some deformation retractions

FibonacciFan · Riemann Hypothesis Offline Joined: 10 Jun 2006 Posts: 434

1. Construct an explicit deformation retraction of the torus with one point deleted onto a graph consisting of two circles intersecting in a point, namely, longitude and meridian circles of the torus. (Hatcher Exercise 0.1)

2. Construct an explicit deformation retraction of $\mathbf{R}^n - \{0\}$ onto $S^{n-1}$ . (Hatcher Exercise 0.2)

Attempted Solutions:

1. I can kind of visualize the deformation retraction: draw lines going radially outward from the point removed; all the points move along those lines to either the meridian or the longitude. But I think some of these lines will intersect and I have no idea how to write that down as a formula (I assume that is what explicitly means).

2. I assume the retraction is x/norm(x). As in 1, I think we want to move all the points radially toward S^{n-1}. I don't know how to pace that points so that this will be continuous. Won't the points have to move "infinitely fast" near t = 0?

**Posted:** Mon Aug 18, 2008 7:49 pm

jmerry

A deformation retract is a homotopy- which means you have to say how it moves. $\frac{x}{\|x\|}$ is a retract, but not a deformation retract.

A way to create a deformation retract: let $f(t,y)=y^t$ (or a similar homotopy between $(0,\infty)$ and $\{1\}$ ). Now, let the homotopy $g(t,x)=f(t,|x|)\frac{x}{|x|}$ .

**Posted:** Mon Aug 18, 2008 8:22 pm

FibonacciFan · Riemann Hypothesis Offline Joined: 10 Jun 2006 Posts: 434

**Posted:** Mon Aug 18, 2008 9:54 pm

jmerry

Interpret the torus as a rectangle with opposite sides identified. Cut out a point somewhere in the middle, and then shrink everything away from that point to the rim.

There's no need to quote the entire post immediately preceding yours, ever.

**Posted:** Mon Aug 18, 2008 10:36 pm

FibonacciFan · Riemann Hypothesis Offline Joined: 10 Jun 2006 Posts: 434

I see thanks. Can I move this thread to the "Solved Problems" thread or does a moderator need to do that?

**Posted:** Tue Aug 19, 2008 7:53 am

ZetaX

$\text{Unsolved Problems} \to \frame{\text{some miracle happens}} \to \text{Solved Problems}$

PS: No you can't Wink

**Posted:** Tue Aug 19, 2008 10:49 am

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