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 space at 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 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 finite complex X 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 is replaced by the require-
ment 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 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 invariants is 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.
______________________________________
This work was supported in part by the National Science Foundation.
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.
Let DI (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, Sq1z = x2 6= 0, and x4 = y2 = z2 = 0. This space DI (4) is an
exotic 2-adic finite loop space.
It is 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 RP 1 = 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 statement that 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.
One indication that BDI (4) might exist comes from Lie theory. If G
is a connected compact Lie group of rank r, 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
Z^2 denote the ring of 2-adic integers. About ten years ago the second
author observed that there exists a finite 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 x
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 indirect way (x4) and make no
explicit reference to the reflection properties 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
work of the authors has carried this idea further by showing that each p-
New Finite Loop Space 3
adic finite loop space determines a p-adic reflection group. On the basis
of the classification of 2-adic reflection groups ([5], [9]) we are therefore
led to conjecture that the classifying space of any connected 2-adic finite
loop space is equivalent to the product of a number of copies of BDI (4)
with the F2 -completion of the classifying space of a connected compact
Lie group.
1.2 The basic technique: Our present method of building BDI (4)
depends on the ideas of [20], [17] and [13]. The starting point is again
the theory of Lie groups, but this time on the homotopy theoretic side.
Let T denote the category of topological spaces. Jackowski and McClure
have proved the following remarkable result:
1.3 Proposition. [17] For any compact Lie group G there exist
(1) a category ALieGwhose objects are the (conjugacy classes of) non-
trivial elementary abelian 2-subgroups of G,
(2) a functor ffLieG: (ALieG)op ! T which up to homotopy assigns to
each such subgroup the classifying space of its centralizer in G,
and
(3) a map ho lim-!ffLieG! BG which is an isomorphism on F2 coho-
mology.
Unless G has a central element of order 2, this proposition provides a
decomposition of BG up to F2 cohomology as a generalized pushout of
classifying spaces of proper subgroups of G. There is also an algebraic
version of the above. Let K denote the category of unstable algebras
over the mod 2 Steenrod algebra.
1.4 Proposition. [13, x2] For any object R of K there exist
(1) a category AalgRin which an object is a pair (V; f *) consisting of
a non-trivial elementary abelian 2-group V together with (1.5) a
suitable K-map f *: R ! H* (BV; F2 ),
(2) a Lannes functor ffalgR: (AalgY)op ! K which sends the pair (V; f *)
to an algebra TfV*(R) (see 1.5), and
(3) a map R ! lim-ffalgRwhich in favorable cases is an isomorphism.
Moreover if R = H* (BG; F2 ) (G compact Lie) then [21] [13] the cate-
gories AalgRand ALieGare equivalent in such a way that the functors ffalgR
and H* (ffLieG; F2 ) correspond.
Now let Y be BDI (4) and let R = H* (Y; F2 ) be the rank 4 Dickson
invariant algebra. It is easy to figure out the category AalgR(see 1.5)
and for each object (V; f *) of AalgRto compute the algebra ffalgR(V; f *).
It is natural to hope that Y is sufficiently similar to the classif*
*ying
4 Dwyer and Wilkerson
space of a compact Lie group that something like the Jackowski-McClure
decomposition might work for Y . With this as motivation, then, we
construct by hand (x7) a functor F : (AalgR)op ! T such that H* (F; F2 )
is naturally equivalent to ffalgR, and verify by direct algebraic calculation
(2.4) that the cohomology of the homotopy direct limit of this functor is
the ring of rank 4 mod 2 Dickson invariants. We obtain BDI (4) as the
F2 -completion of this homotopy direct limit.
To build the functor F we first (x6) construct a parallel functor F with
values in the homotopy category and then use the obstruction theory of
[6] to lift F to a functor F with values in the category of spaces. We are
fortunate that this lifting problem is easy to solve once it is set up: all
of the obstruction groups are zero (7.6).
1.5 The role of Spin(7): The construction of the above functor F
involves careful study of the Lie group Spin (7) and some of its sub-
groups. (Here Spin (7) is the connected double covering group of the
special orthogonal group SO (7).) There is a simple explanation for this.
Let R = H* (BDI (4) ; F2 ) be the rank 4 Dickson invariant algebra. Be-
cause of 1.3 and 1.4 it is useful to refer to an object A = (V; f *) 2 AalgR
with rank (V ) = r as a rank r elementary abelian 2-subgroup of DI (4)
(taken up to conjugacy), and to interpret ffalgR(A ) as the cohomology of
the classifying space of the centralizer of A in DI (4). Explicitly, then,
the elementary abelian 2-subgroups of DI (4) correspond to pairs (V; f *)
where V is an elementary abelian 2-group and f * : R ! H* (BV; F2 )
is a K-map which makes H* (BV; F2 ) into a finitely generated module
over R [13, x2]. The following lemma (due to Lannes) explains how to
enumerate these and calculate the values of the functor ffalgR.
We will let T V denote the T -functor of Lannes [19]; T V is the left ad-
joint on the category of unstable modules (or algebras) over the Steenrod
algebra to the functor given by tensoring with H* (BV; F2 ). If X is a
space the algebra T V(H* (X; F2 )) is closely related to the F2 cohomol-
ogy algebra of the space of maps from BV to X. Recall [11] that to any
object S 2 K and map f *: S ! H* (BV; F2 ) there is associated a quo-
tient object TfV*(S) of T V(S); if S = H* (X; F2 ) then TfV*(S) is closely
related to the F2 cohomology ring of the space of maps BV ! X induc-
ing f * on cohomology. (Note that in [13] the algebra TfV*(S) is denoted
T (V; S)f .)
1.6 Lemma. Let E be an elementary abelian 2-group, a subgroup
of Aut (E), and S the unstable algebra over the Steenrod algebra given
by the fixed point set H* (BE; F2 ) . Let V be an elementary abelian
2-group.
New Finite Loop Space 5
(1) The map which assigns to each homomorphism f : V ! E the
f*
induced composite S !- H* (BE; F2 ) -! H* (BV; F2 ) gives a
bijection between K-maps f *: S!- H* (BV; F2 ) and -orbits of
homomorphisms f : V ! E.
(2) A map f * : S!- H* (BV; F2 ) makes H* (BV; F2 ) into a finitely
generated S module iff the corresponding (1) homomorphism f :
V ! E is a monomorphism.
(3) If f *: S!- H* (BV; F2 ) corresponds to f : V ! E, then TfV*(S)
is naturally isomorphic to H* (BE; F2 )f , where f is the
subgroup of elements which pointwise fix the image of f .
Proof: By exactness of the functor T V [21] there is an isomorphism
T V(S) = T V(H* (BE; F2 ) ) ~= (T V(H* (BE; F2 )) :
Moreover, the algebra T V(H* (BE; F2 )) is isomorphic [21, 3.4.5] as a
-object to the tensor product H0 (Hom (V; E); F2 ) H* (BE; F2 ) with
the diagonal action (the first factor here is the zero-dimensional co-
homology of the discrete space Hom (V; E); the tensor product itself in
fact gives the cohomology of the space of maps BV ! BE). Parts (1)
and (3) follow from the definition of TfV* and inspection. Part (2) can
be obtained for instance from [13, proof of 4.4].
Since the group = GL (4; F2 )acts transitively on the subspaces of
(Z=2 )4 of any given dimension, it follows that up to conjugacy there
is only one rank i elementary abelian 2-subgroup Ai of DI (4) for each
i = 1; : : :; 4. Moreover, the algebra ffalgR(Ai ) is isomorphic to the fixed
point set of the action of i on H* ((BZ=2 )4; F2 ), where i GL (4; F2 )
is the set of matrices which agree with the identity in the first i columns.
A study of morphisms [13, x 2] now implies that the category AalgRis
equivalent to the category A of x2 and that the functor ffalgRis equivalent
to the functor ffR of 2.3. The category A has four objects and the
following shape
A1 ) A2 ) A3 ) A4
| | | |
{1} GL (2; F2 ) GL (3; F2 ) GL (4; F2 )
where under each object is its monoid (in this case group) of self maps
and ")" stands for an appropriate set of morphisms. There are no
morphisms Ai !- Aj for i > j and for i j the group GL (j; F2 )=
Aut (Aj ) acts transitively on the set of morphisms Ai !- Aj .
6 Dwyer and Wilkerson
This calculation, though, reveals something perhaps unexpected: the
cohomology ring ffalgR(A1 ) of the classifying space of the centralizer of
the essentially unique rank 1 elementary abelian 2-subgroup of DI (4)
(the cohomology ring calculated as H* ((BZ=2 )4; F2 )1 above) is exactly
(3.11) the F2 cohomology ring of B Spin (7)! Encouraged by this, we
surmise that this centralizer is Spin (7) and begin the construction of F
by declaring F (A1 ) to be the F2 -completion of B Spin (7). It then stands
to reason that the centralizer in DI (4) of a rank 2 elementary abelian
2-subgroup A , say, should be the same as the centralizer of A inside the
centralizer within DI (4) of any rank 1 subgroup of A , i.e. the same as
the centralizer of A within Spin (7). This can be verified cohomologically
by combining 1.3, 1.4 and 1.6; verification comes down to observing
that the fixed point set of the action of the group i (2 i 4)
on H* ((BZ=2 )4; F2 ) doesn't depend on whether i is encountered as a
subgroup of 1 or as a subgroup of GL (4; F2 ). The conclusion is (3.13)
that for i = 2, 3, 4 the algebra ffalgR(Ai ) is isomorphic to H* (BC(Ei); F2 ),
where C(Ei) is the centralizer in Spin (7) of a rank i elementary abelian
2-group Ei containing the center of Spin (7). This is evidence enough for
us to declare F (Ai ) to be the F2 -completion of BC(Ei).
