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I came a cross an exercise the other day considering the following quotient space: Let $T$ be a torus and let $A, B \hookrightarrow T$ be two parallel circles. Let $X$ be the quotient space collapsing all of $A$ to a point, and all of $B$ to a different point.

What would be the geometric interpretation of $X$? I was thinking something along the lines of a torus pinched at two points? Any ideas?

mNugget
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2 Answers2

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Yes, it would be a "double-pinched" Torus, and here is a picture even though I know images are discouraged.

Also, this was discussed in some detail in this post:

Homology of doubly-pinched torus

enter image description here

AlgTop1854
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    Images are discouraged when an OP uses them in lieu of actually typing the question. They are entirely appropriate in both questions and answers when they contribute to the understanding. – Ethan Bolker Dec 04 '23 at 15:56
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    Not only they are appropriate. In some cases one image can replace thousands words. This is the case. The answer could literally be the image only, and it would still be perfect. – freakish Dec 04 '23 at 15:59
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    Thanks (both comments) - I will not "apologize" next time - images are definitely invaluable in these cases! – AlgTop1854 Dec 04 '23 at 16:00
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    I think it's just screenshots of text that are discouraged, not images in general! – Stef Dec 04 '23 at 16:56
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$X$ is homotopy equivalent to $S^1\vee S^2\vee S^2:$ enter image description here

$$H_0(X)=\Bbb Z, \pi_0(X)=0$$ $$H_1(X)=\Bbb Z, \pi_1(X)=\Bbb Z$$ $$H_2(X)=\Bbb Z\oplus\Bbb Z, \pi_2(X)=\Bbb Z\oplus\Bbb Z$$ For $k\geq3$: $$H_k(X)=0$$ $$\pi_k(X)=\pi_k(S^2)\oplus\pi_k(S^2)\oplus ...$$

Bob Dobbs
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  • @AlgTop1854 Thanks – Bob Dobbs Dec 04 '23 at 17:53
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    @BobDobbs I’m looking again - aren’t the homotopy groups of a wedge far more complicated? – AlgTop1854 Dec 06 '23 at 02:31
  • @AlgTop1854 Ooops. İt seems so. https://math.stackexchange.com/questions/913022/homotopy-groups-of-a-wedge-sum – Bob Dobbs Dec 06 '23 at 18:02
  • @AlgTop1854 İs $\pi_2$ okay? – Bob Dobbs Dec 06 '23 at 18:22
  • Yes - for $pi_2$ it matches $H_2$ by the Hurewicz theorem, and for higher homotopy groups there are additional summands (lots of them!). – AlgTop1854 Dec 06 '23 at 18:38
  • @AlgTop1854 Lots of them? How many? You reply like you know, references. – Bob Dobbs Dec 06 '23 at 18:52
  • I did not mean to be flippant about this. There is paper/theorem of Hilton that covers this, and discussed in this previous post (see second answer). https://math.stackexchange.com/questions/2730707/homotopy-groups-of-sl-vee-sk?rq=1. I have not read it in a long time, but the main idea is that it can be described but as a complicate sum. – AlgTop1854 Dec 06 '23 at 19:07