(The most voted answer to) This question shows spaces of the same dimension can be homotopy equivalent but no homeomorphic. On the other hand "difference in dimension" is still a nice way to tell apart homotopies from homeomorphisms.
I'm not quite sure how to formulate my question precisely, but here goes:
What is the essence of the idea behind the counterexample in the first
answer
to the linked question? (Please don't just say $\mathsf{Y}$ is a deformation
retract of $\mathsf{X}$.)More generally, what kind of "topological differences" can homotopy
equivalences ignore except dimension?
Update: In light of Stefan Hamcke's comment "I think most, if not all local properties can be ignored by homotopy equivalences", I think this is the statement I should try and understand. Thus, I am asking for as-detailed-as-possible (yet formal) explanations of this sentence. Furthermore, since homotopy equivalences ignore local properties, how do plain homotopies behave with them? I'm guessing the same is no longer true, but why?
Best Answer
The essence of the counterexample “X homotopy equivalent to Y but X not homeomorphic to Y” is the following. Consider a space $S$ which is homotopy equivalent to a point but is not a point, then attach $S$ to anoter space $R$, then you’ll get (with some few exceptions) a space $R'$ homotopy equivalent to $R$ but not homeomorphic to $R$. (In the counterexample this is the “fourth leg” attached to Y in order to obtain X.) The attaching procedure can be formalized by choosing two points, one in $S$ and one in $R$ and identifying them.
So a good source of understanding what kinds of properties of a space can be ignored by homotopy equivalences, is to focus on spaces that are homotopy equivalent to points but are not points. Such spaces can intuitively be though as spaces than retract on one of its points. For examples any cone is homotopy equivalent to a point. A cone of a space $X$ is the space obtained first by taking the product $X\times [0,1]$ and then by identifying $X\times\{1\}$ to a single point (the vertex of the cone). This provides a huge class of spaces which are homotopy equivalent to points. Note that Y is the cone of three points and X is the cone of four points. Both are homotopy equivalent to a point.
Properties that are invariant under homotopy equivalence, so that one can distinguish two not homotopy equivalent spaces by means of such properties, are for instance those coming from all homotopy groups (the fundamental group and other). This is the core of the theories of invariants in general.