1.7 Final ingredients: Once we have settled on the spaces which
the functor F will take on as its values, we have to look for homotopy
classes of maps between these spaces satisfying the relations described
in 2.1. The basic maps F (Ai ) ! F (Aj ) (i > j) arise from the inclusion
of one subgroup of Spin (7) in another. No non-identity self-maps of
F (A1 ) are required, and the necessary self-maps of F (A4 ) are easy to
construct, since this space is just (BZ=2 )4. For i = 2 and i = 3, though,
it turns out that inner automorphisms of Spin (7) provide actions of a
certain matrix group (x2) P (1; i - 1) GL (i; F2 )on F (Ai ), whereas the
existence of F as a functor requires an action of the full group GL (i; F2 )
on this space. The case i = 2 of this problem is solved in x5, and the
case i = 3 in x 4 (this last case is particularly interesting because it
involves constructing what is essentially a 2-adic integral form of the
basic reflection representation of WDI (4)). A significant amount of effort
goes into being explicit enough about these solutions that it is possible to
verify concretely that the necessary diagrams commute up to homotopy
and that they have the correct cohomological properties (see x6).
In the end our diagram looks like this:
^BSpin (7) ( B^(SU (2)3={1}) ( ^B((S1 )3x| Z=2 ) ( B(Z=2 *
*)4
| | | |
{1} GL (2; F2 ) GL (3; F2 ) GL (4; F*
*2 )
New Finite Loop Space 7
where "B^X" denotes the F2 completion of "BX". (See x3 for detailed
descriptions of the above groups.) Of course the spaces in the above
diagram exist integrally, but the maps we use exist only after F2 com-
pletion.
Finally, we note that the conflict between Theorem 1.1 and the non-
realizability results of [25] has been resolved in [23].
Lie-like properties: The construction of BDI (4) given in this paper
begins with the hypothesis that this space has properties similar to those
of the classifying space of a compact Lie group. Theorem 1.8, whose
proof is computational but routine, points out a few striking Lie-like
properties that follow from the construction
1.8 Theorem. The space BDI (4) has the following properties:
(1) There is a map f : B Spin (7) ! BDI (4) such that the homotopy
fibre DI (4)= Spin (7) has F2 cohomology which is finite, concen-
trated in even degrees, and has Euler characteristic 24.
Let BT ! B Spin (7) be induced by the inclusion of the (rank 3) maximal
torus, let g denote the composite BT ! B Spin (7) ! BDI (4) and let
DI (4)=T denote the homotopy fibre of g.
(2) H* (DI (4)=T; F2 ) is finite, concentrated in even degrees, and has
Euler characteristic 336 = |WDI (4)|.
(3) The space of self-equivalences of B^ T (see 1.9) over BDI (4) is
homotopically discrete and has WDI (4)as its group of components
(cf. [9, 2.8]).
The space DI (4) = Spin (7) is perhaps an exotic symmetric space at
the prime 2. The space DI (4)=T is the flag manifold (at the prime 2)
corresponding to DI (4).
1.9 Notation and terminology: We will follow the standard con-
vention that composition of maps between sets, groups, etc. works from
right to left: a composite function f . g or f g is defined by the formula
f . g(x) = f (g(x)). In particular, if X is a set, group, etc. the monoid
Hom (X; X) acts in a natural way on X from the left.
Unless otherwise specified, cohomology is taken with coefficients in
the field F2 of two elements. If X and Y are spaces then Map (X; Y )
denotes the function space of maps from X to Y ; the component con-
taining a particular map or homotopy class f is Map (X; Y )f . The space
Emb (X; Y ) is the subspace of Map (X; Y ) containing maps with respect
to which H* X is a finitely generated module over H* Y ; Aut (X) is
the topological monoid of self-homotopy equivalences of X with identity
component Aut (X)1. The symbol B^G will stand for the F2 -completion
of the classifying space BG.
8 Dwyer and Wilkerson
The center of a (Lie) group G is denoted Z(G); if G is given as a
subgroup of some ambient group H then C(G) and N (G) denote respec-
tively the centralizer and normalizer of G in H. For general compact Lie
groups G and H we will let Emb (G; H) stand for the set of H-conjugacy
classes of group monomorphisms G ! H; note [29, 2.4] that there are
natural maps Emb (G; H) ! ss0Emb (BG; BH) ! ss0Emb (B^G; ^BH).
The symbol n will stand for the symmetric group of degree n, i.e.,
the group of automorphisms of the set {1; : : :; n}. Suppose that G is a
(topological) group or monoid. The wreath product n o G is the space
n x Gn with multiplication determined by the formula
[g1; : : :; gn ] . oe = oe . [goe(1); : : :; goe(n)] ;
where oe 2 n and [g1; : : :; gn ] belongs to the product monoid Gn .
The authors would like to thank D. Benson, R. Oliver, R. Kane and
the referee for very helpful comments.
x2. The basic category
In this section we describe the basic index category A which is used
in the paper; then we construct and study the algebraic diagram over A
which we intend eventually to realize geometrically.
Let A be the category whose objects are the F2 vector spaces (Z=2 )i,
i = 1; : : :; 4 and whose morphisms are vector space monomorphisms.
Give the object (Z=2 )i (denoted Ai ) the standard ordered basis ei(j) ,
j = 1; : : :; i, where ei(j) is zero except in the j'th coordinate. It is clear
that Hom A (Ai ; Aj ) is empty if i > j, and that (in view of the chosen
basis for Ai ) the group Hom A (Ai ; Ai) is GL (i; F2 ). We will think of
the elements of Ai as column vectors, so that as usual (1.9) the group
GL (i; F2 )acts on the elements of Ai from the left.
There are standard embeddings fi : Ai ! Ai+1 , i = 1; : : :; 3 in A
determined by the formula fi(ei(j)) = ei+1 (j). Let P (i; j) denote the
subgroup of GL (i + j; F2 )consisting of matrices of the form
M11 M12
0 M22
where M11 2 GL (i; F2 ), and let ri : P (i; j)! GL (i; F2 )be the homo-
morphism which sends the above element of P (i; j)to the matrix M11.
The group P (i; j)is the subgroup of GL (i + j; F2 )given by transforma-
tions which carry to itself the subspace of Ai+j generated by the first i
standard basis elements.
The following lemma is an elementary calculation.
New Finite Loop Space 9
2.1 Lemma. The category A is generated by the groups GL (i; F2 )=
Hom A (Ai ; Ai), i = 1; : : :; 4 and the maps fi : Ai ! Ai+1 , i = 1; : : :;*
* 3
subject to the following relations:
(1) g . f1= f1 . r1(g) g 2 P (1; 1)
(2) g . f2= f2 . r2(g) g 2 P (2; 1)
(3) g . f3= f3 . r3(g) g 2 P (3; 1)
(4) g . f2 . f1= f2 . f1 . r1(g) g 2 P (1; 2)
(5) g . f3 . f2= f3 . f2 . r2(g) g 2 P (2; 2)
(6) g . f3 . f2 . f1= f3 . f2 . f1 . r1(g) g 2 P (1; 3)
2.2 Remark: Lemma 2.1 implies that in order to construct a functor
F : A ! C for some category C, it is enough to find objects F (Ai) = Ci,
i = 1; : : :; 4 of C, homomorphisms GL (i; F2 )! Hom C (Ci; Ci), and
maps F (fi) : Ci ! Ci+1 , i = 1; : : :; 3 such that relations parallel to
2.1(1)-2.1(6) are satisfied.
2.3 A cohomology diagram: Now let V stand for the elementary
abelian group (Z=2 )4. The natural (left) action of Aut (V ) on V induces
a right action of Aut (V ) on the cohomology ring H* BV . The ring of
elements fixed under this action is the algebra R of rank 4 mod 2 Dickson
invariants [32].
If A is an object of A let f"fR(A ) = H* Emb (BA ; BV ); this gives
a functor f"fR from A to the category of right Aut (V )-objects in the
category K (1.2). Note (cf. proof of 1.6) that f"fR(A ) is isomorphic to
the tensor product H0 Emb (A ; V ) H* BV with the diagonal action of
Aut (V ). Let ffR be the fixed point subfunctor of "ffR, so that for each
object Ai of A , ffR (Ai ) = "ffR(Ai )Aut (V ).
2.4 Proposition. There is a natural isomorphism R ~= lim-ffR . The
higher derived functors lim-iffR vanish for i > 0.
Proof: Let = Aut (V ) and let M be the right Aut (V ) module S =
H* BV treated as a left module via the inversion map of Aut (V ). The
category A is equivalent to the category A of x8 in such a way that ffR
corresponds to the functor ff;M ; proposition 8.1 thus gives the desired
result. (Note that in this special case, in which = Aut (V ) is as large
as possible, the categoryA is equivalent to the category B of 8:3 in
such a way that the functor ff;M is equivalent to fi;M .)
Proposition 2.4 also follows from the main theorem of [13], since in
fact the functor ffR of 2.4 is naturally equivalent to the functor ffR of
[13, 1.5].
