What is the difference between homotopy and isotopy at the intuitive level.Some diagrammatic explanation will be helpful for me.
1 Answers
Isotopies are much stricter!
A homotopy is a continuous oneparameter family of continuous functions.
An isotopy is a continuous oneparameter family of homeomorphisms.
You can think of a homotopy between two spaces as a deformation that involves bending, shrinking and stretching, but doesn't have to be onetoone or onto. For example, a punctured torus is homotopy equivalent to a wedge of two circles (a "figure 8"), which can be pictured by sticking your fingers into the puncture and stretching the torus back onto the meridian and longitude lines.
But this map is certainly not a homeomorphism  even the dimension is wrong, not to mention that a wedge of two circles is not a manifold.
An isotopy is a deformation that involves only bending. It must be onetoone and onto at every step. In this way, any two handlebodies of equal genus are isotopic.
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2nice anaswer.just what i wanted to know. – Koushik Feb 07 '13 at 08:14

5Animation of [Deformation Retract of Punctured Torus](https://www.youtube.com/watch?v=j2HxBUaoaPU) into figure 8 – Apiwat Chantawibul Dec 18 '16 at 21:39

figure 8 is a 1dimensional manifold, no? – Yan King Yin Dec 01 '20 at 07:42

2No, the “middle point” is not locally Euclidean. – gmoss Dec 02 '20 at 04:40