A New Finite Loop Space at the Prime Two

From the point of view of homotopy theory a compact Lie group G has the following remarkable combination of properties: (1) G can be given the structure of a finite CW-complex, and (2) there is a pointed space BG and a homotopy equivalence from G to the loop space QB G . Of course the space BG in (2) is the ordinary classifying space of G. In general, a finite complex X together with a chosen equivalence X -+ nBX for some BX is called a finite loop space. If p is a prime number and the geometric finiteness condition on X is replaced by the requirement that X be Fp-complete in the sense of [3] and have finite mod p cohomology, then X is called a p-adic finite loop space or a finite loop space at the prime p. A (p-adic) finite loop space is a strong homotopy-theoretic analogue of a compact Lie group. The study of these spaces is related to many classical questions in topology (for instance, to the problem of determining all spaces with polynomial cohomology rings). Call a p-adic finite loop space X exotic if it is not the Fp-completion of G for a compact Lie group G. There are many known examples of exotic p-adic finite loop spaces at odd primes p [5] and the classification of these spa~es is partially understood [8, 9]. However, until now there have been no known exotic 2-adic finite loop spaces. Recall [32] that the ring of rank 4 mod 2 Dickson invariants is the ring of invariants of the natural action of GL( 4, F 2) on the rank 4 polynomial algebra H*((BZ/2)4, F2); this ring of invariants is a polynomial algebra on classes cs ' C12 ' C14 ' and Cl5 with Sq4Cs = c12 ' Sq2cl2 = c14 ' and Sql Cl4 = C15 • Our main theorem is the following one.