Does there exist a reasonable candidate for such an object as $\pi_{\frac12}(X)$?

5$\begingroup$ Is there a pointed topological space $S^{\frac{1}{2}}$ such that $S^{\frac{1}{2}}\wedge S^{\frac{1}{2}} = S^1$? $\endgroup$ – Michael Albanese Jul 26 '15 at 22:03

6$\begingroup$ If there were a "$1/2$sphere" $S^{1/2}$, you would probably expect it to be a pointed space for which $S^{1/2}\wedge S^{1/2}$ is weak equivalent to $S^1$. It is easy to show (using homology, for instance) that no such space exists. This doesn't prove there is no sensible notion of $\pi_{1/2}$, but it is some evidence against it. $\endgroup$ – Eric Wofsey Jul 26 '15 at 22:04

5$\begingroup$ Obviously $\pi_{3/2}(X)$ is half abelian. $\endgroup$ – Will Sawin Jul 26 '15 at 23:10

20$\begingroup$ People are fond, nowadays, of interpreting the $*$ is $\pi_*$ to be pretty general elements of the Picard group of whatever category you're in. For arbitrary spaces, there's just the integer grading. However, if you're space is K(1)local then you get at least a $p$adics worth of numbers to stick in the $*$ slot. (So if $p \ne 2$ you get a $\pi_{1/2}$). Cf. [Hopkins, Mahowald, Sadofsky] and more recent work Hovey, Goerss, Henn, Rezk, etc. for higher chromatic analogues $\endgroup$ – Dylan Wilson Jul 27 '15 at 0:46

5$\begingroup$ also: if $X$ is a Gspace you can stick in representation spheres for $*$, and if $X$ is a motivic space you can stick in two different types of integers into the $*$ spot one corresponding to the ordinary sphere, and the other to $\mathbb{G}_m$. $\endgroup$ – Dylan Wilson Jul 27 '15 at 0:47
As requested, here's a little more detail in the form of an answer.
Disclaimer: Let me start by saying that it'll be easier for me to talk about stable homotopy groups instead of homotopy groups, just because that's all I know about. By that I mean, I'm not gonna define, say, $\pi_{1/2}S^3$ but rather $\pi_{N+1/2}S^{3+N}$ when $N$ is really big. Throughout I will be lazy and write $\pi_kX$ for what should really be the stable value of $\pi_{k+N}\Sigma^NX$. Feel free to ignore this. Someone who knows more about this than I do can come around and fill in how to translate this back into something about ordinary homotopy groups. Also, unless I say otherwise, my prime is odd. (Contrary to normal practice here at Northwestern.)
Motivation and (pseudo)History
Here is a 'justso' story about how you could have invented $p$adically interpolated homotopy groups. I think it's pretty close to the actual history, but I wasn't in Mike Hopkins's head at the time so I don't know. (But I really wish I had been... what a great sequel to Being John Malkovich that would be).As usual, it starts with $K$ theory. Except I said we're gonna $p$adically interpolate, so we actually want $p$complete $K$ theory, $K_p$. The nice thing about $p$adic $K$theory is that, for any number $m$ coprime to $p$, there's a natural transformation of (multiplicative) cohomology theories, called the Adams operations, $\psi^m: K_p \rightarrow K_p$. This is supposed to act like a linearization of taking exterior powers of vector bundles, and here's all you need to know about it:
 If $v \in K(S^2)$ denotes the Bott periodicity element, then $\psi^m(v) = mv$
 $\psi^m$ induces a graded ring endomorphism of $K^*(pt)$
Now, for various reasons, coming from manifold theory (if you're Sullivan) or just trying to understand some elements in the stable homotopy groups of spheres (if you're Adams), people got interested in understanding the cohomology theory you get by taking the 'fixed points' of $\psi^m$. That is, they wanted to understand the cohomology theory $J$ that has a natural transformation $J \rightarrow K_p$ and such that, for any space $X$, you get a long exact sequence
$$J^n(X)\rightarrow K^n_p(X) \stackrel{\psi^m 1}{\longrightarrow} K^n_p(X) \rightarrow J^{n+1}(X) \rightarrow \cdots$$
When you choose $m$ correctly (I'll say how in a second), this cohomology theory deserves the name $S_{K(1)}$ which stands for '$K(1)$local sphere' because it sees a very special sector of the rainbow that makes up the sphere (or more accurately, the 'stable' phenomena of the sphere).
It is this object which we will now raise to the "1/2 th" power, and maps out of this will constitute elements of $\pi_{1/2}$. The key is to notice the following:
 $K^{2n}_p(pt)= \mathbb{Z}_p$
 The map $x \mapsto (m^{n}x  x)$ on the $p$adics makes sense for (invertible) $p$adic values of $m$.
Now I can tell you how to choose $m$ to get the $K(1)$local sphere: it has to be a topological generator of the $p$adic units. (Those following along at home just saw why my prime was odd.) If we think about Bott periodicity and stare at the formula a bit, it becomes clear that the $\lambda$th suspension, for $\lambda \in \mathbb{Z}_p$, should be obtained by taking the fiber of the map which, on homotopy groups, does $x \mapsto (m^nx  \lambda x)$. [Okay, it takes a little fussing to see this is the right thing to do it helps to interpret $\lambda$ as a unit which is 1 mod $p$].
This fiber behaves like the $\lambda$th suspension of the $K(1)$local sphere, and with it we may probe $K(1)$local spaces and spectra $p$adically. When $p=2$ there's a little more stuff you can probe with, but you get $2$adic things as well.
The general story
All the stuff above is spelled out in HopkinsMahowaldSadofsky's paper here. That's where they launched the program of computing Picard groups. As I said before, the Picard group of a symmetric monoidal category is the set of (isomorphism classes of) $\otimes$invertible objects. These are good for indexing things like homotopy groups in all sorts of settings.
The most mysterious example, however, is the $K(n)$local category (this is the thing that sees the $n$th layer of the rainbow of the sphere, we met $n=1$ earlier). We are still in the very early stages of understanding the Picard group of this category, but it's very desirable to do so. For one, there are mysterious duality phenomena in the $K(n)$local category, analogous to the Poincaré duality example mentioned in the equivariant setting, which require exotic suspensions (see here). On the other hand, it can be useful to index the homotopy groups of something as a function of a $p$adic number, say. Who knows maybe the only way we'll ever get an answer to 'what are the ranks of the homotopy groups of spheres?' is to look for solutions like 'the special values of some horribly incomputable arithmeticpadicyLtype function'.
But that's speculation. For some recent calculations, see some of the papers here by Paul Goerss and his collaborators (Henn, Mahowald, Rezk).

$\begingroup$ 'the special values of some horribly incomputable arithmeticpadicyLtype function'  nice :) $\endgroup$ – David Roberts Jul 28 '15 at 5:47
