# Homotopy equivalence of the two sphere with poles identified

Hello I am having trouble with the following exercise: We are asked to show that the following spaces are homotopic equivalent:

• The two sphere with north and south pole identified: $$\mathbb S^2/\{N ∼ S\}$$. (Here N and S represent the north and south pole.

• The two sphere with a interval glueing the north and south pole together: $$(\mathbb S^2\cup[0, 1])/\{N ∼ 0, S ∼1\}$$.

I need to write down two functions $$f,g$$ s.t. they are homotopic inverses of each other.

Intuitively it makes sense to me since we can retract the interval to a single point, and the end points of the interval are identified with the north and south pole. And since the interval is contractible it is homotopic equivalent to a single point. But writing explicit functions seems kinda hard.

• Definition: Two spaces $$X, Y$$ are homotopic equivalent if there exists $$f:X\to Y$$, and $$g:Y\to X$$ s.t. $$g\circ f ∼ id_X$$ and $$f \circ g ∼ id_Y$$. The composition of the maps is homotopic to the identity.

We were also given a hint to write the two sphere as two copies of discs with the boundary glued together. I don’t really see how this helps with finding the functions $$f,g$$.

Can someone give a hint on how to define the functions $$f$$ and $$g$$?

Let $$X= (\mathbb S^2\cup[0, 1])/\{N ∼ 0, S ∼1\},\qquad \qquad Y=\mathbb S^2/\{N ∼ S\}.$$

Then $$f\colon X\to Y$$ is easy to define. Following your intuition of retracting the interval to a point, just set $$f|_{\mathbb S^2}=1_{\mathbb S^2}$$ and $$f(t)=N=S$$ for all $$t\in [0,1]$$.

Now we need to define $$g\colon Y\to X$$. The advantage of regarding $$\mathbb S^2$$ as a union of a pair of (unit) disks, is that on each disk we can parameterize using polar co-ordinates, with $$S$$ or $$N$$ as the origin.

The idea now is simple: Firstly $$g\colon N,S\mapsto 0.5\in [0,1]$$. Then for a point $$(r,\theta)$$, on either disk of $$Y$$, if $$r\leq0.5$$ we map $$g\colon (r,\theta)\mapsto 0.5\pm r\in [0,1].$$ Here the sign $$\pm$$ is determined by which disk $$(r,\theta)$$ lies on.

Finally if $$r\geq0.5$$ we map $$g\colon (r,\theta)\mapsto (2r-1,\theta).$$

Now you have to construct the homotopies $$g\circ f ∼ id_X$$ and $$f \circ g ∼ id_Y$$. Let me know if you need help with that.

Let's start with the composition $$f \circ g$$. This collapses points of radius less than $$0.5$$ to $$N=S$$, and applies the function $$2r-1$$ to radii greater than $$0.5$$.

We need to construct $$H_t\colon Y\to Y$$ for $$t\in [0,1]$$ so that $$H_0=1_Y$$ and $$H_1=f \circ g$$.

It makes sense then for $$H_t$$ to collapse points of radius less than $$t/2$$ to $$N=S$$. Then radii from $$t/2$$ to $$1$$ should get mapped to radii from $$0$$ to $$1$$, which we can do linearly:

$$H_t(r,\theta)=H_t\left(\left(\frac {r-\frac t2}{1-\frac t2},\theta\right)\right),\qquad r\geq t/2,$$

and $$H_t(r,\theta)=N=S$$ if $$r\leq t/2$$.

Now check these two definitions agree at the boundary $$r=t/2$$, and that $$H_0=1_Y$$ and $$H_1=f \circ g$$. Check also that points of radius $$1$$ get mapped to points of radius $$1$$ for all $$t$$, as this is needed to ensure continuity where the disks are glued.

• To be honest I find it hard to compare those maps when composed because we have so much going on with identifying the sphere with two discs glued together and $g$ is not very trivial. Could u show me how to construct the homotopies? I never saw any non trivial examples of spaces being homotopic equivalent. Its like I have no tools in my bag, maybe seeing more examples will give me a better understanding. Thanks!
– user1027529
Commented Feb 19, 2022 at 19:00
• OK I will do one to start with. Before I do that - have you checked that the maps I defined are continuous e.g. the local definitions agree at the boundaries?
– tkf
Commented Feb 19, 2022 at 19:02
• I think f is continuous due to the pasting lemma. And g looks well defined on the boundary of either disk and is continuous aswel!
– user1027529
Commented Feb 19, 2022 at 19:04
• Yes, but make sure you have checked that the definitions agree at the boundary $r=0.5$ and that $g(N)=g(S)$.
– tkf
Commented Feb 19, 2022 at 19:15
• Yes I just wrote it out! Im still struggling to make sense that the maps composed are homotopic to the identity.
– user1027529
Commented Feb 19, 2022 at 19:51