10 Dwyer and Wilkerson
Suppose that F : Aop ! T is a functor, where T is the category of
topological spaces. According to [3, XII, 5.8] there is a spectral sequence
for the cohomology of the homotopy direct limit ho lim-!F which has the
form
{Ep;q2= lim-pHq (F )} ) Hp+q (ho lim-!F ) :
If H* (F ) ~= ffR , then by 2.4 the spectral sequence collapses into the
isomorphism H* (ho lim-!F ) ~= R and by [3, VII, 3.2] the F2 -completion
of ho lim-!F is an F2 -complete space with cohomology given by R. In this
way the following result implies Theorem 1.1.
2.5 Theorem. There exists a functor F : Aop ! T such that H* (F )
is naturally equivalent to ffR .
For the rest of the paper we will concentrate on proving 2.5.
x3. Subgroups of Spin (7)
In x2 we described a very small category A which up to equiva-
lence should be the category of conjugacy classes of elementary abelian
subgroups (1.5) of DI (4), and we reduced the problem of constructing
BDI (4) to that of constructing a suitable functor F : Aop !- T . As ex-
plained in 1.5, the spaces which the functor F will take on as its values
will be of the form B^K, where K is the centralizer in Spin (7) of an ele-
mentary abelian 2-group containing the center of Spin (7). In this section
we take the first steps toward constructing F (or its homotopy version F )
by identifying (3.2) within Spin (7) the appropriate elementary abelian
2-subgroups Ei, (i = 1; : : :; 4) and collecting needed information about
(1) the centralizers C(Ei) of the groups Ei (3.3),
(2) the normalizers N (C(Ei)) of the groups C(Ei) (3.5), and
(3) the cohomology rings H* BC(Ei) (3.8).
The normalizers (2) determine certain key self-equivalences of the spaces
B^C(Ei) (the "internal Spin actions" of x6). In order for us to prove that
the functor F constructed in x6 has the correct cohomological behavior,
it is necessary to understand the cohomology rings (3) in a functorial,
geometric way (3.13).
Let T be the maximal torus in SO (7) given by block matrices
0 M1 0 0 0 1
B@ 0 M2 0 0 C
0 0 M3 0 A
0 0 0 1
New Finite Loop Space 11
where each Mi is of the form cossiin - sini . The Lie algebra of
i cos i
T is a subalgebra of the Lie algebra so(7) of 7 x 7 skew-symmetric real
matrices. Let L denote the kernel of the exponential map on the Lie
algebra of T ; L is a free Z module of rank 3 with basis `i (i = 1; : : :; 3),
where 0 1
M1i 0 0 0
i
`i = B@ 0 M2 0 0 CA
0 0 M3i 0
0 0 0 0
with Mji= 20ss -2ss0 if i = j and Mji= 0 otherwise. Let T be the
torus in Spin (7) which is the inverse image of T and L the kernel of
the exponential map on the Lie algebra of T . The natural map L !
L exhibits L as the index two subgroup of L consisting of elements
n1`1 + n2`2 + n3`3 such that n1 + n2 + n3 is even; this is spanned by the
set {`1; `2; `3} where `1 = 2`1, `2 = `1 - `2, and `3 = `1 + `3. The group
L is naturally isomorphic to the fundamental group ss1T .
Recall that the map B Spin (7) ! B SO (7) is up to homotopy a
principal fibration with fibre K(Z=2 ; 1) classified by the second Stiefel-
Whitney class w2 2 H2 B SO (7). We will implicitly use the following
lemma several times.
3.1 Lemma. Let V be an elementary abelian 2-subgroup of SO (7), V
its inverse image in Spin (7), and w2(V ) 2 H2 BV the restriction of w2.
Then V is an abelian group iff w2(V ) is the square of a one-dimensional
class and an elementary abelian group iff w2(V ) = 0.
Remark: Often the best way to compute the class w2(V ) of 3.1 is to
express the representation V ! SO (7) as a sum of one-dimensional
representations of V .
3.2 Elementary abelian subgroups of Spin (7). We will let E be
the rank three elementary abelian 2-subgroup of SO (7) generated by the
matrices diag (- - - - + + + ), diag (- - + + - - + ) and diag (- + - + - + - ),
where diag (. . .) indicates a diagonal matrix with the specified pattern
of 1 entries. By 3.1 the inverse image E4 of E in Spin (7) is a rank
4 elementary abelian 2-group, and it is clear that E4 contains a rank 3
elementary abelian subgroup E3 generated by the elements of order 2 in
the torus T . The exponential exact sequence L ! LR ! T shows that
E3 is naturally isomorphic to ( 1_2L)=L ~= L=2L; under this isomorphism
the basis {`1; `2; `3} for L provides a basis for E3. Let E2 and E1 denote
the (based) subgroups of E3 corresponding to the subgroups of L=2L
12 Dwyer and Wilkerson
generated respectively by {`1; `2} and {`1}. The group E1 is the center
of Spin (7), i.e. the kernel of the epimorphism Spin (7) ! SO (7).
3.3 Centralizers of elementary abelian subgroups. We need to
describe the centralizers of the above groups very explicitly (see 1.5).
Let Q stand for the multiplicative group of unit quaternions; this is
isomorphic to the special unitary group SU (2). The center of Q is {1},
and we will use Qnredto stand for the quotient of Qn by the diagonal
subgroup generated by (-1; : : :; -1). Let Tenxtdenote the semidirect
product of the n-torus (R=Z)n with {1}, where the generator of {1}
acts on (R=Z)n by inversion. Recall (1.9) that C(Ei) (resp. N (Ei)) is
the centralizer (resp. normalizer) in Spin (7) of Ei.
3.4 Lemma. The groups C(Ei), i = 1; : : :; 4 are isomorphic respectively
to Spin (7), Q3red, Te3xt, and E4. The quotient group N (E4)=C(E4) has
order 8 . |GL (3; F2 )|.
Proof: We will use the fact that if K Spin (7) is the inverse image
of the subgroup K SO (7), then the normalizer of K in Spin (7) is
the inverse image of the normalizer of K . Case i = 1 of the lemma is
clear. For i = 2, note that the image of E2 in SO (7) is the group E2 of
order 2 generated by diag (- - - - + + + ). The normalizer N (E2 ) of E2 in
SO (7) is a two-component group with identity component SO (4)xSO (3);
one element in the non-identity component is diag (- - - + + + - ). By
3.1, C(E2) is the inverse image in Spin (7) of the identity component of
N (E2 ), and it is easy to check that this inverse image, as a two-fold
covering group of SO (4) x SO (3) ~= Q2redx Q1red, has the indicated form.
For i = 3 observe that the image E3 of E3 in SO (7) is the group gen-
erated by E2 and diag (- - + + - - + ); the representation E3 ! SO (7)
is the sum of the trivial one-dimensional representation of E3 with two
copies of each non-trivial one-dimensional representation. The central-
izer C(E3 ) of E3 in SO (7) is a group isomorphic to O (2)3 generated by
the torus T and the rank 6 elementary abelian group D of diagonal ma-
trices. A check with 3.1 shows that C(E3) is generated by the torus T
and the inverse image in Spin (7) of diag (- + - + - + - ).
For i = 4, observe that the image E4 of E4 in SO (7) is generated
by E3 and diag (- + - + - + - ); the representation E4 ! SO (7) is the
sum of all seven non-trivial one-dimensional representations of E4 . The
centralizer C(E4 ) is generated by E4 and the group D above; the quotient
N (E4 )=C(E4 ) is isomorphic in its conjugation action on E4 to Aut (E4 ).
It is now easy to see that E4 is its own centralizer in Spin (7), since it is
clear from the above that E4 is its own centralizer in C(E3). The desired
result follows.
New Finite Loop Space 13
3.5 Normalizers of centralizers. We now study the normalizers in
Spin (7) of each one of above centralizers. This is important to us because
(3.6) the normalizers determine which self homotopy equivalences of
the classifying spaces of the centralizers are realized by conjugations in
Spin (7). In the end (see 1.7) we will have to augment these particular self
homotopy equivalences by others in order to construct the functor F , but
it is necessary (x6) in verifying the commutative diagrams that constitute
F to recognize that certain equivalences are internal to Spin (7). We
approach the problem of naming the equivalences (3.7) by labeling them
with integral matrices (see 5.5 and the proof of 6.1 for an indication of
how effective this is).
Note that for i = 1; : : :; 4 the group Ei is the center Z(C(Ei)), so that
the normalizer N (Ei) of Ei in Spin (7) is also the normalizer N (C(Ei))
of C(Ei) in Spin (7). For i = 1; 2; 3 let Wi be the Weyl group of C(Ei)
(i.e., the quotient by T of the normalizer of T in C(Ei)), and let Wi+ be
the quotient by T of the normalizer of T in N (Ei). The groups Wi and
Wi+ are all in a natural way subgroups of the Weyl group W1 of Spin (7);
in fact there are inclusions W3 W2 W2+ W3+ = W1 = W1+.
3.6 Remark: Since inner automorphisms of a group G act trivially up
to homotopy on BG, there are natural homotopy actions of Wi+=Wi on
BC(Ei) , i = 2; 3.
Given the chosen basis {`1; `2; `3} for L, the conjugation action of
W1 on T provides a homomorphism aeW : W1 ! GL (3; Z). Let :
GL (3; Z) ! GL (3; F2 )stand for reduction mod 2 and det : GL (3; Z) !
{1} for the determinant map. Recall from x 2 the definition of the
subgroup P (1; 2)of GL (3; F2 ). Denote by U (2; 1) the subgroup of P (2; 1)
consisting of matrices which agree with the identity except in the last
column and by U GL (3; F2 )the group of upper-triangular matrices.
