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Generalities on Homotopy Classes of Mappings

  • George W. Whitehead
Part of the Graduate Texts in Mathematics book series (GTM, volume 61)

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 ) = [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 [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.

Keywords

Exact Sequence Base Point Hopf Algebra Homotopy Class Finite Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1978

Authors and Affiliations

  • George W. Whitehead
    • 1
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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