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An Operad Action on Infinite Loop Space Multiplication

Published online by Cambridge University Press:  20 November 2018

Thomas Lada
Thomas Lada*
Affiliation:
North Carolina State University, Raleigh, North Carolina
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It is well-known that an infinite loop space is an H-space whose multiplication enjoys nice properties concerning associativity and commutativity. A practical way of identifying infinite loop spaces is the utilization of May's recognition principle [3; 4]. To apply this principle, one requires an E∞-operad action on a space X; this action gives rise to various multiplications on X. In this note, it is shown that such multiplications enjoy an operad action up to homotopy that encodes the associativity and commutativity information, and that May's delooping theorem may be applied to them. We refer to [3] for the terminology of operads and monads.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 29 , Issue 6 , 01 December 1977 , pp. 1208 - 1216
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Boardman, J. M. and Vogt, R. M., Homotopy-everything H-spaces, Bull. Amer. Math. Soc. 74 (1968), 1117–1122.Google Scholar
2. Cohen, F., Lada, T., and May, J. P., The homology of iterated loop spaces, Lecture Notes in Mathematics 533 (Springer-Verlag, 1976).Google Scholar
3. May, J. P., The geometry of iterated loop spaces, Lecture Notes in Mathematics 271 (Springer- Verlag, 1972).Google Scholar
3. May, J. P. Ex spaces, group completions, and permutative categories, London Mathematical Society Lecture Note Series II, p. 61–94 (Cambridge University Press, 1974).Google Scholar