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Recall the mapping cone of a map $f: X\rightarrow Y$ is defined as the space $C_f: X\times [0, 1]\dot{\cup} Y/\sim$, where $\sim$ is the equivalence relation given by $(x, 1)\sim f(x)$ and $(x, 0)\sim(x', 0) $ for all $x, x'\in X$. Show that if $f, g: X\rightarrow Y$ are maps such that $f$ is homotopic to $g$, then the spaces $C_f$ and $C_g$ are homotopic.

In the first attempt to solve it, I tried to define natural maps between $C_f$ and $C_g$ that could induce homotopic inverse of each other, but I have not got the answer. When I check Hatcher's book ``Algebraic Topology", I find that I could directly use proposition 0.18 on page 17. But after I read the proof, I find it still does not provide us the homotopic maps between $C_f$ and $C_g$ directly.

So, my first question is:

Can we find a natural homotopic equivalence between $C_f$ and $C_g$ without using the process provided in proof of proposition 0.17 mentioned above?

I want to ask a more general or soft problem, since I am a beginner to learn algebraic topology, when I am asked to prove some spaces are homeomorphic (or homotopic) or some maps are homotopic, the first thing come to mind is apply the definition of homeomophism or homotopic etc, so I have to find certain specific good mapsm, but it seems really hard to me in some situations, just as the situation above.

Take another problem for example, prove that $\frac{S^1\times [0, 1]\dot{\cup}S^1}{(z, 0) \sim z^2}$ is homeomorphic to the Möbius band. My first attempt is to prove $\frac{S^1\times [0, 1]\dot{\cup}S^1}{(z, 0) \sim z^2}$ is homeomorphic to $\frac{S^1\times [0, 1]}{(z,0) \sim(-z,0)}$, this time I can easily give the homeomorphism, but when I tried to find some natural homeomorphism from the latter to the Möbius band, which is defined as $\frac{[0, 1]\times[0, 1]}{(x, 0)\sim(1-x,1)}$, it is very hard for me, and finially I have to do some cutting and gluing operation to prove they are homeomorphic without giving a specific map.

So, My second question is:

Are there any suggestions when facing these situations? To be more specific, I mean when one tries to find some natural maps, but it is not easy at all.

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I replaced your \bigudot, which didn't work, with \dot{\cup} - just to let you know, in case it isn't what you actually wanted. –  Matt Pressland Oct 8 '12 at 12:47
    
@Matt, many thanks! –  ougao Oct 8 '12 at 13:01
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You may find helpful the more leisurely approach in Chapter 7 of my book "Topology and Groupoids" (pages.bangor.ac.uk/~mas010/topgpds.html) See also the comments in mathoverflow.net/questions/96071 –  Ronnie Brown Oct 8 '12 at 15:54
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I'd like to point out that this is proved in much detail in proposition 3.2.15 in Arkowitz's Introduction to Homotopy Theory. –  lentic catachresis Mar 25 at 19:23

1 Answer 1

Hint for the first question. Given the cone $\Gamma X$ of a topological space $X$, you can cut it in the middle : $\Gamma X \simeq X_1 \cup X_2$ with $X_1 \simeq \Gamma X$, $X_2 \simeq X\times[0,1]$ and $X_1 \cap X_2 \simeq X$.

Given $f,g \colon X \to Y$ homotopic by $H \colon X \times [0,1] \to Y$, you could now be able to define a map $$M_g \to M_f$$ which will be (easily) a homotopy equivalence.

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