The th homotopy group has as elements equivalence classes of spheres in where two such are regarded as equivalent if there is a left homotopy between them, fixing the base point. The group operation is given by gluing of two spheres at their basepoint.
Lower homotopy groups act on higher homotopy groups; the nonabelian group cohomology of this gives the Postnikov invariants of the space. All of this data put together allows one to reconstruct the original space, at least up to weak homotopy type, through its Postnikov system.
Let be a topological space, let be a point, to be called the base point.
For , write be the pointed -sphere.
where two such are regarded as equivalent if there is a basepoint-preserving left homotopy between them.
Now we will put some structure on that set.
There are independent equators through the basepoint of . Given two maps , form their copairing in the category of pointed spaces to get a map
(where indicates the wedge sum); then combine this with a map that maps the th equator to the basepoint and each hemisphere to one copy of the sphere. The result is a map , called the th concatenation of and :
One can check that each of these operations respects homotopy equivalence and hence equips with the structure of a magma.
These magmas are in fact groups; in particular:
This may seem like quite a complicated kind of structure, but it is actually quite simple up to homotopy. First of all, all concatenations of given maps and are homotopic, so we speak of simply a single concatenation for (and none for ). By the Eckmann–Hilton argument, this concatenation will be commutative up to homotopy for . In any case, it is associative and invertible up to homotopy, and the null element is an identity up to homotopy.
The result is that the set of equivalence classes is an abelian group for , a group for , and a pointed set for (when the null element is the only structure).
If is path-connected, then all of the are isomorphic. Accordingly, it's traditional to just write in that case. (This is why we must use for the homotopy -groupoid.) However, there may be many different isomorphisms between and (given by ), so a more careful treatment requires keeping track of the basepoint even in the connected case.
This is described in detail at
a space is -truncated? if all homotopy groups above degree are trivial.
Vacuously, every space is -truncated, and precisely the inhabited spaces are -connected. On the other end, precisely the weakly contractible spaces are -connected, and a space is -truncated iff it is weakly contractible if inhabited. (So classically, using excluded middle, a space is -truncated iff it is either empty or weakly contractible.)
To extend one step further in negative thinking, every space (even the empty space) is -connected, and precisely a weakly contractible space (but not the empty space) is -truncated.
A weakly contractible space is an Eilenberg–MacLane space in every degree, and these are the only Eilenberg–MacLane spaces in degree or . In degree , they are the pointed discrete spaces (and those weakly homotopy equivalent to such). In degree , they are (up to weak homotopy equivalence) precisely the classifying spaces of groups. And so on.
The th homotopy ‘group’ can be identified with the set of all path components of , with the component containing as the basepoint. Similarly, the fundamental 0-groupoid is the set of all path components without a chosen basepoint. Note that is traditionally written , even without a basepoint.
The st homotopy group is precisely the fundamental group of at . This is the original example from which all others derived. It was once written simply with the standing for Poincaré, who invented it.
At least, that's where I think that it comes from … —Toby
In the early years of the 20th century it was known that the nonabelian fundamental group of a space with base point was useful in geometry and complex analysis. It was also known that the abelian homology groups existed for all and that if is connected then is isomorphic to the abelianisation of any .
Consequently it was hoped to generalise the fundamental group to higher dimensions, producing nonabelian groups whose abelianisations would be the homology groups.
In 1932, E. Čech proposed a definition of higher homotopy groups using maps of spheres, but the paper was rejected for the Zurich ICM since it was found that these groups were abelian for , and so do not generalise the fundamental group in the way that was originally desired. Nonetheless, they have proved to be extremely important in homotopy theory, although more difficult to compute in general than homology groups. See weak homotopy equivalence.
It was early realised that the fundamental groupoid operates on the family of groups which should thus together be regarded as a module over .
A key property of homotopy groups is the Whitehead theorem: if is a map of connected m-cofibrant spaces (spaces each of the homotopy type of a CW complex), and induces isomorphisms for some and all , then is a homotopy equivalence.
However, the homotopy groups by themselves, even considering the operations of , do not characterise homotopy types. See also algebraic homotopy theory.
See also the Freudenthal suspension theorem.
From that perspective we might say that:
where denotes the homotopy category of .
The standard example is that where is the -sphere. This naturally comes with an co-group structure up to homotopy, which is precisely the structure underlying the co-category structure of the interval object and more generally that underlying the mechanism of the Trimble n-category.
Homotopy groups and their properties can naturally be formalized in homotopy type theory. In this context a proof that is in
and a proof that