A new finite loop space at the prime two

Abstract

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, F 2 ); this ring of invariants is a polynomial algebra on classes c s ' C 12 ' C 14 ' and C l5 with Sq 4 C s = c 12 ' Sq 2 c l2 = c 14 ' and Sql C l4 = C 15 • Our main theorem is the following one.1.1. Theorem. There exists an F 2-complete space B DI( 4) such that H*(BDI(4) , F 2 ) is isomorphic as an algebra over the Steenrod algebra to the ring of rank 4 mod 2 Dickson invariants.Let DI(4) = QBDI(4). Standard methods show that H*(DI(4) , F 2 ) is multiplicatively generated by elements x 7 , y 11' and z I3' with S q 4 X = y, Sq2y = z, Sql Z = x 2 =1= 0, and X4 = i = z2 = O. This space DI(4) is an exotic 2-adic finite loop space.

On the 2-compact group DI(4)

“…In both cases the higher derived limits all vanish [8], the Bousfield spectral sequence collapses and is concentrated in a singular line. This allows to prove the following proposition: Proof.…”

“…If H * (A; F 2 ) ∼ = H * ((BZ/2) 4 ; F 2 ) Gl(4,F 2 ) , standard methods show that ΩA is F 2 -finite (see [8]). The pair (ΩA, A) gives rise to a 2-compact group.…”

“…In this section we recall the construction of DI(4) and other material from [8]. The general linear group Gl(4, F 2 ) acts on the 4-dimensional F 2 -vector space E 4 = (F 2 ) 4 and therefore on the algebra of formal polynomial functions…”

“…Sullivan showed that, if n divides p − 1, the sphere S 2n−1 gives rise to a connected p-compact group ((S 2n−1 ) ∧ p , B(S 2n−1 ) ∧ p ) [27]. At the prime 2, Dwyer and Wilkerson constructed the only known example DI(4) of a connected 2-compact group, which is not a Lie group [8]. The classifying space BDI(4) realizes the rank 4 mod-2 Dickson invariants, namely 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 ).…”

Triangles of groups

“…Here Sol is a (solvable) group of order 210 • 34, sol is a maximal subgroup of C03 of order 210-33,and Spin7(p) is the non-split double cover of the simple orthogonal group £li(p), p an odd prime. Much more recently, David Benson [2] has investigated an "exotic 2-adic finite loop space" discovered by Dwyer and Wilkerson [6]. In exploring the structure of this object, Benson demonstrates the existence of a triangle of groups (which I will call (Tf)) of the following type:…”