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Samelson Products In Spaces of Self-Homotopy Equivalences

Published online by Cambridge University Press:  20 November 2018

Jesper Michael Møller
Jesper Michael Møller*
Affiliation:
Københavns Universitet, Københavns, Denmark
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The homotopy groups of any group-like space are equipped with a Samelson product satisfying, up to sign, the identities of a graded Lie bracket. We shall compute the Samelson product in two kinds of spaces of selfhomotopy equivalences arising when adding a homotopy or a homology group to a space.

First, let A→ X be a cofibration with a Moore space M(G,n) as cofibre. For the monoid autA (X) of maps under A homotopic (rel. A) to the identity, the Samelson product is a pairing

πn+i(G;X)⨂πn+j(G;X) → πn+i+j(G;X)

of homotopy groups with coefficients [1] in G. Theorem 2.1 computes this pairing in terms of a homomorphism associated to a α ∈ πi(autAX)).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 42 , Issue 1 , 01 February 1990 , pp. 95 - 108
Copyright
Copyright © Canadian Mathematical Society 1990

References

1. Hilton, P., Homotopy theory and duality (Gordon and Breach, New York, 1965).Google Scholar
2. Mϕller, J.M., Spaces of sections of Eilenberg-MacLane fibrations, Pacific J. Math. 130 (1987), 171–186.Google Scholar
3. Samelson, H., A connection between the Whitehead and the Pontryagin product, Amer. J. Math 75 (1953), 744–752.Google Scholar
4. Switzer, R.M., Counting elements in homotopy sets, Math. Z. 178 (1981), 527-554Google Scholar
5. Algebraic topology — homotopy and homology, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 212 (Springer-Verlag, Berlin-Heidelberg-New York, 1975).Google Scholar
6. Whitehead, G.W., On mappings into group-like spaces, Comment. Math. Helv. 28 (1954), 320–328.Google Scholar
7. Whitehead, G.W., Elements of homotopy theory, Graduate Texts in Mathematics 61 (Springer-Verlag, Berlin-Heidelberg-New York, 1978).Google Scholar