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Loop-theoretic properties of H-spaces

Published online by Cambridge University Press:  24 October 2008

Martin Arkowitz
Affiliation:
Department of Mathematics and Computer Science, Dartmouth College, Hanover, NH 03755, U.S.A.
Gregory Lupton
Affiliation:
Department of Mathematics, Cleveland State University, Cleveland, OH 44115, U.S.A.

Extract

If X is a topological space with base point, then a based map m: X × XX is called a multiplication if m restricted to each factor of X × X is homotopic to the identity map of X. The pair (X, m) is then called an H-space. If A is a based topological space and (X, m) is an H-space, then the multiplication m induces a binary operation on the set [A, X] of based homotopy classes of maps of A into X. A classical result due to James [6, theorem 1·1 asserts that if A is a CW-complex and (X, m) is an H-space, then the binary operation gives [A, X] the structure of an algebraic loop. That is, [A, X] has a two-sided identity element and if a, b ∈[A, X], then the equations ax = b and ya = b have unique solutions x, y ∈ [A, X]. Thus it is meaningful to consider loop-theoretic properties of H-spaces. In this paper we make a detailed study of the following loop-theoretic notions applied to H-spaces: inversivity, power-associativity, quasi-commutativity and the Moufang property – see Section 2 for the definitions. If an H-space is Moufang, then it has the other three properties. Moreover, an associative loop is Moufang, and so a homotopy-associative H-space has all four properties. Since many of the standard H-spaces are homotopy-associative, we are particularly interested in determining when an H-space, in particular a finite CW H-space, does not have one of these properties.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

REFERENCES

[1]Arkowitz, M. and Lupton, G. M.. Rational co-H-spaces. Comment. Math. Helv. (to appear).Google Scholar
[2]Bruck, R. H.. Contributions to the theory of loops. Trans. Amer. Math. Soc. 60 (1946), 245354.CrossRefGoogle Scholar
[3]Douglas, R. R.. H-equivalence and quasi-commutativity. J. Indian Math. Soc. 34 (1970), 109113.Google Scholar
[4]Hilton, P. J., Mislin, G. and Roitberg, J.. Localization of Nilpotent Groups and Spaces. Notas de Matemática vol. 15 (North-Holland, 1975).CrossRefGoogle Scholar
[5]James, I. M.. Multiplications on spheres (II). Trans. Amer. Math. Soc. 84 (1957), 545558.Google Scholar
[6]James, I. M.. On H-spaces and their homotopy groups. Quart. J. Math. Oxford Ser. (2) 11 (1960), 161179.CrossRefGoogle Scholar
[7]Kane, R. M.. The Homology of Hopf Spaces. North-Holland Mathematical Library (North-Holland, 1988).Google Scholar
[8]Milnor, J. W. and Moore, J. C.. On the Structure of Hopf algebras. Ann. of Math. 81 (1965), 211264.CrossRefGoogle Scholar
[9]Norman, C. W.. Homotopy loops. Topology 2 (1963), 2343.CrossRefGoogle Scholar
[10]O'Neill, R. C.. On H-spaces that are CW-complexes I. Illinois J. Math. 8 (1964), 280290.CrossRefGoogle Scholar
[11]Scheerer, H.. On rationalized H- and co-H-spaces, with an appendix on decomposable H-and co-H-spaces. Manuscripta Math. 51 (1984), 6387.CrossRefGoogle Scholar
[12]Spanier, E. H.. Algebraic Topology (McGraw-Hill, 1966).Google Scholar
[13]Stasheff, J. D.. H-Space Problems, H-Spaces (Neuchâtel). Lecture Notes in Math. vol. 196 (Springer-Verlag, 1971), 122136.Google Scholar
[14]Whitehead, G. W.. Elements of Homotopy Theory. Graduate Texts in Math. no. 61 (Springer-Verlag, 1978).CrossRefGoogle Scholar
[15]Williams, F. D.. Quasi-commutativity of H-spaces. Michigan Math. J. 19 (1972), 209213.CrossRefGoogle Scholar