In homotopy theory there are lots of nice constructions that seem designed to have some effect on the homotopy of a space, i.e. completing, localizing, and taking various homotopy (co)limits. It seems like some of these (for instance Quillen's + construction) have moderately visual interpretations by attaching cells and things like that. In general, does anyone out there have interest in constructing such things geometrically, i.e. drawing a picture of what a sphere looks like after you, I don't know, complete it at a prime, or something like that? Are there sensible ways to visually represent mod p Moore spaces? Do such constructions usually cause things to have to leap into higher, non-visual, dimensions or something? Or is it simply that this kind of thing doesn't really serve any purpose, and we can just prove that it works and that it exists? In general it also seems very interesting to me to know what a sphere localized at a given homology theory might look like.
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$\begingroup$ I like the spirit of this question: visual intuition is often crucial in topology and geometry, and I would certainly benefit from good answers. However, I think it could be much improved if you focus on a question of the type: "are there good visual interpretations of homotopy theoretic constructions such as (blah), for instance by attaching cells?" rather than ask people whether they would personally find such constructions useful. $\endgroup$– Vidit NandaCommented Jun 30, 2012 at 21:01
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$\begingroup$ I think Kathryn Hess discusses how you might think of the rationalized sphere in her notes on rational homotopy theory. As is alluded to below, i don't know that it is helpful geometrically. (she presents it as a cell complex, but it is huge) $\endgroup$– Sean TilsonCommented Jul 2, 2012 at 4:56
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$\begingroup$ That's sort of what I thought might happen, you necessarily get really massive cell complexes or something. Geometric constructions of such things would be really interesting to see (in low dimensions at least). $\endgroup$– Jonathan BeardsleyCommented Jul 2, 2012 at 18:41
2 Answers
Though it is often satisfying and useful to have an appropriate way to visualize a construction, in fact the real innovation in homotopy theory that lead to the constructions you mention, among many others, was the realization that you need not limit your attention to geometric, visualizable operations and spaces but can be content knowing only certain relevant properties.
This realization led to what Mike Hopkins (and perhaps others before him) called designer homotopy theory. The simplest example of a designer homotopy type is the Eilenberg-MacLane spaces---once you convince yourself they exist, you don't concern yourself with what they look like (a horrendous infinite CW complex, perhaps), but use only their characteristic relationship to homotopy groups. Needless to say this particular example led to spectacular advances in the hands of Serre and others.
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1$\begingroup$ Thanks, I think that's a really good answer, and sort of in the spirit of what I was thinking about. I had not heard of the term "designer homotopy theory." However, I should hope that, after swinging to the side of purely algebraic knowledge, in time, we could come up with nice ideas of what such things look like. However, after writing this question it occurred to me, knowing what such a thing "looks like" may in some sense be the same as knowing AT LEAST its homotopy groups, which we don't always know! $\endgroup$ Commented Jul 1, 2012 at 3:56
Many operations on the homotopy groups of a CW complex can be realized by a "geometric" (rather, topological) construction on the space itself. A popular example is the rationalization, studied in rational homotopy theory.
The general idea is to write down a CW model for your space, with explicit attaching maps, and doing the construction on spheres, then re-glue by the attaching maps.
An easy example to look at is the following construction on the 1-sphere: take the covering $z \mapsto z^n$ from the 1-sphere to itself. A path on the "lower" circle that generates the fundamental group $\pi_1(S^1,1)=\mathbb{Z}$ can be multiplied with $n$ (going around $n$ times) and then lifted to the covering space and there it corresponds to the generator. Attaching a 2-cell along this covering map kills this $n$-times going around map on the homotopy level (since it is liftable, so contractible in the newly attached 2-cell). The result on the fundamental group is that you get $\pi_1(S^1 \cup_{z^n} e^2) = \mathbb{Z}/n\mathbb{Z}$.
You can iterate this with a mapping telescope construction to get an inverse limit over these groups, thus the fundamental group becomes the profinite completion $\hat{\mathbb{Z}}$ of the integers.
Since you asked for how to visualize this, I tend to draw pictures on the blackboard and imagine the paths and the contractions quite visually.
To address the question whether this is useful or just good to know it exists: Sometimes it can be crucial to know how many cells in which dimensions you need to attach to change the homotopy groups, and knowledge about the attaching maps can be important too. For example, from all attaching maps you can compute CW (co)homology, so you can look at how the (co)homology changes if you, say localize at a prime. Maybe the Quillen + construction is a good example for this, since one really has to attach only a 2-cell and then a 3-cell, to get it right.