3.7 Lemma. The product map OE = ( . aeW ; det .aeW ) induces isomor-
phism W1 ~= P (1; 2)x {1}, W2+ ~= U x {1}, W2 ~= U (2; 1) x {1}, and
W3 ~= {1}. Under these isomorphisms the map aeW sends -1 2 {1}
to the negative of the identity matrix in GL (3; Z).
Proof: The group 3 o {1} acts on L by the formula
oe . [ffl1; ffl2;.ffl3](a1`1 + a2`2 + a3`3 ) = ffl1a1`oe(1)+ ffl2a2`oe(2)+ ff*
*l3a3`oe(3)
= ffloe-1 (1)aoe-1 (1)`1+ ffloe-1 (2)aoe-1 (2)`2+ ffloe-1 (3)ao*
*e-1 (3)`3
where oe 2 3 , ffli = 1 and ai 2 Z. Under the isomorphism ss1T ~= L
this is the conjugation action of the Weyl group of SO (7) on L . The
action restricts to an action of W1 ~= 3 o {1} on the sublattice L of L ,
14 Dwyer and Wilkerson
and the stated formula for OE(W1) can be derived by a straightforward
calculation. The rest of the lemma follows from the way in which the
groups involved are defined. For example, W1 acts on E3 by conjugation
and it is clear that W2 is the subgroup of W1 consisting of elements
which pointwise fix the first two basis elements of E3. Since W1 acts on
E3 ~= L=2L as P (1; 2), the desired identification W2 follows immediately.
3.8 Cohomology of centralizers. To conclude this section we will
give a description (3.13) of the cohomology rings of the spaces BC(Ei)
which is functorial enough for the purposes of x6. All of these cohomol-
ogy rings are rings of invariants (3.11).
3.9 Lemma. (Lannes) Let G be a compact Lie group, E G an ele-
mentary abelian 2-subgroup and W 0the quotient N (E)=C(E).0Suppose
that the natural restriction map H* BG ! (H* BE)W is an isomor-
phism. Let V be an elementary abelian 2-group. Then in the natural
commutative square
Hom (V; E)=W 0 --- - ! Hom (V; G)=G
?? ?
y ?y
(ss0Map (BV; BE))=W 0 --- - ! ss0Map (BV; BG)
all of the arrows are bijections. Moreover if h : BV ! BG is a
homotopy class of maps, k : BV ! BE is a homotopy class cov-
ering h, and W 0(k) W 0 is the subgroup of elements w such that
w . k = k, then the0 evident restriction map H* Map (BV; BG)h !
(H* Map (BV; BE)k )W (k) is an isomorphism.
Remark: Here Hom (V; E)=W 0denotes the collection of orbits of the
action of W 0on the set Hom (V; E), and Hom (V; G)=G is the set of conju-
gacy classes of homomorphisms V ! G. Note that for any k : BV ! BE
evaluation at the basepoint gives an equivalence Map (BV; BE)k ! BE
which is equivariant with respect to0W 0(k); this equivalence0induces the
isomorphism (H* Map (BV; BE)k )W (k) ~=(H* BE)W (k) .
Proof of 3.9: This follows from 1.6 and the fact [20] [14] that for any
compact Lie group G and elementary abelian 2-group V the map
T VH* (BG)!- H* Map (BV; BE)
is an isomorphism.
For i = 1; : : :; 4 let Wi0denote the quotient group Ni(E4)=E4, where
Ni(E4) is the normalizer of E4 in C(Ei). Choose a basis for E4 which
extends the basis given above for E3, so that the conjugation action of
Wi0on E4 gives a homomorphism Wi0! GL (4; F2 ). Let P0(i; j) denote
the kernel of the map ri : P (i; j)! GL (i; F2 ).
New Finite Loop Space 15
3.10 Lemma. The conjugation action of Wi0on E4 (i = 1; : : :; 4) induces
an isomorphism Wi0~= P0(i; 4 - i).
Proof: The map Wi0! GL (4; F2 )is clearly a monomorphism; it has
image contained in P0(i; 4 - i) because the first i basis vectors of E4
belong to the center of C(Ei). Case 1 is proved by observing (3.4) that
W10and P0(1; 3) have the same order. The rest follows from the fact
that essentially by definition Wi0(2 i 4) is the group of elements in
W10which fix the first i basis vectors of E4.
3.11 Lemma. For i = 1; : : :; 4 the restriction0map H* BC(Ei) ! H* BE4
induces an isomorphism H* BC(Ei) ~= (H* BE4)Wi .
Proof: We will first treat the case i = 1. According to [2] or [28] the
algebra H* B Spin (7) is a polynomial algebra on classes x4, x6, x7, x8.
The restriction map H* B Spin (7) ! H* BE4 makes H* BE4 a finitely
generated module over H* B Spin (7) [29, 2.4] and, since H* BE4 is itself a
polynomial algebra on four (one-dimensional) generators, it follows from
a transcendence degree argument [1, 11.21] that this restriction map
must be injective. The ring H* BE4 is integral [1, 5.1] over H* B Spin (7)
(this is an easy consequence of the fact that H* B Spin (7) is Noetherian
and H* BE4 is finitely generated as a module over H* B Spin (7)). Since
the product of the degrees of the polynomial generators of H* B Spin (7)
equals the order of W10, the desired isomorphism follows from [32, 3.2].
Cases i = 2; 3; 4 now follow from 3.9, since BC(Ei) is naturally equiv-
alent [14] at the prime 2 to Map (BEi; B Spin (7))f , where the map
f : BEi ! B Spin (7) is induced by the inclusion Ei ! Spin (7).
Let ffii : BEi ! BC(Ei) be induced by the inclusion Ei ! C(Ei).
3.12 Lemma. Let V be a rank 4 elementary abelian 2-group and let
h : Ei ! V be a monomorphism (i 2 {1; : : :; 4}). Then up to homo-
topy there is a unique map k 2 Emb (BV; BC(Ei) ) such that k . Bh is
homotopic to ffii.
Proof: This is an immediate consequence of 3.11, 3.9 and 3.10, since
up to the action of Wi0there is only one monomorphism V ! E4 which
when composed with h gives the inclusion Ei ! E4. Note that by 1.6(2)
the bijection (3.9) Hom (V; E4)=Wi0~= ss0Map (BV; BC(Ei)) restricts to
a bijection Emb (V; E4)=Wi0~= ss0Emb (BV; BC(Ei)).
Now let V be a rank 4 elementary abelian 2-group and choose i 2
{1; : : :; 4}. Lemma 3.12 shows that up to homotopy there is a unique
map
i;V
Emb (BEi; BV ) ~= Emb (Ei; V ) x BV --- ! BC(Ei)
16 Dwyer and Wilkerson
with the property that for each f 2 Emb (BEi; BV ) the diagram
ffii
BEi ~= Aut (Ei)1 --- - ! BC(Ei)
? ?
f.(-)?y ?y=
i;V
Emb (BEi; BV ) --- - ! BC(Ei)
homotopy commutes and such that *i;V makes H* Emb (BEi; BV ) a
finitely generated module over H* BC(Ei). (The equivalence Aut (Ei)1 ~=
BEi in the above diagram is obtained by evaluation at the basepoint
BEi, as is each equivalence Emb (BEi; BV ) ~= BV .) By uniqueness the
map i;V is equivariant with respect to the evident action of Aut (V ) on
its domain and the trivial action of Aut (V ) on its range. It follows that
the cohomology map *i;Vinduces a map from H* BC(Ei) to the fixed
point set of the action of Aut (V ) on H* Emb (BEi; BV ).
3.13 Proposition. Let V be a rank 4 elementary abelian 2-group.
Then for each i 2 {1; : : :; 4} the map
H* BC(Ei) ! (H* Emb (BEi; BV ))Aut (V )
induced by *i;Vis an isomorphism.
Proof: Let h : Ei ! V be a monomorphism and k : BV ! BC(Ei)
the extension of 3.12. Let W 0 Aut (V ) be the subgroup consisting of
elements w such that k . Bw is homotopic to k. By Shapiro's lemma
the desired0statement amounts to a claim that the map H* BC(Ei) !
(H* BV )W is an isomorphism. This follows from 3.10 and 3.11.
x4. A 2-adic representation
In this section we will show that the conjugation action of the Weyl
group of Spin (7) on the fundamental group of the torus actually extends
(after tensoring with ^Z2) to an action of a larger group. This will allow
us in x6 to construct an action of GL (3; F2 )on the space B^C(E3) (see
3.4).
Under the isomorphism W1 ~= P (1; 2)x {1} provided by 3.7, the
homomorphism aeW gives a map aeP : P (1; 2)! GL (3; Z). Let ^aePdenote
the composite of aeP with the natural inclusion GL (3; Z) ! GL (3; ^Z2)
and ^ : GL (3; ^Z2)! GL (3; F2 )the map given by reduction mod 2.
4.1 Theorem. There is a homomorphism ^aeGL: GL (3; F2 )! GL (3; ^Z2)
such that
(1) the composite ^ . ^aeGLis the identity map of GL (3; F2 ), and
(2) the restriction of ^aeGLto P (1; 2) is ^aeP.
New Finite Loop Space 17
4.2 Remark: By construction the composition ^ . ^aePgives the identity
map of P (1; 2). Theorem 4.1 leads immediately to a homomorphism
GL (3; F2 )x {1} ! GL (3; ^Z2)which in an evident sense (3.7) extends
aeW .
Let sl(3; F2 ) denote the additive group of trace zero 3 x 3 matrices
over F2 . The group GL (3; F2 )acts on sl(3; F2 ) by conjugation.
