Elements of Homotopy Theory pp 96-156 | Cite as

# Generalities on Homotopy Classes of Mappings

## Abstract

The set [*X, Y]* of homotopy classes of maps between two compactly generated spaces*X, Y* has no particular algebraic structure. This Chapter is devoted to the study of conditions on one or both spaces in order that [*X, Y*] support additional structure of interest. Guided by the fact that π_{1} (*X*,_{ 0 }*) = [S*^{ 1 }*, y*_{ 0 }*; X, x*_{ 0 }] is a group, while*[S*^{ 1 }*, X*] is in one-to-one correspondence with the set of all conjugacy classes in*π*_{ 1 }*(X*, x_{0}), and the latter set has no algebraic structure of interest, we discuss in §1 the way in which [*X, x*_{ 0 }*; Y*,*y*_{ 0 }] depends on the base points. It turns out that under reasonable conditions the sets [*X, x*_{ 0 }*; Y*_{ 0 }*, y*_{ 0 }] and*X, x*_{ 0 }*; Y, y*_{ 1 }*]* are isomorphic. However, there is an isomorphism between them for every homotopy class of paths in*Y* from*y*_{ 1 } to*y*_{ 0 }. In particular, the group π_{1}*(Y, y*_{ 0 }*)* operates on*[X, x*_{ 0 };*Y*,*y*_{ 0 }*]*, and [*X, Y*] can be identified with the quotient of the latter set under the action of the group. This action for the case*X* = S^{n} was first studied by Eilenberg [1] in 1939; it, and an analogous action of*π*_{ 1 }*(B, y*_{ 0 }*)* on the set [*X*, *A*, x_{0};*B, y*_{ 0 }], are discussed in §1.

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