Abstract
Let R be a subring of the rationals with 1/2, 1/3∈R; let S nR denote the R-local n-sphere and define Ω nR :=S nR for n odd, Ω nR :=ΩΣS nR for n>0 even. An H-space (resp. a 1-conn. co-H-space) is “decomposable over R”, if it is homotopy equivalent to a weak product of spaces Ω nR (resp. to a wedge of R-local spheres). We prove that, if E is grouplike decomposable of finite type over R, the functor [-,E] is determined on finite dim. complexes by the Hopf algebra M*(E;R); here M* denotes the unstable cohomotopy functor of H.J. Baues. If C is cogrouplike decomposable over R, the functor [C,-] is determined on 1-conn. R-local spaces by π*(ΩC) as a cogroup in the category of M-Lie algebras. For R = Φ the functor [-,E] is also determined by the Lie algebra π*(E) and [C,-] by the Berstein coalgebra associated to the comultiplication of C.
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Scheerer, H. On rationalized H- and co-H-spaces with an appendix on decomposable H- and co-H-spaces. Manuscripta Math 51, 63–87 (1985). https://doi.org/10.1007/BF01168347
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DOI: https://doi.org/10.1007/BF01168347