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Some properties of Hopf-type constructions

Published online by Cambridge University Press:  24 October 2008

Martin Arkowitz  and
Paul Silberbush
Martin Arkowitz
Affiliation:
Department of Mathematics, Dartmouth College, Hanover, NH 03755, U.S.A.
Paul Silberbush
Affiliation:
Department of Mathematics, Dartmouth College, Hanover, NH 03755, U.S.A.
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Extract

If f: X × Y → Z is a map, then the classical Hopf construction associates to f a map hf: X * Y → ΣZ, where X * Y is the join of X and Y and ΣZ the suspension of Z. Since X * Y has the homotopy type of Σ(X Λ Y), the suspension of the smash product of X and Y, the homotopy class of hf can be regarded as an element Hf ↦ [Σ(X Λ Y), ΣZ]. Now elements of [Σ(X Λ Y), ] are in one to one correspondence with homotopy classes in the group [σ(X Λ Y), ΣZ] which are trivial on the suspension of the wedge Σ(X ≷ Y).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 117 , Issue 2 , March 1995 , pp. 287 - 301
Copyright
Copyright © Cambridge Philosophical Society 1995

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