Pedagogical challenge: Homeomorphic vs. Homotopy equivalent vs. Homologous?

Question

I believe it is the case that, between spaces, homeomorphism is stronger than homotopy equivalence which is stronger than having isomorphic homology groups.

For example, the annulus and the circle are not homeomorphic but they have the same homotopy type. For the comparison between homotopy and homology, I am less clear, but it was addressed in an MSE question.

My question is:

Q. What is the best way to explain the differences among the three concepts to students, beyond simply pointing to the definitions?

I am interested in approaches that would clearly convey the intuitive differences and yet be responsive to the technical differences. The audience is undergraduates learning topology. I am primarily interested in simplicial complexes, but perhaps it is not necessary to specialize to get across the concepts.

Answer 1

Homotopy equivalence v. Homeomorphism. I believe an accessible difference between homotopy equivalence and homeomorphism is that one preserves an intuitive (though hard-to-define) topological invariant, while the other almost never does: the invariant of dimension.

For instance, any $\mathbb R^n$ is homotopy equivalent to a point. Similarly, $\mathbb R^n - \{0\}$ is homotopy equivalent to the $(n-1)$-sphere.

Back when I was learning this stuff, this was one of the examples where I learned quickly that homotopy equivalence does not preserve rigid invariants like dimension. And if, like me, some of your students are not so used to the "flexibility" or "wiggly-ness" of homotopical invariants, this might be taken as a sign that homology seems like a less-ambitious invariant of detecting when spaces are different. For instance, if the space of our universe had a single hole in it, homology would not be able to tell the difference between the whole of our universe and the surface of the earth.

As an aside, it is probably easiest for a student to visualize a homeomorphism, while it takes some getting used to to learn about homotopy equivalence. I believe most useful examples of homotopy equivalences are strong deformation retracts, though a notable non-example is in proving the contractibility of EG for a group G.

Preserving homology groups. This is a great place to hammer home the notion that invariants aren't just groups that simply emerge from having a space: Most useful invariants are functorial, in that maps of spaces induce maps of invariants.

It might be useful to talk about the Whitehead-Hurewicz theorem (as a "Coming Attraction" for their future mathematics), which tells you that if two simply-connected, reasonable spaces admit a map inducing isomorphisms on their homology groups, then those two spaces are homotopy equivalent via that map. This might also encourage your students to stay away from simply-connected spaces if you're looking for these kinds of examples.

(Regardless, a possibly accessible example of two simply-connected spaces with isomorphic homology groups, that do not admit a homotopy equivalence between them, is $CP^2$ and $S^2 \vee S^4$. This may be a bit advanced for undergraduates, as $CP^2$ is not always covered. Also, the proof that they aren't homotopy equivalent usually utilizes cohomology, which is also very advanced. Undergraduates sometimes enjoy examples even if they can't be proven, though.)

Homology groups v. Homotopy equivalence This is hard, because proving two things aren't homotopy equivalences usually requires some invariant like cohomology or homotopy groups (if the spaces begin with isomorphic homology groups).

Regardless, a good example might be $S^2$ wedged with $2g$ circles, compared to the surface of genus $g$ (for $g \geq 2$). The fundamental groups of these differ so they can't be homotopy equivalent, but they have isomorphic homology groups.

Answer 2

I am not entirely sure on the best way to convey the difference between homotopy equivalence and isomorphic homology groups (or even isomorphic homotopy groups, though on CW-complexes I guess this isn't as big of a concern), except by way of examples. I remember my algebraic topology exam had an explicit example of spaces with all the same homology and fundamental groups. I'm pretty sure one was $S^1 \vee S^2 \vee S^3$, and the other was $S^2 \times S^1$. However the proof these were not homotopy equivalent required showing through a guided exercise that for homotopy equivalent nice spaces the equivalence lifts to the universal covers, which in this case are clearly different by considering the third homology group.

For homotopy and homeomorphism, my favorite exercise I've ever seen was identifying which letters of the standard print alphabet are homotopy equivalent and homeomorphic. It gives you a real idea for the types of topological manipulation the two equivalencies permit.

Answer 3

(Moved from comment...) Examples in the plane: a twice-punctured annulus shows the difference between homotopy and homology, because the fundamental group (free on two generators) has a much smaller abelianization. That is, for $s,t$ two loops, one around one hole, the other around the other, $sts^{-1}t^{-1}$ is non-zero in homotopy, but is homologous to $0$.

Similarly, two circles attached at a single point (equivalently, a figure-eight) has the homotopy type of a twice-punctured disk, but is visibly of lower dimension.