Here is some background. In 1965 I noticed that the proof of the van Kampen theorem for the fundamental groupoid seemed to generalise to dimension 2, but there was a lack of a suitable gadget, a homotopy double groupoid. I also noticed that a proof due to J.F. Adams that any map $ S^r \to S^n$ for $ r < n $ is inessential (7.6.1 of Topology and Groupoids) should have algebraic consequences, but again there was no appropriate algebraic gadget. Nine years later we had found out a lot about double groupoids and crossed modules, but were still lacking the homotopy double groupoid! Then Philip Higgins and I did a strategic analysis which went as follows:
J.H.C. Whitehead had a subtle theorem on $\pi_2(X \cup_\lambda e^2_\lambda,X,x)$ as a free crossed $\pi_1(X,x)$-module. This was an example, maybe the only then example, of a $2$-dimensional universal property in homotopy theory. (here is a link to an exposition of Whitehead's proof).
If our conjectured $2$-dimensional van Kampen theorem was to be any good it should have Whitehead's theorem as a corollary.
Whitehead's theorem was about relative homotopy groups.
So we should look for homotopy double groupoids in a relative situation, i.e. a space $X$ and a subspace $A$.
The simplest way of doing this we could think of was to consider
as in the above diagram maps of a square into $X$ which takes the edges into $A$ and the vertices to a subset $C$ of $A$ and to take homotopy classes of these rel vertices of the square. The proof that this works and gives a strict double groupoid is not entirely trivial! To my knowledge, this was the first example of a strict higher homotopy groupoid.
To our delight, this went swimmingly, and we were able to prove a $2$-d van Kampen theorem which had Whitehead's theorem as a Corollary. In fact we computed for example $\pi_2(X \cup_f CA,X,x)$ with Whitehead's theorem the case $A$ (not now a subspace) was a wedge of circles.
R. Brown amd P.J. Higgins, ``On the connection between the second
relative homotopy groups of some related spaces'', Proc.
London Math. Soc. (3) 36 (1978) 193-212.
Another surprise was that the submitted paper was asked to be withdrawn, in order not to embarrass the editor and two international authorities! A request for more information got another referee and a request (order?) to cut the paper by one third. So the final paper had no pictures, and some slicker arguments.
We also managed to work out the results for filtered spaces, and so all dimensions, and these were published in JPAA, 1981. See the book on Nonabelian algebraic topology.
I emphasise that the basic methods were cubical, and we were unlikely to conjecture let alone prove these results using globular or simplicial methods.
I like to think that these methods fulfill the dreams of the topologists of the early 20th century to find higher dimensional versions of the nonabelian fundamental group, since the nonabelian nature of the fundamental group was known to be useful in geometry and analysis.
Over to you, reader, to get such applications!
Added 28 April: The contrast with what are called in the literature "fundamental higher groupoids'' is that:(i) those do not generalise the usual fundamental groupoid since they are not strict, and are just singular complexes; (ii) while they do satisfy some version of what is called the "small simplex theorem" that does not directly imply strict colimit theorems in higher dimensions of a nonabelian type; (iii) the versions of higher groupoids we have worked with are strict structures, are defined non trivially for filtered spaces or $n$-cubes of spaces, and satisfy nonabelian colimit theorems with consequences not so far obtainable by other means. See for example the nonabelian tensor product of groups.
These ideas relate to and are aimed at relative homotopy theory, and $n$-adic homotopy theory, and I hope are seen in low dimensions as relevant to geometric group theory and to geometric topology.
The point is that one needs to evaluate what different approaches do and do not do, to compare and contrast.
See also the question and answer to Infinity-categories vs Kan complexes
Aug 5, 2014. The diagram there suggests how convenient cubical methods are for multiple compositions, compared with simplicial or globular methods.
July 7, 2014: A presentation I gave to a workshop at the IHP, Paris, June 5, 2014, entitled "Intuitions for cubical methods in nonabelian algebraic topology" is available on my preprint page.
See also this stackexchange answer on homotopical excision in dim 2.
Aug 5, 2014 A point should be made about the book on Nonabelian Algebraic Topology referred to above, and the foundations of algebraic topology, and in particular of homology theory. A key idea there is that of formal sums of geometric elements, an idea introduced by Poincaré. A definition of boundary then allows the notion of cycle, and boundary. The more geometric method, as in this book, is that a "chain" is defined for a filtered space, and in dimension > $1$ is an element of a relative homotopy group. The "composition" of such chains is the composition in the relative homotopy groups. However cubical homotopy groupoids are used to prove many crucial properties of such chains.
Best Answer
If the dimension of $M$ is low relative to that of $G$ then the calculation of $[M,G]$ typically reduces to stable homotopy theory or generalised cohomology, for which many methods are known. For example, if $\dim(M)<2n$ then $$[M,U(n)]\simeq [M,U(\infty)] \simeq [\Sigma M,BU(\infty)] \simeq K(\Sigma M) $$ (where $K$ denotes complex $K$-theory). This can often be understood using explicit constructions with vector bundles, or using the Atiyah-Hirzebruch spectral sequence $H^*(M;K^*)\Longrightarrow K^*(M)$. Similar methods work for $[M,O(n)]$, $[M,SU(n)]$, $[M,Sp(n)]$ and so on, provided that $n$ is large enough. If you want to consider small $n$ then it may be possible to work back from large $n$ using fibrations like $U(n)\to U(n+1)\to S^{2n+1}$.