Abstract
We introduce the notion of a relatively homotopy associative and homotopy commutative H-space, construct one for any path-connected space X, and describe several useful properties, including exponent properties.
Highlights
If X is a pointed, path-connected space having the homotopy type of a C W -complex, Y is a homotopy associative H -space, and f : X −→ Y is a continuous map, the James construction shows that f extends to an H -map f : X −→ Y
If X is as above, Z is homotopy abelian, and f : X −→ Z is a continuous map, it is natural to ask if there is an analogue of the James construction that extends f to a homotopy abelian H -space constructed functorially from X
If Z is homotopy associative a choice of Hopf construction can be made so that the H -map Z −→ ∂ Z in Corollary 2.6 appears as the fibration connecting map
Summary
If X is a pointed, path-connected space having the homotopy type of a C W -complex, Y is a homotopy associative H -space, and f : X −→ Y is a continuous map, the James construction shows that f extends to an H -map f : X −→ Y. Fibration sequences X −→ δ T −→ ∗ R −→ φ X have been studied in many contexts: to construct finite H -spaces in [3], to produce functorial retracts of X in [15], to establish a universal property for particular H -spaces in [6,10,19], and to analyze the “bottom” indecomposable factor of X in [7] in the case when. An example is when X is the odd primary Moore space P2n+1( pr ); the space T was constructed in [2] and shown to be neither homotopy associative nor homotopy commutative, but as φ factors through Whitehead products it will satisfy part (b). The authors would like to thank the referee for giving many constructive comments
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