Homotopy commutativity of finite loop spaces (Q1814124)

Suppose that \(p\) is a prime and \(X\) a finite loop space of the homotopy type of a 1-connected \(p\)-localized CW-complex. The main result of the paper is to consider the known, so-called standard, examples of finite loop spaces and determine whether there is a homotopy commutative multiplication on \(X\). The standard examples are products of spaces of the following type: (1) \(p\)-localizations of compact 1-connected Lie groups, (2) the spaces \(\Omega(X(G)_{(p)})\) for \(G\) an irreducible complex reflection group of order prime to \(p\) (from the Clark-Ewing list), (3) \(\Omega(X_{(p)})\) where \(X\) is a \(p\)-adic Grassmannian constructed by Quillen, that is, \(H^*(X;F_ p)\cong F_ p[x_ 1,\dots,x_ n]\) and \(\hbox{deg} x_ i=2mi\) for \(m\) a divisor of \(p-1\), (4) two spaces constructed by Zabrodsky of type (12,16) for \(p=3\) and of type (16,24,40,48) for \(p=5\), and (5) two spaces constructed by Aguadé of type (8,16,24,40) for \(p=5\) and of type (12,24,36,48,60,84) for \(p=7\). By splitting the space into products of spheres and sphere bundles over spheres, that is, taking the space to be \(p\)-regular or \(p\)-quasi- regular, the author proves that a standard finite loop space \(X\) has a homotopy commutative multiplication if and only if it is a product of (i) finite loop spaces of type \((2n_ 1,\dots,2n_ l)\) and \(p>2n_ l\), (ii) the sphere bundles \(B_ 1(p)\) for \(p\) odd, \(B_ 7(17)\) for \(p=17\), \(B_ 5(19)\) for \(p=19\), \(B_{19}(41)\) for \(p=41\), \(B_{11}(19)\) for \(p=19\), \(B_ 3(11)\times S^{11}_{(11)}\) for \(p=11\), and \(B_ 1(19)\times B_{11}(19)\) for \(p=19\) (with any loop multiplication). The proof is by exhaustive analysis involving the application of results of McGibbon and careful analysis of Whitehead products and Steenrod algebra module structures.