deformation retract and strong deformation retract

Question

I am trying to gain some intuition about retracts, deformation retracts and strong deformation retracts (see http://en.wikipedia.org/wiki/Deformation_retract for definitions). We have that any strong deformation retract is a deformation retract and any deformation retract is a retract (of course, by definition). Also, it is easy to think in a retract which is not a deformation retract.

Which are some examples of deformation retracts which are not strong deformation retracts?

See exercise 6 on page 18 of Hatcher's book. Note the difference in terminology to Wikipedia! — t.b., May 05 '11 at 21:56
Similar question to this one has been asked again and an answer can be found here. The same example but with more details can be found here. — Kal S., Apr 27 '14 at 06:59

score 0 · Answer 1 · answered Dec 13 '15 at 19:58

Take the closed unit ball In the complex plane. For time $t=0$ apply the identity map. For $t \in (0,1/2]$ apply the map $e^{t i \pi} z$. For $t \in (1/2, 1]$ contract the ball to a point, $p$, other than the center. Since for $t$ equal to say $1/3$ the image of $p$ is not $p$ this map is not a strong deformation retraction, but meets the requirements for a deformation retraction.

deformation retract and strong deformation retract

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