strong retract. Is it wrong?

share|improve this question
    
Cross-posted on math.SE: math.stackexchange.com/questions/198319/… –  Qiaochu Yuan Sep 18 '12 at 3:44
    
Sorry. My intention was to post only in one of the foruns. How I do I delete the other question? Thank you –  Nunes Sep 18 '12 at 6:21
add comment

2 Answers 2

up vote 4 down vote accepted

I don't trust anything with degenerate basepoints, but I do think you are right about the homotopy type via the parenthetical "(or only using an explicit homotopy)''. However, I was working in compactly generated spaces, denoted $\mathcal{U}$, including the weak Hausdorff property. I don't think you can prove that $Mf$, constructed as usual, is in $\mathcal{U}$ without assuming nondegenerate basepoints. Also, without nondegenerate base points, you cannot be sure that $j$ is a cofibration (at least not in the unbased sense; see p. 56 of Concise). In More Concise, Ponto and I take $\mathcal{T}$ to mean nondegenerately based spaces and $\mathcal{U}_*$ to mean based spaces in $\mathcal{U}$. This is justified model theoretically by noting that $\mathcal{T}$ is the full subcategory of $h$-cofibrant objects in $\mathcal{U}_{*}$ (that is, cofibrant in the based Hurewicz model structure on $\mathcal{U}_{*}$). Incidentally, one reason to start work in $\mathcal{U}_{*}$ and not $Top_{\ast}$ is that the smash product in $Top_{\ast}$ is not associative (a published source for a very old counterexample is Parametrized homotopy theory, by Sigurdsson and myself).

share|improve this answer
    
Thank you, professor! Sorry, professor! When I say, $ Top_* $, I mean compactly generated spaces ( I am very familiar with this differences, thanks your book ) But I am still not understanding. I thought $ M_f $ is a compactly generated space by the second proposition (pag. 38). Is the inclusion $ X\to X\wedge I_+ $ closed? And does this closure imply $ M_f $ compactly generated? –  Nunes Sep 18 '12 at 6:01
add comment

I'm lousy at point-set topology, always was, but I don't see that $X$ is closed in $X\wedge I_+$. This is the sort of question a working algebraic topologist does not want to think about. Cofibrant approximation in the $h$-model structure on $\mathcal{U}_{*}$ takes $X$ to the whiskered space $X\vee I$ with new basepoint at $1$ if the basepoint of $I$ is taken to be $0$. If $X$ has a nasty degenerate basepoint, take the nasty thing away: I do not want to think about it.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.