New Finite Loop Space 3 adic

We construct a space BDI(4) whose mod 2 cohomology ring is the ring of rank 4 mod 2 Dickson invariants. The loop space on BDI(4) is the rst example of an exotic nite loop space at 2. We conjecture that it is also the last one. 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 nite CW-complex, and (2) there is a pointed space BG and a homotopy equivalence from G to the loop space BG. Of course the space BG in (2) is the ordinary classifying space of G. In general, a nite complex X together with a chosen equivalence X ! BX for some BX is called a nite loop space. If p is a prime number and the geometric niteness condition on X is replaced by the requirement that X be F p-complete in the sense of 3] and have nite mod p cohomology, then X is called a p-adic nite loop space or a nite loop space at the prime p. A (p-adic) nite 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 nite loop space X exotic if it is not the F p-completion of G for a compact Lie group G. There are many known examples of exotic p-adic nite loop spaces at odd primes p 5] and the classiication of these spaces is partially understood 8] 9]. However, until now there have been no known exotic 2-adic nite 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 ; F 2); this ring of invariants is a polynomial algebra on classes c 8 , c 12 , c 14 , and c 15 with Sq 4 c 8 = c 12 , Sq 2 c 12 = c 14 , and Sq 1 c 14 = c 15. Our main theorem is the following one.