Higher homotopy groups and sphere maps

For $n\ge1$, define $\pi_n(X,x_0)$ using maps $(I^n,\partial I^n)\to(X,x_0)$, with homotopies keeping the entire boundary at $x_0$. Collapsing that boundary identifies the quotient with a sphere, so this is equivalent to based maps from $S^n$ with based homotopies. For n equal to one the definition recovers the fundamental group. For n equal to zero it is customary to use the set of path components, which generally is not a group. The group operations therefore should not be applied indiscriminately to every symbol π with an index.

Concatenate cube maps along the first coordinate by compressing each into half the cube. The boundary condition makes the joining face map to the basepoint. Reversing that coordinate gives an inverse class, and the constant map is the identity. Based continuous maps induce homomorphisms by composition, while based homotopies preserve the induced homomorphisms. Moving the basepoint along a path gives an identification, but for a non-simply-connected target the choice can affect higher groups through the fundamental group's action. These are based invariants even when their underlying groups happen to be abelian.

When n is at least two, concatenate along either of two independent cube coordinates. Both operations have the same constant-map identity, and subdividing a square of pieces gives an interchange law between them. The Eckmann–Hilton argument shows the operations coincide and are commutative on homotopy classes. The conclusion concerns the group operation on classes, not a pointwise ability to slide arbitrary embedded spheres past one another without intersection. It also does not force the fundamental group itself to be abelian, since a loop has only one available concatenation direction.

For a sphere of dimension n, its nth homotopy group is the integers, identified by degree, with a chosen orientation fixing the generator. This degree classification is a theorem, not a consequence of drawing an approximate number of wraps. Groups below the sphere dimension vanish; cellular approximation will explain that fact. Above the sphere dimension, nontrivial groups can remain. In particular, the Hopf fibration will relate maps from a three-sphere to a two-sphere to an integer invariant, showing why higher-dimensional maps cannot be dismissed as automatically deformable to a point.

A covering map induces isomorphisms on $\pi_n$ for every $n\ge2$, after choosing compatible basepoints. A based sphere map lifts because its source is simply connected, and the same lifting property for based homotopies gives injectivity as well as surjectivity. This is why a universal cover removes π₁ while retaining higher homotopy. For the circle and torus, their contractible universal covers force all higher groups to vanish. For a two-sphere, the universal cover is itself and its nonzero second group survives.

An abelian higher group should not be confused with homology in the same dimension for every space. Hurewicz theorems require connectivity hypotheses before identifying them, and the fundamental group can act on the higher groups of a non-simply-connected space. Knowing the fundamental group alone therefore does not determine the full homotopy type. A circle and a circle wedged with a two-sphere have the same fundamental group but different higher structure. Computations should identify which lifting, degree, connectivity, or exact-sequence theorem supplies each asserted group value.