Intuition Regarding Homotopic Spaces

Question

I am just starting to do some algebraic topology (very basic stuff) so have obviously just been introduced to the notion of homotopys, contractible spaces and homotopic spaces. It's this last one that I am having a bit of trouble with.

For a lot of things that I have covered in topology/analysis there are intuitive ways to think about concepts. For example we can think of two spaces being homeomorphic if we can in some sense "bend and stretch" one space into the other. We can think of a homotopy between two functions to be "Continuously sliding" one function into the other.

There are also some obvious invariants for some of these things like number of connected components/connectedness....

However I have no way of thinking about wither two spaces are homotopic apart from trying to find functions between them that are homotopic equivalents beyond the defintion so I was looking for some help with this. As ever I apologise if this is a poor question or and if I am asking/saying something stupid I am always happy to be told so!

Thanks for any help

Answer 1

Please note that the first time I wrote this response, I was writing about isotopies and not homotopies!!! A gross error on my part, now fixed.

Homotopic spaces are those that can be deformed into one another by a series of "bending, stretching, and squeezing" operations which do not necessarily have to be one-to-one or onto. So, every homeomorphism is a homotopy, but certainly not the other way around.

For example, consider removing a disk from a torus $\overline{S^1\times S^1-D^2}$, you can think of the homotopy to the wedge of two circles by putting your fingers into the hole and stretching and squeezing the entire torus back onto its meridian and longitude lines. The end result, a figure-8 shape, is not the same dimension as the torus, and is not even a manifold!