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I am looking for any resources (book/online) that teaches and further elaborates on how the "cut and glue" technique works for $\Delta$-complexes.

To be precise, I am looking for techniques and at least some semi-rigorous theoretical framework explained nicely in a textbook style to solve questions like this in Hatcher:

Show that the $\Delta$-complex obtained from $\Delta^3$ by performing the edge identifications $[v_0,v_1]\sim [v_1,v_3]$ and $[v_0,v_2]\sim [v_2,v_3]$ deformation retracts onto a Klein bottle.

Hatcher doesn't explain it in his book as far as I read (does he?), and online solutions like Show that the $\Delta$-complex obtained from $\Delta^3$ by performing edge identifications deformation retracts onto a Klein bottle. unfortunately is hard to understand for one, and don't look very rigorous to me.

Thanks for any help! Explanations of how to do the question above will also be upvoted and accepted.

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yoyostein
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1 Answers1

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You will have to get used to the fact that sometimes pictures suffice to give a proof. The idea is that a picture can be made "rigorous" by writing down an explit homeomorphism from the picture. A lemma that is often implicitly used here is the following:

Let $Y$ be a Hausdorff space, and let $X$ be a compact, and $\sim$ and equivalence relation on $X$. Let $f:X\rightarrow Y$ be a surjective continuous mapping that is constant on the equivalence classes. Then $f$ induces a surjective map $\overline{f}:X/\sim \rightarrow Y$. If $\overline{f}$ is bijective, then it is a homeomorphism.

This is best understood with a very simple example: For example to show that $[0,1]$ with endpoints identified is homeomorphic to $S^1=\{(x,y)\in\mathbb{R}^2\,|\,x^2+y^2=1\}$, we define the equivalence relation $$ t\equiv t':\Longleftrightarrow t=t'\quad \text{or} \quad t=0,t'=1, \quad\text{or} \quad t=1,t'=0 $$

one writes down the map

$$ f(t)=(cos(2\pi(t),\sin(2\pi t)) $$

Then $f$ satisfies the hypothesis so $[0,1]/\sim\cong S^1$.

Thomas Rot
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