## Higher forms of homotopy commutativity and finite loop spaces.(English)Zbl 0682.55006

Let X be a loop space. Its classifying space BX is an H-space iff X is strongly homotopy commutative in the sense of Sugawara. Strong homotopy commutativity is defined by an infinite sequence of coherent homotopies $$C_ n: X^{2n}\times I^ n\to X$$ starting with the commuting homotopy for $$n=1$$, and is equivalent to the existence of an $$A_{\infty}$$ structure on the multiplication map $$X\times X\to X$$. A loop space X is called a $$C^ n$$-space if the coherent homotopies $$C_ k$$ exist for $$1\leq k<n$$. The author investigates finite $$C^ p$$-spaces, p a prime, and proves that a non-contractible, 1-connected, p-local finite loop space is not a $$C^ p$$-space. This extends a result of John Hubbuck who studied homotopy commutative finite H-spaces. The proof is inspired by Hubbuck’s argument: A $$C^ p$$-structure on X makes the k-fold multiplication $$X^ k\to X$$ and the k-th power map $$X\to X$$ an $$A_ p$$- map. The result is obtained by studying the induced map of a suitable power map on the K-theory of p-th projective space $$P_ p(X)$$. In addition, the author compares $$C^ n$$-structures with the notion of higher homotopy commutativity due to Frank Williams.
Reviewer: R.Vogt

### MSC:

 55P45 $$H$$-spaces and duals 55P35 Loop spaces
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### References:

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