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(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:

  1. 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 $\mathsf{X}$.)

  2. 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?

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    I think most, if not all local properties can be ignored by homotopy equivalences. in contrast to homeomorphisms, which preserve all local properties. For example $\mathsf X$ has a different local topology than $\mathsf Y$ at the point in the middle. – Stefan Hamcke Feb 17 '15 at 20:12
  • @StefanHamcke what are some examples of interesting such local properties? –  Feb 17 '15 at 20:28
  • Well, local compactness, local path-connectedness, locally Euclidean, ... There's a precise definition: A space $X$ is locally $P$ if every point has a neighborhood base of sets which as a subspace of $X$ have property $P$. But I'm not sure, maybe there's a local property such that $X$ and $Y$ cannot be homotopy equivalent, when $X$ is locally $P$ and $Y$ is not. – Stefan Hamcke Feb 17 '15 at 20:43
  • Seems like none of the local properties $P$, where $P$ is a property dealing with (higher) connectivity, is an obstacle. If $X$ is locally $P$, then its cone $CX$ is also locally $P$. But any map between cones $CX\to CY$ is a homotopy equivalence. For example, one could construct a homotopy equivalence between a locally simply connected space and a space which is not. – Stefan Hamcke Feb 17 '15 at 21:03
  • @StefanHamcke: I think you need to ba bit more specific: e.g., what if $P$ is the property "not simply connected"? – Rob Arthan Feb 17 '15 at 21:53
  • @RobArthan: I think it should be a property that every contractible space has. Then the cone $CX$ has this local property as well since the apex of $CX$ has a local base of contractible sets. – Stefan Hamcke Feb 17 '15 at 22:10
  • @StefanHamcke: but that excludes some your examples like locally compact or locally Euclidean. – Rob Arthan Feb 17 '15 at 22:17
  • @RobArthan: One can of course find other spaces of the same homotopy type with different local topologies, I'm not saying that every counterexample involves cones. – Stefan Hamcke Feb 17 '15 at 22:25
  • A more general example is that of mapping cylinders. – Pandora Feb 24 '15 at 16:22
  • A basic idea is: If a path connected topological space is contractible then its homotopy type is the same as that of a point whereas obviously the homeomorphism is very much dimension dependent. – DBS Mar 23 '15 at 19:48
  • So from a categorical perspective: Homeomorphism is just isomorphism in the topological category where maps are continuous maps. From a categorical perspective:It is not much different than isomorphism of sets or isomorphism of vector spaces. – DBS Mar 23 '15 at 20:01
  • On the other hand homotopy is a far more subtle notion. It amounts to declaring an equivalence relation on a category i.e. (you are declaring that two spaces are isomorphic if after contracting all the contractible parts they are the same). So here the same space might have several incarnations (upto adding contractible parts) and given a problem you have the option of choosing the most suitable model for your problem. – DBS Mar 23 '15 at 20:07
  • The classical algebraic topological invariants like homology and cohomology are essentially homeomorphic invariants and they are easier to compute. It is well known that homotopic invariants (the higher homotopic groups) are much more difficult to compute [the fundamental group is a misleading example]. – DBS Mar 23 '15 at 20:09
  • What do you mean by "On the other hand "difference in dimension" is still a nice way to tell apart homotopies from homeomorphisms."? – Bruno Joyal Mar 27 '15 at 05:05
  • @BrunoJoyal I mean that different lebesgue covering dimension is often a useful test to rule out homeomorphic-ness of spaces –  Mar 28 '15 at 10:52

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  1. 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.

  2. 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.

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