# Does the reduced Mapping cylinder have the same homotopy type of unreduced Mapping cylinder?

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?

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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

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).

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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

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.

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