A New Finite Loop Space
at the Prime Two
W. G. Dwyer and C. W. Wilkerson
University of Notre Dame
Purdue University
Abstract. 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 first example of an exotic finite loop spaceat 2. We conjecture
that it is also the last one.
x1. Introduction
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 isa p ointed space BG and a homotopy equivalence from G
to the lo op space B G.
Of course the space BG in (2) is the ordinary classifying space of G.
In general, a finite complexX together with a chosen equivalence X !
BX for some BX is called a finite loop space. If p is a prime number
and the geometric finiteness condition on X isreplaced by the require-
ment that X be Fp -complete in the sense of [3] andhave 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 spaces 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 invariantsis the ring
of!invariants!of the natural action of GL (4; F2 )on the rank 4 polynomial
algebra!H ((BZ=2 )4;F2 ); this ring of invariants is a polynomial algebra
on!classes!c8, c12, c14, and c15 with Sq4c8 = c12, Sq2c12 = c14, and
Sq1c14!=!c15. Our main theorem is the following one.
!
2 Dwyer and Wilkerson
1.1 Theorem. There exists an F2 -complete space BDI (4) such that
H (BDI (4) ;F2 ) is isomorphic as an algebra over the Steenrod algebra
to the ring of rank 4 mod 2 Dickson invariants.
LetDI (4) = BDI (4). Standard methods show that H (DI (4); F2 )is
multiplicatively generated by elements x7, y11, and z13, with Sq4x = y,
Sq2y = z,S q1z= x2 6= 0, and x4 = y2 = z2 = 0. This spaceDI (4) is an
exotic 2-adic finite loop space.
Itis natural to ask about the realizability of other Dickson invari-
ant algebras. Say that a space Y is of type BDI (n) if H (Y; F2 ) is
isomorphic, as an algebra over the Steenrod algebra, to the algebra of
rank n mod 2 Dickson invariants [32]. Then RP1 = BZ=2 is of type
BDI (1), B SO (3) is of type BDI (2) and the classifying space B G 2 of
the exceptional Lie group G 2 is of type BDI (3). It is known that no
space of type BDI (n) can exist for n 6 [31]. Lannes has recently used
methods similar to ours to show the non-existence of a space of type
BDI (5), while [24] proves the stronger statementthat an H -space"of
type DI (5)" does not exist (see also [18]). The construction of BDI (4)
in this paper completes the determination of the set of integers n for
which a space of type BDI (n) exists.
Oneindication that BDI (4) might exist comes from Lie theory. If G
is a connected compact Lie group of rankr ,then the Weyl group WG is
a finite subgroup of GL (r;Q) generated by reflections, and the rational
cohomology ring of BG is naturally isomorphic to the ring of polynomial
invariants of WG. Finite reflection groups are relatively uncommon,and
in fact G is close to being determined by its Weyl group WG . Let
^Z2 denote the ring of 2-adic integers. About ten years ago the second
author observed that there exists afinite reflection subgroup WDI(4) of
GL (3;Q ^Z2) such that the ring Q H (BDI (4); ^Z2) (which is easily
seen to be a polynomial algebra on generators of dimensions 8, 12, and
28) is isomorphic to the ring of polynomial invariants of WDI(4). In a
sense this group WDI(4) is a plausible "Weyl group" for a 2-adic finite
loop space of type DI (4). The group WDI(4) is isomorphic to Z=2
GL (3;F2 ) and arises as complex reflection group (number 24 on the list
of [5]) whose reflection representation can be defined over a subfield of
C which embeds in Q ^Z2[5, p. 431]. In 1980 there was no evident
way to use the existence of WDI(4) to construct BDI (4) ; even now we
use the existence of WDI(4) only in an indirectway (x4) and make no
explicit reference to the reflection prop erties of the group. On the whole,
though, the results of this paper suggest that there is a very close link
between p-adic finite loop spaces and p-adic reflection groups. Recent