4.3 Lemma. For any subgroup K of GL (3; F2 )and i 1 the cohomol-
ogy group Hi(K; sl(3; F2 )) vanishes.
Proof: Let U GL (3; F2 )be the group of upper-triangular matrices;
the group U has order 8 and is a 2-Sylow subgroup of GL (3; F2 ). A
calculation shows that the vector space H0(U; sl(3; F2 )) is of dimension
one. Let x 2 sl(3; F2 ) be an element which projects to a generator of
H0(U; sl(3; F2 )) and let f : F2 [U ] ! sl(3; F2 ) be the F2 [U ] module map
which sends 1 to x. Since the augmentation ideal of F2 [U ] is nilpotent
[4, VI, 8.3] Nakayama's lemma [4, VI, 8.4] shows that the map f is
surjective. By a dimension count, then, f is an isomorphism. Now let
K GL (3; F2 ) be any subgroup and K2 K its 2-Sylow subgroup;
note that K2 is conjugate in GL (3; F2 )to a subgroup of U . It is clear
that sl(3; F2 ) is free as a module over F2 [K2] and hence [4, VI, 8.5] that
the group Hi(K2; sl(3; F2 )) vanishes for i 1. The lemma follows from
a transfer argument [4, III, 10.3].
4.4 Lemma. Suppose that : K ! GL (3; F2 )= SL (3; F2 )is a subgroup
inclusion. Then
(1) there exists a group homomorphism ae : K ! SL (3; ^Z2)such that
^ . ae = , and
(2) if ae0 is another group homomorphism K ! SL (3; ^Z2)with ^ . ae =
, then there exists an element C 2 SL (3; ^Z2)such that ^ (C) is
the identity matrix and CaeC-1 = ae0.
Proof: Let G denote SL (3; ^Z2) and let Gn G (n 1) be the nor-
mal subgroup consisting of matrices which are congruent to the identity
matrix modulo 2n . It is easy to see that for each n 1 the quotient
group Gn =Gn+1 is abelian and is isomorphic as a module over G=Gn to
sl(3; F2 ), where the action of G=Gn on sl(3; F2 ) is obtained by compos-
ing the quotient map G=Gn ! GL (3; F2 )with the conjugation action
that appears in 4.3.
Suppose that has been lifted to a homomorphism aen : K ! G=Gn
for some n, and let q : K0 ! K be the pullback over aen of the surjection
G=Gn+1 ! G=Gn . The kernel of q is an abelian group which is iso-
morphic as a K module to sl(3; F2 ). By 4.3 the group H2 (K; sl(3; F2 ))
18 Dwyer and Wilkerson
vanishes and so [4, IV, 3.12] the epimorphism q can be split; moreover,
since H1 (K; sl(3; F2 )) also vanishes, any two such splittings are conju-
gate by an element in the kernel of q [4, IV, 2.3]. It follows that the map
aen lifts to a homomorphism aen+1 : K ! G=Gn+1 and that, if ae0n+1 is
another such lift, the maps aen+1 and ae0n+1 are conjugate by an element
in the kernel of G=Gn+1 ! G=Gn .
By induction, then, the map lifts for any n to a homomorphism
aen : K ! G=Gn , and any two such lifts are conjugate via an element
Cn 2 G=Gn which projects to the identity element of GL (3; F2 ). Let
L(n) be the set {aen } of all such lifts. There are natural surjections
L(n) ! L(n - 1), and L(1) = lim-L(n) is the set of all homomorphisms
ae : K ! G such that ^ . ae = . It is clear that L(1) is nonempty. This
proves (1).
Now suppose that ae and ae0 are elements of L(1). Let aen and ae0n
denote the images of these homomorphisms in L(n), and let C(n) denote
the set of all elements of G=Gn which conjugate aen to ae0n and project
trivially to GL (3; F2 ). There are natural maps C(n) ! C(n - 1), and
C(1) = lim-C(n) is the set of all elements of G which conjugate ae to ae0
and project trivially to GL (3; F2 ). Part (2) of the lemma follows from
the fact that C(1) is nonempty, since each C(n) is finite and nonempty.
Proof of 4.1: By part (1) of 4.4 applied to K = GL (3; F2 ), there
exists a homomorphism ^aeGL: GL (3; F2 )! GL (3; ^Z2)which satisfies (1)
of 4.1. By part (2) of 4.4 applied to K = P (1; 2)(see 4.2), the restriction
of ^aeGLto P (1; 2)is conjugate to ^aePvia an element C 2 GL (3; ^Z2)which
projects trivially to GL (3; F2 ). The proof is completed by replacing ^aeGL
with its conjugate by C.
x5. Self-maps of completed classifying spaces
In this section we will calculate (5.5) the space of self homotopy equiv-
alences of B^Qnred(see 3.4). This will allow us in x6 to construct an action
of GL (2; F2 )on the space B^C(E2) ~= ^BQ3red. There is an evident permu-
tation action of GL (2; F2 )~= 3 on B^Q3red, but the action which comes
up in x6 is another one.
Some of what we do is implicit in [14], [10] [27], although the partic-
ular results we need don't appear in these references. For the rest of the
section let G denote a compact Lie group with chosen maximal torus T
and Weyl group W .
The set Emb (T; G) (see 1.9) has a distinguished basepoint given by
the inclusion of T in G. Since T is unique up to conjugacy the discrete
group Aut (T ) acts transitively by composition on Emb (T; G); it is clear
that the isotropy group of the distinguished basepoint is the image of
New Finite Loop Space 19
the map W ! Aut (T ) given by the conjugation action of W on T .
We will need a homotopy theoretic version of this fact. Note that the
set ss0Emb (B^T; ^BG) has a distinguished basepoint given by the map
B^T ! B^G induced by the inclusion T ! G, and that the discrete group
ss0Aut (B^T )acts on ss0Emb (B^T; ^BG) by composition.
5.1 Proposition. If G is a connected compact Lie group then the
natural action of ss0Aut (B^T ) on ss0Emb (B^T; ^BG) is transitive; *
*the
isotropy group of the distinguished basepoint is the image of the map
W ! ss0Aut (B^T )given by the conjugation action of W on T .
5.2 Remark: This will be proven below. Note that choice of a ba-
sis for ss1T gives isomorphisms Aut (T ) ~= GL (r; Z) and ss0Aut (B^T )~=
GL (r; ^Z2), where r is the rank of G.
5.3 Lemma. Suppose that G is connected and that ss is a finite two-
group. Then
(1) the natural map Map (Bss; BG) ! Map (Bss; ^BG) induces an iso-
morphism on mod 2 cohomology, and
(2) the space Map (Bss; ^BG) is F2 -complete in the sense of [3].
Remark: Lemma 5.3 implies that Map (Bss; ^BG) is homotopy equiva-
lent to the F2 -completion of Map (Bss; BG) .
Proof of 5.3: Part (1) is proved by the argument of [6, 4.5]; the main
step is to note that the homotopy fibre of the map BG ! B^G is a simple
space with uniquely 2-divisible homotopy groups. Part (2) now follows
from [3, Ch. VII, 5.1] and the fact [14, 1.1] that each component of
Map (Bss; BG) has a finite fundamental group.
Proof of 5.1:n Let Tn be the subgroup of T consisting of elements x
such that x2 = 1, and let T1 denote the discrete group [n Tn . The
composite map BT1 ! BT ! B^T is a cohomology isomorphism and
so induces an isomorphism ss0Emb (B^T; ^BG) ~= ss0Emb (BT1 ; ^BG). It is
easy to see by explicit calculation that the functorial map Aut (T1 ) !
ss0Aut (B^T1 ) ~= ss0Aut (B^T )is an isomorphism. To prove 5.1, then, it
is enough to show that Aut (T1 ) acts transitively on ss0Emb (BT1 ; ^BG)
and that the isotropy group of the evident distinguished basepoint is the
image of the map W ! Aut (T1 ) given by the conjugation action of W
on T1 .
Since BT1 is equivalent to the homotopy direct limit ho lim-!BTn [3,
XII, 3.6], the mapping space Emb (BT1 ; ^BG) is equivalent to the ho-
motopy inverse limit ho lim-Emb (BTn ; ^BG) (see [3, XII, 4.1]). Choose
20 Dwyer and Wilkerson
n0 >> 0 so that for n > n0
(1) the map Emb (Tn ; T ) ! Emb (Tn ; G) is surjective, and
(2) the centralizer of Tn in G is exactly T .
(It is easy to obtain (1) for large n by observing that any f : Tn ! G has
image contained up to conjugacy in the normalizer N (T ) of T [30] and
then using the fact that W = T =N (T ) is a finite group. For (2), let Cn
be the centralizer in G of Tn , observe that T is self-centralizing and T1 is
dense in T so that \n Cn = T , and use a dimension argument to conclude
that Cn = T for sufficiently large n). By [14, 1.1] and 5.3 the natural
map Emb (Tn ; G) ! ss0Emb (BTn ; ^BG) is an isomorphism for any n and
for n > n0 each component of Emb (BTn ; ^BG) is naturally homotopy
equivalent to B^ T . It follows [3, IX, 3.5] that lim-1 ss1Emb (BTn ; ^BG)
vanishes for every choice of a compatible sequence of basepoints and
[3, XI, 4.1; IX 3.1] that there is an isomorphism ss0Emb (BT1 ; ^BG) ~=
lim-Emb (Tn ; G). The desired result can now be proven by combining
the isomorphism Aut (T1 ) ~= lim-Aut (Tn ) with the fact that for n > n0
the action of Aut (Tn ) on Emb (Tn ; G) is clearly transitive and has the
conjugation image of W in Aut (Tn ) as the isotropy group of the natural
basepoint.
