Explicit expression for homeomorphism and homotopy equivalence

Question

In many cases of topology when one needs to show two spaces $X$ and $Y$ are homeomorphic or homotopy equivalent, one uses some description instead of constructing an explicit homeomorphism or homotopy equivalence. An example is to show the "House with two rooms" is contractible.

For some beginners in topology they may not feel "comfortable" with this kind of proof: "Is it really a strict proof?" might be a question some beginners have in their mind.

Another example is an easy exercise showing that $S^{n-1}\subseteq S^n-\{N,S\}$ is a strong deformation retract, where $N=\{0,\cdots, 0, 1\}$ and $S=\{0,\cdots, 0, -1\}$. Intuitively it looks obvious (at least for $n=2$); but to find an explicit strong deformation retraction expression one needs to do some work. Is it really necessary to find an explicit expression at all for exercises like this?

If one is satisfied with a non-strict visualisation, which is quite straightforward for $n=2$, then what about larger $n$'s? It may not be trivial to beginners to visualise $S^3$ and $S^4$, for example.

Answer 1

I have some sympathy with zuriel! In the 1960s When I was writing the first edition of what is now Topology and Groupoids I was baffled by some papers where a complicated diagram was written down with formulae for a homotopy in each bit, and I felt that this was difficult to develop for myself, or to explain to readers. So I gradually explored the underlying formalism, and came up for example with a gluing theorem for homotopy equivalences, which is now part of abstract homotopy theory, although the original proof has some advantages in giving control of the homotopies involved. See this mathoverflow for a reference.

Being able to give an algebra for homotopies is also part of the motivation for higher homotopy groupoids. An example of this is the "rotations" which you will find in Section 6.4 of the book on Nonabelian algebraic topology.

Part of the motivation for the work for that book was to give expression for "algebraic inverses to subdivision", and that was a major reason for the cubical approach.

This discussion is also relevant to this mathoverflow discussion.

July 8, 2014: I add that in 1958 at the ICM at Edinburgh I heard Raoul Bott comment: "Grothendieck was prepared to work very hard in order to make things tautological!" This seems to me an admirable aim, that proofs should make it clear why things are true, not just follow "by a calculation" or from a seemingly ad hoc diagram.

Answer 2

For students first learning the subject, in my opinion it is indeed very important to do the work to write down explicit expressions.

It is important to emphasize that the statement "$X$ is homeomorphic to $Y$" or "$X$ is homotopy equivalent to $Y$" is at its root an existence statement: some function, satisfying some properties, exists. And the proof of an existence statement at its root is to write down an explicit expression.

I also like to emphasize the point that the most imaginative and creative part of mathematics is dreaming up new things the existence of which was not previously imagined: a house with two rooms! Who woulda thunk it? Once you have had a successful dream, then the work starts, proving rigorously that this thing you dreamed up truly exists, and for that you must write down an explicit expression.

These expressions often turn out to be rather straightforward in the end, being formulas involving standard stuff from a calculus course. Your example of $S^{n-1} \subset S^n$, in the case $n=2$, involves just spherical coordinates. Then the students get to ponder how to generalize spherical coordinates to higher dimensions.

As we get more experienced, we learn to take productive shortcuts in our existence proofs. But that only comes after we have learned to be very explicit in writing down the necessary expressions.