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The homotopy groups $\pi_{n}(X)$ arise from considering equivalence classes of based maps from the $n$-sphere $S^{n}$ to the space $X$. As is well known, these maps can be composed, giving arise to a group operation. The resulting group contains a great deal of information about the given space. My question is: is there any extra information about a space that can be discovered by considering equivalence classes of based maps from the $n$-tori $T^{n}=S^{1}\times S^{1}\times \cdots \times S^{1}$. In the case of $T^{2}$, it would seem that since any path $S^{1}\to X$ can be "thickened" to create a path $T^{2}\to X$ if $X$ is three-dimensional, the group arising from based paths $T^{2}\to X$ would contain $\pi_{1}(X)$. Perhaps more generally, can useful information be gained by examining equivalence classes of based maps from some arbitrary space $Y$ to a given space $X$.

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Well, if you examine homotopy classes of based maps from all based spaces Y at once, then you get enough information to characterize the space X up to homotopy equivalence, by the Yoneda lemma in the homotopy category. (-: – Mike Shulman Sep 5 '10 at 17:25
There's something similar to this that I thought of once: given any two spaces $X$ and $Y$, the set of homotopy classes of maps $X \times I \to Y$ sending all $(x, 0)$ and $(0,x)$ to a fixed base point form a group. If $X = I^{n-1}$, I believe the result contains, at least in some cases, all the homotopy groups $\pi_1$ through $\pi_n$. But given how hard the latter are to compute, I doubt that this construction is all that useful. [But if it is, I am not in a position to know.] – Charles Staats Sep 5 '10 at 20:39
Mike- isn't this true even if we restricted to based maps for all based <i>co-Moore</i> spaces at once to X? Or am I completely off-base? – Daniel Moskovich Sep 5 '10 at 21:53 Groups of Flagged Homotopies and Higher Gauge Theory Valery V.Dolotin - there is some generalization of the homotopy groups. – Alexander Chervov Feb 16 '13 at 14:33
up vote 32 down vote accepted

There's always information to be got. But in this case:

  • Based homotopy classes of maps $T^2\to X$ don't form a group! To define a natural function $\mu\colon [T,X]_*\times [T,X]_*\to [T,X]_*$, you need a map $c\colon T\to T\vee T$ (where $\vee$ is one point union). And if you want $\mu$ to be unital, associative, etc., you'll want $c$ to be counital, coassociative, etc. For $T=T^n$ with $n\geq2$, there is no $c$ that is counital. (The usual way to see this is to think about the cohomology $H^*T$ with its cup-product structure.)

  • The inclusion $S^1\vee S^1\to T^2$ gives a map $$r\colon [T^2,X]_* \to [S^1\vee S^1,X]_*\approx \pi_1X\times \pi_1X.$$ The image of this map will be pairs $(a,b)$ of elements in $\pi_1X$ which commute: $ab=ba$. It won't usually be injective; so there might be something interesting to think about the in preimages $r^{-1}(a,b)$.

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Back in the 1940's, Ralph Fox defined something called the torus homotopy group. For a based space $(Y,y_0)$ and natural number $r$, the $r$-dimensional torus homotopy group $\tau_r(Y,y_0)$ is just the fundamental group of the mapping space ${\rm map}(T^{r-1},Y)$, based at the constant map (where $T^{r-1}$ is of course a torus).

The group $\tau_r(Y,y_0)$ contains isomorphic copies of $\pi_n(Y,y_0)$ for all $n\leq r$. Also, Whitehead products become commutators in the torus homotopy group. By passing to the limit over $r$ one obtains the (infinite) torus homotopy group $\tau(Y, y_0)$, which contains all of the homotopy information of $Y$ in one place!

Unfortunately for Fox, the idea doesn't seem to have caught on (although I hear he had a few others which did). MathSciNet only turns up 11 papers containing the phrase "torus homotopy groups" (although the most recent is from 2007).

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Your problem is that $T^n$ is not in general a co-Moore space. Therefore Eckmann-Hilton duality breaks down, as the dual spaces no longer form a spectrum, and there would be no (co)homology theory dual to such a "homotopy theory". Thus, a theory of homotopy classes of pointed maps from $T^n$ to $X$ would be much less interesting than a theory of homotopy classes of pointed maps from $S^n$ to $X$.

On the other hand, the study of homotopy classes of pointed maps from a co-Moore space other than $S^n$ to $X$ does lead to useful theories of homotopy with coefficients. I believe these classify $X$ up to homotopy equivalence.

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I was told by Brian Griffiths that Fox was hoping to obtain a generalisation of the van Kampen theorem and so continue work of J.H.C Whitehead on adding relations to homotopy groups (see his 1941 paper with that title).

However if one frees oneself from the base point fixation one might be led to consider Loday's cat$^n$-group of a based $(n+1)$-ad, $X_*=(X;X_1, \ldots, X_n)$; let $\Phi X_*$ be the space of maps $I^n \to X$ which take the faces of the $n$-cube $I^n$ in direction $i$ into $X_i$ and the vertices to the base point. Then $\Phi$ has compositions $+_i$ in direction $i$ which form a lax $n$-fold groupoid. However the group $\Pi X_*= \pi_1(\Phi, x)$, where $x$ is the constant map at the base point $x$, inherits these compositions to become a cat$^n$-group, i.e. a strict $n$-fold groupoid internal to the category of groups (the proof is non trivial).

There is a Higher Homotopy van Kampen Theorem for this functor $\Pi$ which enables some new nonabelian calculations in homotopy theory (see our paper in Topology 26 (1987) 311-334).

So a key step is to move from spaces with base point to certain structured spaces.

Comment Feb 16, 2013: The workers in algebraic topology near the beginning of the 20th century were looking for higher dimensional versions of the fundamental group, since they knew that the nonabelian fundamental group was useful in problems of analysis and geometry. In 1932, Cech submitted a paper on Higher Homotopy Groups to the ICM at Zurich, but Alexandroff and Hopf quickly proved the groups were abelian for $n >1$ and on these grounds persuaded Cech to withdraw his paper, so that only a small paragraph appeared in the Proceedings. It is reported that Hurewicz attended that conference. In due course, the idea of higher versions of the fundamental group came to be seen as a mirage.

One explanation of the abelian nature of the higher homotopy groups is that group objects in the category of groups are abelian groups, as a result of the interchange law, also called the Eckmann-Hilton argument. However group objects in the category of groupoids are equivalent to crossed modules, and so are in some sense "more nonabelian" than groups. Crossed modules were first defined by J.H.C. Whitehead, 1946, in relation to second relative homotopy groups. This leads to the possibility, now realised, of "higher homotopy groupoids", Higher Homotopy Seifert-van Kampen Theorems, and the notions of higher dimensional group theory.

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