[Math] Have there been (successful) attempts to use something other than spheres for homotopy groups

algebraic-topologyhomotopy-theory

Homotopy groups are famous invariants in algebraic topology. They have a myriad of wonderful properties:

  • For $n \ge 1$, $\pi_n(X,*)$ is a group; for $n \ge 2$, this group is abelian.
  • $\pi_n$ defines a functor from based spaces to (abelian) groups.
  • $\pi_n$ is invariant under homotopy equivalences, and homotopic maps induce the same morphism on homotopy groups.
  • The Seifert–van Kampen theorem allows one to compute the fundamental group of $X \cup_A Y$.
  • A fibration $F \to E \to B$ yields a long exact sequence of homotopy groups.
  • There is a natural suspension morphism $\pi_n(X) \to \pi_{n+1}(\Sigma X)$, and it eventually stabilizes as described by the Freudenthal suspension theorem.
  • If $f : X \to Y$ is a map between two CW-complex that induces an isomorphism on all homotopy groups, then it's a homotopy equivalence.
  • The Whitehead product endows $\pi_*(X)$ with a shifted Lie algebra structure.
  • Etc., etc.

But after all, $\pi_n(X,*)$ is "merely" the set of morphisms in the homotopy category of pointed spaces from the $n$-sphere to $X$: $\pi_n(X,*) = \hom_{\mathsf{hTop}_*}((S^n,*), (X, *))$. This has always seemed somewhat "biased" to me; spheres are given a preponderant role in the definition. They are certainly a fundamental type of spaces, but they're not all there is to life. So my question is:

Have there been (successful) attempts to use something other than spheres for homotopy groups, in a way that some part of the structure described above is retained (in one form or the other)?

For example, maybe I'm interested in totally disconnected spaces, and I'd consider something like $\{ [0,1]^n \cap \mathbb{Q} \to X \}$ (with boundary conditions) up to "homotopy". And even here we see spheres (or at least Euclidean spaces) lurking in the background: the definition of a homotopy involves a path $[0,1] \to Y^X$, so why not consider "homotopies" of the type $[0,1] \cap \mathbb{Q} \to Y^X$…? One problem that could crop up here is that $[0,1] \cap \mathbb{Q}$ is not compact, so a lot of the theory goes out of the window.

Or maybe I'm interested in what happens in infinite dimension, so why not replace the spheres with some compactification of an infinite dimensional Banach space, for example? I have no idea what such a thing would look like.

So far, I've found three things in this direction:

  • Homotopy groups with coefficients replace the sphere with a Moore space $M(G,n)$ (i.e. a space with a single nonzero reduced homology group). A sphere is in particular a Moore space of type $M(\mathbb{Z},n)$. Unfortunately such Moore spaces are not, in general, co-H-spaces, so $[M(G,n), X]$ is not a group, merely a pointed set. One reference for this seems to be Weibel K-book, but as far as I can tell they are only used as motivation for the definition of K-theory with coefficients (I haven't had the chance to read the book in detail).
  • Hazewinkel, in Encyclopaedia of Mathematics, mentions something called "toroidal homotopy groups". I haven't been able to find much information about it anywhere and I can only guess it has something to do with tori. (I don't have access to the book, only to a few pages on Google Books (and other people may not even be able to see the same pages as I do))
  • Last week I saw this preprint about "big fundamental groups" (the interval is apparently replaced with an interval of large cardinality) in the daily arXiv email, which I think pushed me towards this question.

Best Answer

But after all, $\pi_n(X,*)$ is "merely" the set of morphisms in the homotopy category of pointed spaces from the $n$-sphere to $X$: $\pi_n(X,*) = \hom_{\mathsf{hTop}_*}((S^n,*), (X, *))$. This has always seemed somewhat "biased" to me; spheres are given a preponderant role in the definition.

Spheres emerge naturally out of the relationship between homotopy theory and higher category theory. One example of this is that you might object that the study of $\pi_1$ somehow arbitrarily picks out $S^1$, but in fact you can define $\pi_1$ without mentioning $S^1$ at all, using the theory of covering spaces or locally constant sheaves. One definition is the following: $\pi_1(X, x)$ is the automorphism group of the functor given by taking the stalk of a locally constant sheaf on $X$ at $x$. This is a functor on the homotopy category of reasonable pointed spaces and the circle naturally emerges as the object representing this functor.

Similarly you can describe the fundamental $n$-groupoid $\Pi_n(X)$ of $X$ (which knows about the first $n$ homotopy groups) without mentioning paths at all, using a theory of "higher" locally constant sheaves (locally constant sheaves of $(n-1)$-groupoids, rather than sets).

If you believe one way or another that fundamental $n$-groupoids are reasonable things to study (inductively: if you believe that homotopies are reasonable, then you believe that homotopies between homotopies are reasonable, etc.; this means you care about the interval $[0, 1]$ but it doesn't yet obligate you to care about spheres), then the higher homotopy groups naturally emerge by repeatedly taking automorphisms:

  • $\pi_1(X, x)$ is the automorphism group of $x$ as an object in the fundamental groupoid $\Pi_1(X)$,
  • $\pi_2(X, x)$ is the automorphism group of the identity path $x \to x$ as an morphism in the fundamental $2$-groupoid $\Pi_2(X)$,

etc. None of this explicitly requires talking about spheres, but again, these are all very natural functors on pointed homotopy types and spheres represent them.

Said another way, among homotopy types, the $n$-sphere $S^n$ has a universal property: it is the free homotopy type on an automorphism of an automorphism of... of a point. These aren't just arbitrary spaces we picked because we like them; they're fundamental to homotopy theory in the same way that $\mathbb{Z}$ is fundamental to group theory.

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