Isotopy and Homotopy

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

Solution 1:

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.