Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

What is the difference between homotopy and isotopy at the intuitive level.Some diagrammatic explanation will be helpful for me.

share|cite|improve this question
up vote 8 down vote accepted

Isotopies are much stricter!

A homotopy is a continuous one-parameter family of continuous functions.

An isotopy is a continuous one-parameter 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 one-to-one 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 one-to-one and onto at every step. In this way, any two handlebodies of equal genus are isotopic.

share|cite|improve this answer
nice anaswer.just what i wanted to know. – Koushik Feb 7 '13 at 8:14

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.