Now suppose that G is connected. Since T is self-centralizing the
conjugation action of W on B^T is faithful, and we will use this action
to identify W with a subgroup of ss0Aut (B^T ). Let = G denote the
normalizer of W in ss0Aut (B^T )and = G the quotient of by W . By
5.1, is isomorphic to the group of automorphisms of ss0Emb (B^T; ^BG)
as a set with an action of ss0Aut (B^T ). Under this isomorphism, the
left composition action of ss0Aut (B^G) on ss0Emb (B^T; ^BG) gives a map
O = OG : ss0Aut (B^G) ! .
Recall that Z(G) denotes the center of the compact Lie group G.
As in [14, x1] the group homomorphism Z(G) x G ! G gives a map
BZ(G) ! Aut (B^G)1.
5.4 Lemma. The space Map (B^Q; ^BQ) is the union of Aut (B^Q) with
the component Map (B^Q; ^BQ)0 of null homotopic maps. Moreover
(1) the map OQ : ss0Aut (B^Q) ! Q is an isomorphism,
(2) the natural map BZ(Q) ! Aut (B^Q)1 is an equivalence, and
(3) the basepoint evaluation map Map (B^Q; ^BQ)0 ! B^Q is an equiv-
alence.
Remark: The group Z(Q) is {1} and the group Q is canonically
isomorphic to ^Zx2={1}.
New Finite Loop Space 21
Proof: Let O"48 and Q16 be the subgroups of Q constructed in [8, x4]
and let N2T Q be the subgroup of the normalizer of the torus in Q
consisting of elements with order a power of 2. It follows from [8, 4.1]
that there is a diagram BO"48 BQ16 ! BN2T induced by subgroup
inclusions and a map from the homotopy pushout P of this diagram to
B^Q which induces an isomorphism on cohomology. As a consequence
the space Map (B^Q; ^BQ) ~= Map (P; ^BQ) is equivalent to the homotopy
pullback of the induced diagram
Map (BO"48 ; ^BQ) ! Map (BQ16; ^BQ) Map (BN2T; ^BQ) :
By [14, 1.1], 5.3, and a limiting argument based on the fact that N2T
is a union of finite two-groups (cf. proof of 5.1), it is possible to identify
Map (BN2T; ^BQ) up to homotopy as a disjoint union involving one copy
of B^Q corresponding to the component of null homotopic maps together
with multiple copies of B{1} (one for each element Q ). Similarly,
the subspace of Map (BQ16; ^BQ) consisting of maps which extend up
to homotopy to BN2T is up to homotopy the disjoint union of B^ Q
with two copies of B{1}. Finally, both Q16 and O"48 contain a normal
subgroup Q8 of order 8 (essentially the classical quaternion group); the
subspace Me of M = Map (BQ8; ^BQ) consisting of maps which extend
up to homotopy to BN2T is up to homotopy the disjoint union of B^Q
with one copy of B{1}. As in [14, 5.1] there are equivalences
Map (BO"48 ; ^BQ) ~= M hfl
0
Map (BQ16; ^BQ) ~= M hfl
where we have used the homotopy fixed point set notation (- )h- of [14],
fl = O"48=Q8 and fl0 = Q16=Q8. Now fl is isomorphic to the symmetric
group 3 and so has a normal subgroup fl00of order 3 with fl=fl00~= fl0. It
is easy to see that fl00 acts trivially0up0to homotopy on Me and thus by
the argument of [14, 2.3] that Mehfl ~=0Me.0By0the argument of [14, 5.1]0
there is an equivalence Mehfl~=(Mehfl)hfl. It follows that Mehfl~=Mehfl,
which implies that the above homotopy pullback collapses to an equiv-
alence Map (B^Q; ^BQ) ~= Map (BN2T; ^BQ). The desired result follows
directly.
5.5 Proposition. Suppose that G is Qn or Qnred(n > 0). Then
(1) the map OG : ss0Aut (B^G) ! G is an isomorphism, and
(2) the natural map BZ(G) ! Aut (B^G)1 is an equivalence.
22 Dwyer and Wilkerson
Proof: It is clear that evaluation at the basepoint gives an equivalence
Map (B^Q; BZ(Q)) ! BZ(Q). It follows from an inductive argument
using 5.4 and the formula
Map (B^Qn ; ^BQ) ~= Map (B^Q; Map (B^Qn-1 ; ^BQ))
that any map B^Qn ! B^Q factors up to homotopy through one of the
n evident projections and that via these projections Map (B^Qn ; ^BQ) is
equivalent to an amalgamated union of n copies of Map (B^Q; ^BQ) (the
amalgamation amounts to identifying the subspaces of null homotopic
maps). An argument along the lines of [16, x4] now shows that the
obvious map n o Aut (B^Q) ! Aut (B^Qn ) is an equivalence. Since
Qn ~= n o Q and Z(Qn ) ~= Z(Q)n the desired result follows easily in
the case G = Qn .
For the case G = Qnred, let K denote the Eilenberg-MacLane space
K(Z=2 ; 2) and note that there is a fibration p : B^Qnred! K (with fibre
B^Qn ) which is the projection of B^Qnredto its second Postnikov stage.
Since the formation of Postnikov stages is functorial, the topological
monoid of self-equivalences of p in the sense of [7, 6.1] is equivalent to
Aut (B^Qnred) (cf. [7, 6.3]). By [7, 6.2], then, there is a natural weak
equivalence
BAut (B^Qnred) ~= EAut (K) xAut (K) Map (K; BAut (B^Qn ))p
(where the subscript p on the right denotes the component of the in-
dicated mapping space which contains a classifying map for p). Since
Aut (K) ~= K and Map (K; BAut (B^Qn ))p ~= BAut (B^Qn ) (both via
basepoint evaluation maps), it is not hard to calculate directly that the
natural map Z(Qnred) ! ss2BAut (B^Qnred) is an isomorphism. It is also
clear that every self-equivalence of B^Qnredis covered up to homotopy
by a unique self-equivalence of B^Qn . The proof is completed by check-
ing that if T" is a maximal torus of Qn which covers the maximal torus
T of Qnred, then each automorphism of T which normalizes the action
of the Weyl group of Qnredlifts to a unique automorphism of T" which
normalizes the action of the Weyl group of Qn .
x6. A construction in the homotopy category
Let Ho (T ) be the homotopy category of T . In this section we will
define a functor F : Aop ! Ho (T ) such that H* F is naturally equivalent
to the functor ffR of 2.5. We will continue to use the notation of x3.
New Finite Loop Space 23
By construction each vector space Ei, i = 1; : : :; 4 comes with a basis
and therefore an action of GL (i; F2 ); we will refer to the induced action
of GL (i; F2 )on BEi as the basis action of GL (i; F2 )on BEi. By 3.6 and
3.7 the internal structure of Spin (7) provides actions up to homotopy of
P (1; 1) ~= W2+=W2 on B^C(E2) and of P (1; 2) ~= W3+=W3 on B^C(E3);
we will refer to these as the internal Spin actions of these groups. Let
^ffii: BEi ! B^C(Ei) be the map induced by the inclusion Ei ! C(Ei).
6.1 Proposition. There are homotopy actions of GL (i; F2 )on ^BC(Ei),
i = 2; 3 with the following properties.
(1) The map ^ffii: BEi ! B^C(Ei) is equivariant with respect to the
basis action of GL (i; F2 )on BEi for i = 2; 3.
(2) The restriction to P (1; 2) of the action of GL (3; F2 )agrees with
the internal Spin action of P (1; 2) on B^C(E3).
(3) The restriction to P (1; 1) of the action of GL (2; F2 )agrees with
the internal Spin action of P (1; 1) on B^C(E2).
Restricting the action of GL (3; F2 )to P (2; 1) gives an action of P (2; 1)
on ^BC(E3) and composing the action of GL (2; F2 ) with the natural
projection P (2; 1) ! P (2; 1)=U (2; 1) ~= GL (2; F2 )gives an action of
P (2; 1) on B^C(E2).
(4) The map B^C(E3) ! B^C(E2) induced by the inclusion C(E3) !
C(E2) is equivariant with respect to these actions of P (2; 1).
Proof: Let Oi : ss0Aut (B^C(Ei)) ! i be the map OG of x5 for G =
C(Ei), i = 2; 3. The map O2 is an isomorphism by 5.5. It is easy
to see that O3 is well-defined (in spite of the fact that C(E3) is not
connected) and induces an isomorphism from ss0Aut (B^C(E3)) to the
quotient of GL (3; ^Z2)by the subgroup {I3} generated by the negative
of the identity matrix. (Here and in what follows we use the specific
basis {`1; `2; `3} of L ~= ss2BT to identify ss0Aut (B^T )with GL (3; ^Z2).)
The representation ^aeGLof 4.1 can be composed with the quotient map
GL (3; ^Z2)! GL (3; ^Z2)={I3} and the inverse of O3 to give an action
of GL (3; F2 ) on B^C(E3). Condition (1) (i = 3) follows from the iso-
morphism E3 ~= L=2L together with 4.1(1), condition (2) from 3.7 and
4.1(2).
