Generalities on Homotopy Classes of Mappings
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 ) = [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 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  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.
KeywordsExact Sequence Base Point Hopf Algebra Homotopy Class Finite Type
Unable to display preview. Download preview PDF.