Homotopy commutativity of finite loop spaces

Abstract

Let p be a fixed prime. We say that X is a finite loop space if X has the homotopy type of a 1-connected p-localized finite CW-complex and X has a classifying space BX. As examples of finite loop spaces we can consider: (1) The p-localizations of the compact 1-connected Lie groups. (2) For each irreducible complex reflection group G of order prime to p, whose unitary representation restricts to the p-adics, we can construct a space X(G) where mod p cohomology is the ring of invariants of G, and it is therefore a polynomial algebra. Since by [-2] this space can be "pulled back" to a space of finite type, then (2(X(G)(p)) is a finite loop space. These irreducible types are listed in Clark-Ewing list. (3) In [9] Quillen showed that there is a space X such that H*(X,F~)~-Fp[x I . . . . . xn] where deg(xl)=2mi and m is any divisor of p 1 . These spaces are called "p-adic Grasmannians" and, also by [2], f2(X(p)) are finite loop spaces. (4) Zabrodsky constructed in [13] two more spaces with polynomial mod p cohomology which also produce finite loop spaces in the same way as above. (5) In [-1] Aguad6 proved that the types 29 and 34 of the Clark-Ewing list are realizables for p = 5 and p = 7 respectively, and consequently their rood p cohomology is the ring of invariants. Furthermore, for the type 29 and p--5 , Aguad6 has also proved that its m o d p cohomology is a polynomial algebra, hence its loop space is a finite loop space. Since for the type 34 and p = 7 the proof of the same is in progress, we also include its loop space as a finite loop space. (6) Any product of the spaces above is also a finite loop space.