Higher forms of homotopy commutativity and finite loop spaces (Q1824241)

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.