Homotopy Equivalence intuition Can somebody tell me intuitively what does it mean geometrically when we say two spaces are homotopy equivalent ? I understand the technincal definition.
 A: Homotopy equivalence is a weaker version of equivalence than homeomorphism. A homotopy equivalence can be thought of as a continuous squishing and stretching (i.e. a deformation) of the space. I'd like to give some examples of homotopy equivalences which are and are not homeomorphisms.
1) Every pair of spaces that are homeomorphic are homotopy equivalent. This is because you can take a family of maps parameterized by $t \in [0,1]$ as the homotopy which start at the identity and end at the homeomorphism. If $f$ is the homeomorphism, the family might be something like $(1-t)Id + tf$.
2) Spaces that are not homeomorphic might be homotopy equivalent. Consider the letter X. We can contract this space to its center point by sucking up the horizontal lines on the legs, and then pulling the legs in to the center point. However, X is not homeomorphic to a point, because this map is not bijective (and more generally, X minus its center point has 4 connected components, but the point minus a point is just empty).
3) For compact surfaces (without boundary), homotopy equivalence and homeomorphism are actually the same thing. If you're not familiar with surface theory, there are an abundance of good references on the basics that give the classification theorem which should clear this point up. I think Munkres is the standard reference.
A: It's easier to first understand what a deformation retraction is. Namely if $A\subset X$, $X$ deformation retracts onto $A$ if there is a homotopy from the identity on $X$ to a retraction $X\to A$ which fixes $A$ the whole time. Intuitively, you are continuously shrinking the parts of $X$ not in $A$ until they all land in $A$.
On the other hand a general homotopy equivalence can always be written as a composition of deformation retractions, so once you understand this, you also understand the general case, in some sense.
