Generalities on Homotopy Classes of Mappings

Abstract

The set [X, Y] of homotopy classes of maps between two compactly generated spacesX, 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 π₁ (X, ₀ ) = [S¹, y ₀ ; X, x ₀ ] is a group, while[S¹, X] is in one-to-one correspondence with the set of all conjugacy classes inπ ₁ (X, x₀), 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 ] andX, x ₀ ; Y, y ₁ ] are isomorphic. However, there is an isomorphism between them for every homotopy class of paths inY fromy ₁ toy ₀ . In particular, the group π₁(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 caseX = Sⁿ was first studied by Eilenberg [1] in 1939; it, and an analogous action ofπ ₁ (B, y ₀ ) on the set [X, A, x₀;B, y ₀ ], are discussed in §1.