Homotopy group

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Definition

As homotopy classes of based maps

Suppose (X,x_0) is a based topological space and n is a positive integer. The homotopy group \pi_n(X,x_0) is defined as follows:

  • Consider the based n-sphere (S^n,p) where p is a chosen basepoint. As a set, \pi_n(X,x_0) is the set of homotopy classes of all based maps from (S^n,p) to (X,x_0), where the homotopy classes are with respect to homotopies that preserve basepoints.
  • Two maps f_1,f_2 are composed as follows. S^n \setminus \{ p \} is identified with the open northern and open southern hemisphere of a new sphere via homeomorphic identifications \varphi_1 and \varphi_2 from these hemispheres to S^n \setminus \{ p \} (these identifications are universally fixed, independent of X; there's a natural choice for them). The composite map is now defined as follows: as f_1 \circ \varphi_1 on the northern hemisphere, as f_2 \circ \varphi_2 on the southern hemisphere, and as the constant map to p on the equator. The basepoint is a fixed point on the equator (again, this choice is independent of X and is universally fixed).

The multiplication defined above can be viewed as arising from the corresponding comultiplication on the n-sphere S^n, because of the contravariant nature of maps from.

Proof that this gives a group

The case n = 0

The definition of homotopy group still gives a set definition for n = 0. S^0 is a two-point space, and one of these points must go to a fixed basepoint, while the other can go anywhere. Thus, the set of all based maps is the set of points in X, and the set of homotopy classes is the set of path components. Thus, \pi_0(X,x_0) is the set of path components in X. Note that it is independent of x_0 because S^0 being discrete, the image of the basepoint does not affect where the other point goes.

However, the composition operation does not make sense for n = 0, because S^0 has an empty equator. Hence, \pi_0 is only a set and has no group structure for arbitary topological spaces. (It does have a group structure when the topological space is a H-space, induced by the multiplication in the topological space).

The case n = 1

In this case, we get the fundamental group \pi_1(X,x_0). Recall that for f_1,f_2 based maps from the circle to (X,x_0), we think of f_1,f_2 as maps from [0,1] to X with f_1(0) = f_1(1) = f_2(0) = f_2(1) = x_0. The usual way of composing is to define:

(f_1 * f_2)(t) := \lbrace \begin{array}{rl} f_1(2t), & 0 \le t \le 1/2 \\ f_2(2t - 1), & 1/2 < t \le 1 \\\end{array}

Here, the definition on (0,1/2) can be viewed as the northern hemisphere definition, the definition on (1/2,1) can be viewed as the southern hemisphere definition, with the equator corresponding to the two points 1/2 and \! 0 \sim 1, of which we choose the latter as basepoint.

Omission of basepoint

For a path-connected space, the homotopy groups \pi_n for all basepoints are isomorphic. In fact, any choice of path between two points can be used to define an isomorphism between the \pi_ns at these basepoints. The key fact that we need to use here is that the inclusion of a point in S^n is a cofibration (which is easily seen by noting that S^n is the boundary of D^n, or more generally from the fact that manifold implies nondegenerate).

In general, the homotopy group \pi_n may differ for different path components. For a homogeneous space, or more generally for a space where all the path components are homeomorphic, the isomorphism class of \pi_n does not depend upon the choice of basepoint.

Dependence on homotopy type

The homotopy groups \pi_n depend only on the homotopy type of the based topological space. In fact, they depend only on the homotopy type of the path component of the basepoint in the topological space.

However, knowledge of the homotopy groups does not determine the homotopy type, or even the weak homotopy type, of the topological space.

Facts

  • For a H-space, all homotopy groups, including the fundamental group, are abelian groups. This is a consequence of the Eckmann-Hilton principle. In fact, the group operation coincides with the operation induced by pointwise multiplication of loops (when we go down to homotopy classes). Also, \pi_0 gets the structure of a (not necessarily abelian) group. Further information: fundamental group of H-space is abelian
  • For any topological space, all higher homotopy groups, i.e., all \pi_n, n \ge 2, are abelian groups. This can be viewed as a consequence of the Eckmann-Hilton principle, or the fact for n \ge 2, S^n can be rotated about any equatorial axis to interchange the roles of the north and the south poles. Further information: higher homotopy groups are abelian