The group U (2; 1) is a subgroup of P (1; 2). By 3.7 the image *
*of
W2 in GL (3; ^Z2) is generated by {I3} and (4.1) a^eP(U (2; 1)), where
by 4.1(2) this last group is equal to a^eGL(U (2; 1)), Since U (2; 1) is a
normal subgroup of P (2; 1), it follows that ^aeGL(P (2; 1)) is contained in
the normalizer of the image of W2 and hence that ^aeGL induces a map
P (2; 1)=U (2; 1) ~= GL (2; F2 ) ! 2 ~= ss0Aut (B^C(E2)) . Use this map to
24 Dwyer and Wilkerson
give an action of GL (2; F2 )on B^C(E2). Condition (1) (i = 2) follows
from 4.1(1) and the natural isomorphism between E2 and the subgroup
of L=2L generated by the residue classes of `1 and `2.
Restricting the above action of GL (2; F2 )on B^C(E2) to P (1; 1) and
applying O2 gives an monomorphism
P (1; 1) = U=U (2; 1) ! 2 = N (^aeGL(U (2; 1)))=^aeGL(U (2; 1))
(see 3.7) which is covered by the map U ! GL (3; ^Z2) given by the
restriction to U of ^aeGL. The internal Spin action of P (1; 1) on B^C(E2)
induces a parallel monomorphism P (1; 1) ! 2 which is covered by the
map U ! GL (3; ^Z2) given by the restriction to U of ^aeP(3.7, 4.1).
Condition (3) is thus a consequence of 4.1(2).
Finally, property (4) is clear by construction.
Lemma 2.1 describes what is needed to build F . Choose F (Ai ) =
B^C(Ei), i = 1; : : :; 4. The maps F(fi) (i = 1; 2; 3) are induced *
* by
the inclusions C(Ei+1 ) ! C(Ei). The right actions of GL (2; F2 ) and
GL (3; F2 )on F (A2 ) and F (A3 ) respectively are obtained by composing
the left actions of 6.1 with the inverse map. The right action of GL (4; F2 )
on F (A4 ) is obtained in a similar way from the natural left action of
GL (4; F2 )on the based vector space E4 = C(E4). By 6.1(3) the subgroup
P (1; 1) GL (2; F2 )acts on F (A2 ) by automorphisms of C(E2) which
extend to inner automorphisms of Spin (7); this (cf. 3.6) gives relation
2.1(1). Relation 2.1(2) is 6.1(4). By 6.1(2), the restriction of the action
of GL (3; F2 )on F (A3 ) to P (1; 2)is induced by automorphisms of C(E3)
which extend to inner automorphisms of Spin (7); this establishes relation
2.1(4). Finally, relation 2.1(6) follows from 3.10, since the action of
P (1; 3) on B^C(E4) is induced by group automorphisms which become
inner in Spin (7).
Relations 2.1(3) and 2.1(5) require a little more work. For 2.1(3), let
Tenxt(2) denote the group of elements of order a power of 2 in Tenxt(see
3.4), so that Tenxt(2) is a semidirect product of (Z21 )n with Z=2 . The
inclusion Tenxt(2) ! Tenxtinduces a cohomology isomorphism H* Tenxt!
H* Tenxt(2) and therefore an equivalence B^Tenxt(2) ~= B^Tenxt. Under this
equivalence the above action of GL (3; F2 ) on B^ C(E3) ~= ^BTe3xtis by
construction induced by an action of GL (3; F2 )on Tenxt(2) by group ho-
momorphisms. This last action is obtained by letting GL (3; F2 )act via
^aeGL on (Z21 )3 (note that Aut ((Z21 )3) ~= GL (3; ^Z2)) and then extend-
ing to Te3xt(2) by using the fact that -I3 2 GL (3; ^Z2)commutes with
^aeGL(GL (3; F2 )). In a similar way, the map F (f3 ) : B^E4 ! B^C(E3) is
induced by a group homomorphism E4 ! Te3xt(2) which sends the first
New Finite Loop Space 25
three basis vectors of E4 to elements of order 2 in the subgroup (Z21 )3
and the last basis vector to an element of order 2 which does not lie
in (Z21 )3. Relation 2.1(3) now follows from a straightforward group
theoretic calculation involving 6.1(1); note in particular that the auto-
morphisms of E4 provided by U (3; 1) become inner automorphisms in
Te3xt(2) (or in C(E3)).
For 2.1(5) observe (3.10) that the automorphisms of E4 provided by
P0(2; 2) become inner automorphisms in C(E2), so it is enough to check
2.1(5) for matrices h 2 GL (4; F2 ) which agree with the identity ex-
cept in the upper left-hand 2 x 2 block. Such a matrix h belongs to
P (3; 1), so that by 2.1(3) F (f3 )F (h) = F (r3(h)) and hence by 2.1(2)
F (f2 )F (f3 )F (h) = F (r2r3(h))F (f2 )F (f3 ). This immediately gives 2.1(5)
and completes the proof that F is a functor Aop ! Ho (T ).
Next we show that the functor H* F is naturally equivalent to ffR .
Let EA : A ! T be the functor which assigns to each object Ai the
classifying space BEi. A morphism f : Ai ! Aj is sent by EA to the map
BEi ! BEj induced by the homomorphism Ei ! Ej corresponding to
f under the unique basis-preserving isomorphisms Ai ~= Ei and Aj ~=Ej.
6.2 Lemma. The maps ^ffii: BEi ! B^C(Ei), i = 1; : : :; 4 (x3) are the
components [26, p. 215] of a dinatural transformation ^ffi: EA !..F , i.e.,
for each map f : Ai ! Aj in A there is a commutative diagram
^ffii
EA (Ai ) --- - ! F (Ai )
? x
EA (f)?y F (f)?? :
^ffij
EA (Aj ) --- - ! F (Aj )
Proof: If f is an automorphism of Ai the above diagram commutes by
6.1(1), while if f is one of the basic maps fi (2.1) it commutes by the
definition of F (fi). The lemma follows from the fact that these maps
generate A .
Now let V be a rank four elementary abelian 2-group. Let MA :
Aop ! Ho (T ) be the functor which assigns to Ai the function space
Emb (BEi; BV ) ~= Emb (Ei; V ) x BV . As in the discussion preceding
3.13 there are canonical maps i;V : MA (Ei) ! B^C(Ei).
6.3 Lemma. The maps i;V, i = 1; : : :; 4 are the components of a nat-
ural transformation V : MA ! F .
Proof: We must show that for each morphism f : Ai ! Aj in A
and each g 2 Emb (BEj; BV ) the following diagram in the homotopy
26 Dwyer and Wilkerson
category
(i;V)g0
Emb (BEi; BV )g0 --- - - ! ^BC(Ei)
x x
(-).EA f?? F(f)??
Emb (BEj; BV )g --- - ! B^ C(Ej)
(j;V)g
commutes, where g0 = g . EA (f ) and subscripting a morphism denotes
restriction to the indicated component of the domain. Each of the spaces
on the left is equivalent by evaluation at the basepoint to BV , so the
question is one of comparing two maps BV ! B^C(Ei). The upper map
(i;V)g0 is the unique homotopy class which at the same time makes
H* BV a finitely generated module over H* BC(Ei) and satisfies (i;V)g0.
g0 = ^ffii. The other map, F (f ) . (j;V)g, is the composite with Ff of the
unique homotopy class which makes H* BV a finitely generated module
over H* ^BC(Ej) and satisfies (j;V)g . g = ^ffij. The two maps agree
because F (f ) . ^ffij. EA (f ) = ^ffii(see 6.2) and H* F(f ) makes H* ^BC(Ei)
a finitely generated module over H* ^BC(Ej). (This last follows from the
fact that F (f ) differs only by homotopy equivalences in the domain and
range from a map induced by an inclusion K1 ! K2 of compact Lie
groups: any such map makes H* ^BK1 a finitely generated module over
H* ^BK2 [29, 2.4].)
In the same way as in 3.13, the transformation V is equivariant with
respect to the natural action of Aut (V ) on MA and the trivial action
of Aut (V ) on F ; it follows from 3.13 that the induced cohomology map
*V : H* F ! (H* MA )Aut (V )is an isomorphism. The equivalence H* F ~=
ffR follows from the fact that (H* MA )Aut (V )is clearly up to equivalence
the functor ffR of 2.3.
x 7. Lifting to the category of spaces
In this section we will use the results of [6] to show that the functor
F : Aop ! Ho (T ) can be lifted to a functor F : Aop ! T . This proves
2.5 and hence 1.1.
Recall that a homotopy class of maps f : X ! Y is said to be cen-
tric [6, x 1] if precomposition with f induces a homotopy equivalence
Aut (X)1 ! Map (X; Y )f . A functor : C ! Ho (T ) is centric if (f )
is a centric map for each morphism f of C.
7.1 Proposition. The functor F : Aop ! Ho (T ) is centric.
This is [6, 4.2]. (Note that the partial completion functor which ap-
pears in the statement of [6, 4.2] agrees with the F2 -completion functor
New Finite Loop Space 27
in the cases at hand because the fundamental groups of the spaces in-
volved are finite 2-groups [3, II, 5.2(iv)].)
7.2 Lemma. The natural map (cf. [6, 4.5])
BEi ~= BZ(C(Ei)) ! Aut (B^C(Ei))1
is an equivalence for i = 2; : : :; 4.
Proof: The case i = 2 is 5.5. Cases i = 3 and i = 4 can be handled by
explicit calculation (cf. [6, 5.3]).
Let fljF (j 1) be the functor [6, x1] from A to the category of abelian
groups with which assigns to A the group ssjAut (F (A ))1; there is no
basepoint problem with these homotopy groups because Aut (F (A ))1 is
a simple space. For each map f : A ! A 0 in A the induced map
fljF (f ) : fljF (A ) ! fljF (A 0) is the composite
# )-1
ssjAut (F (A ))1 -h#! ssjMap (F (A 0); F(A ))h (h-! ssjAut (F (A 0))1
where h is F (f ), h# is induced by postcomposition with h and h# by
precomposition.
