structural:expansion, quality:correction, stylistic:style_transfer
Homology vs. homotopy

The nth homotopy group π n (X) {\displaystyle \pi _{n}(X)} of a topological space X {\displaystyle X} is the group of homotopy classes of basepoint-preserving maps from the n {\displaystyle n} -sphere S n {\displaystyle S^{n}} to X {\displaystyle X}, under the group operation of concatenation. The most fundamental homotopy group is the fundamental group π 1 (X) {\displaystyle \pi _{1}(X)}. For connected X {\displaystyle X}, the Hurewicz theorem describes a homomorphism h ∗: π n (X) → H n (X) {\displaystyle h_{*}:\pi _{n}(X)\to H_{n}(X)} called the Hurewicz homomorphism. For n > 1 {\displaystyle n>1}, this homomorphism can be complicated, but when n = 1 {\displaystyle n=1}, the Hurewicz homomorphism coincides with abelianization. That is, h ∗: π 1 (X) → H 1 (X) {\displaystyle h_{*}:\pi _{1}(X)\to H_{1}(X)} is surjective and its kernel is the commutator subgroup of π 1 (X) {\displaystyle \pi _{1}(X)}, with the consequence that H 1 (X) {\displaystyle H_{1}(X)} is isomorphic to the abelianization of π 1 (X) {\displaystyle \pi _{1}(X)}. Higher homotopy groups are sometimes difficult to compute. For instance, the homotopy groups of spheres are poorly understood and are not known in general, in contrast to the straightforward description given above for the homology groups.

For an n = 1 {\displaystyle n=1} example, suppose X {\displaystyle X} is the figure eight. As usual, its first homotopy group, or fundamental group, π 1 (X) {\displaystyle \pi _{1}(X)} is the group of homotopy classes of directed loops starting and ending at a predetermined point (e.g. its center). It is isomorphic to the free group of rank 2, π 1 (X) ≅ Z ∗ Z {\displaystyle \pi _{1}(X)\cong \mathbb {Z} *\mathbb {Z} }, which is not commutative: looping around the lefthand cycle and then around the righthand cycle is different from looping around the righthand cycle and then looping around the lefthand cycle. By contrast, the figure eight's first homology group H 1 (X) ≅ Z × Z {\displaystyle H_{1}(X)\cong \mathbb {Z} \times \mathbb {Z} } is abelian. To express this explicitly in terms of homology classes of cycles, one could take the homology class l {\displaystyle l} of the lefthand cycle and the homology class r {\displaystyle r} of the righthand cycle as basis elements of H 1 (X) {\displaystyle H_{1}(X)}, allowing us to write H 1 (X) = { a l l + a r r | a l, a r ∈ Z } {\displaystyle H_{1}(X)=\{a_{l}l+a_{r}r\,|\;a_{l},a_{r}\in \mathbb {Z} \}}.