Recall from x6 that EA is a functor which sends the object Ai of A
to BEi. The natural action (7.2) of BEi on B^C(Ei) gives rise to maps
ssjBEi ! ssjAut (B^C(Ei))1.
7.3 Lemma. The above maps ssjBEi ! ssjAut (B^C(Ei))1 are the com-
ponents of a natural transformation ssjEA ! fljF .
Proof: An examination of the definition of flk F shows that by 7.2 it is
enough to check that for each morphism f : Aff ! Afi of A the following
diagram in Ho (T ) commutes:
1xF (f)
BEffx ^BC(Efi) --- - ! BEffx ^BC(Eff)
? ?
EA (f)x1 ?y m ?y :
F (f).m
BEfix ^BC(Efi) --- - ! B^ C(Eff)
Here m stands for the map BEi x B^C(Ei) ! B^C(Ei) induced by the
group homomorphism Ei x C(Ei) ! C(Ei). Restricting to the axes
BEffx pt and pt x ^BC(Efi) respectively gives the diagrams
1 F(f)
BEff --- - ! BEff B^C(Efi) --- - ! B^C(Eff)
? ? ? ?
EA (f)?y ?y^ffiff 1?y 1?y
F(f).^ffifi F(f)
BEfi --- - - ! B^C(Eff) B^C(Efi) --- - ! B^C(Eff)
28 Dwyer and Wilkerson
the first of which commutes by 6.2 and the second of which trivially
commutes. The lemma follows from the fact that the basepoint evalua-
tion map Map (BEff; ^BC(Eff))^ffiff! B^C(Eff) is an equivalence ([14] and
5.3), so that a map BEffx ^BC(Efi) ! B^C(Eff) which agrees with ^ffiffon
BEffx pt is actually determined up to homotopy by its restriction to
pt x ^BC(Efi).
7.4 Lemma. The abelian groups lim-kssjEA vanish for j; k > 0.
Proof: This is trivial if j 6= 1, since in that case the functor ssjEA
itself vanishes. Let = Aut (A4 ) and let M be the module A4 itself.
Inspection then shows that the category A is equivalent to the category
A of x8 in such a way that the functor ss1EA corresponds to the functor
ff;M of 8.1. Case j = 1 of the lemma thus follows from 8.1.
7.5 Lemma. If fl is a functor from A to the category of abelian groups
such that fl(Ai ) = {0} for i 6= 1, then lim-kfl vanishes for k > 0.
Proof: This is an easy calculation with the standard cochain complex
[3, XI, 6.2] for computing lim-*fl; in dimension k this cochain complex
contains a product Y
fl(ak )
a0-! a1-! ...-!ak
indexed by k-simplices of the nerve of A . Under the given circumstances
this is the complex for computing the singular cohomology of a point
with coefficients in fl(A0 ).
7.6 Lemma. The abelian groups lim-kfljF vanish for j; k > 0.
Proof: By 7.2 the natural transformation of 7.3 is an isomorphism at
the objects Ai , i 6= 1, of A . It is easy to check that the natural transfor-
mation is a monomorphism at A1 ; for j > 1 this is the case because the
domain group is zero, while for j = 1 it follows from naturality and the
fact that for any morphism A1 ! A2 of A the induced map E1 ! E2
is injective. If fl is the cokernel of this natural transformation, then by
7.5 the groups lim-kfl vanish for k > 0. The lemma now follows from 7.4
and the long exact lim-* sequence [3, XI, 6.1] associated to a short exact
sequence of functors.
Remark: Jackowski, McClure and Oliver have recently shown that 7.2
also holds for i = 1; given this result, which is difficult, 7.6 follows
directly from 7.4.
New Finite Loop Space 29
7.7 Proposition. The functor F : Aop ! Ho (T ) can be lifted to a
functor F : Aop ! T .
Proof: Given 7.6, this is a consequence of the main theorem of [6]. In
fact, the results of [6] show that up to a strong notion of equivalence
there is only one lift F of F .
Remark: Given a centric diagram F in the homotopy category such
that the realization obstruction groups [6] for F vanish (7.6) and such
that H* F is equivalent to ffR (x2), Oliver has shown how to construct
BDI (4) directly; this provides a proof of 1.1 which does not use 7.7.
x8. Derived functors of the inverse limit functor
In this section we will prove a certain vanishing theorem for the higher
derived functors lim-i [3, XI, x6] of the inverse limit functor. This van-
ishing theorem figures in the proof of 2.4 and in the proof of 7.4. We
work in greater generality than we strictly need in order to emphasize
the analogy with [13].
Let p be a prime number (not necessarily 2), V a finite-dimensional
vector space over Fp , Aut (V ) a subgroup and M a module over
Fp []. For any Fp vector space A the group acts by composition on
Hom (A; V ). Define A to be the category of pairs (A; O) where A is a
non-zero Fp vector space and O is a orbit of monomorphisms A ! V ;
a map (A; O) ! (A0; O0) is a monomorphism f : A ! A0 such that
g O f 2 O whenever g 2 O0.
Let ff;M be the functor from A to abelian groups which assigns to
(A; O) the vector space of -maps O ! M . For any object x of A
there is an evident map M ! ff;M (x); taken together these induce a
map M ! lim-ff;M .
8.1 Proposition. The above map M ! lim-ff;M is an isomorphism.
The groups lim-iff;M vanish for i 1.
Remark: Let ffk;M be the functor from A to abelian groups which
assigns to (A; O) the group Ext kFp[](Fp [O]; M ). (Note that ff0;M =
ff;M and that ffk;M (A; O) is isomorphic to Hk (f ; M ), where f
is the isotropy subgroup of any f 2 O.) The proof of 8.1 shows that
lim-ffk;M ~= Hk (; M ) and that lim-iffk;M = 0 for i 1.
8.2 Lemma. The conclusion of 8.1 is true under the additional assump-
tion that is a p-group.
Proof: (cf. [13, proof of 4.10]) Since is a finite p-group, V 6= {0}.
Denote by F the object of A corresponding to the inclusion : V ! V .
30 Dwyer and Wilkerson
Let F # A be the under category [26, p. 46] and j : F # A ! A the
forgetful functor. The category F # A has the identity map of F as
an initial object and it is clear that the natural map M ! ff;M (F ) =
1
ff;M (j(F !- F )) is an isomorphism; it follows [3, XI, 7.2 and 9.*
*2]
that the natural map M ! lim-(ff;M . j) is an isomorphism and that
lim-i(ff;M . j) vanishes for i 1. Given an arbitrary object x = (A; O)
of A choose f 2 O and let oe(x) be the pair (A0; O0) where A0 is the
quotient of A V by the kernel of the sum map f + : A V ! V and
O0 is the orbit of the monomorphism A0 ! V . It is easy to check that
oe(x) does not depend on the choice of orbit element f , that x 7! oe(x)
produces a functor oe : A ! F # A , and that the map A ! A V !
A0 induces a natural transformation o from the identity functor of A
to the composite j . oe. By inspection the natural transformation from
ff;M to ff;M . j . oe induced by o is a natural equivalence (cf. 8.1).
Now let x be an object of F # A , y an object of A and oe(y) ! x
a map of F # A . It is clear that there is a unique map w : y ! j(x)
in A such that the given map oe(y) ! x is the composite of oe(w)
with the evident natural isomorphism oe(j(x)) ~= x (in fact, oe is left
adjoint to j). This shows that the over category oe # x has a terminal
object and therefore a contractible nerve [3, XI, x2]; consequently, the
functor oe is left cofinal [3, XI, x9]. The proposition now follows directly
from the fact [3, XI, 7.2 and 9.2] that the natural maps lim-i(ff;M .
j) ! lim-i(ff;M . j . oe) ~= lim-iff;M are isomorphisms for all i. (The
isomorphism lim-i(ff;M . j) ~= lim-iff;M is in fact a general consequence
of the fact that j has a left adjoint [17]).
Let B be the category whose objects are non-trivial vector spaces of
dimension dim (V ) and whose morphisms are vector space monomor-
phisms. Let fi;M be the functor from B to abelian groups which assigns
to each object A the vector space of -maps Emb (A; V ) ! M (here as
in x5 Emb (A; V ) is the set of monomorphisms A ! V ). The unique
-map Emb (A; V ) ! * induces for each A a map M ! fi;M (A); these
combine to give a map M ! lim-fi;M .
8.3 Lemma. There are natural isomorphisms lim-iff;M ! lim-ifi;M for
all i 0. Under these isomorphisms the natural map M ! lim-ff;M
corresponds to the natural map M ! lim-fi;M .
The proof of this is identical to the proof of [13, 2.3]. The standard
cochain complexes which compute the two higher limits involved are
isomorphic.
New Finite Loop Space 31
Proof of 8.1: (cf. [13, proof of 1.2]) By 8.3 it is enough to prove that
the natural map M ! lim-fi;M is an isomorphism and that lim-ifi;M
vanishes for i 1. Let be a p-Sylow subgroup. Since the index of
in is prime to p, the transfer construction [4, p. 80] exhibits fi;M
as a natural retract of fi;M . This implies that lim-ifi;M is a retract
of lim-ifi;M (i 0). The desired result now follows from 8.2, since a
retract of an isomorphism is an isomorphism (i = 0) and a retract of
{0} is {0} (i 1).
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