Relations in the Homotopy of
Simplicial Abelian Hopf Algebras
by
James M. Turner
Submitted to the Department of Mathematics on May 10, 1994
in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy in Mathematics
Abstract
In this paper, we analyze the structure possessed by the homotopy groups of
a simplicial abelian Hopf algebra over the field F2. Specifically, we review the
higherorder structure that the homotopy groups of a simplicial commutative
algebra and simplicial cocommutative coalgebra possess. We then demonstrate
how these structures interact under the added assumptions present in a Hopf
algebra.
Thesis Supervisor: Haynes R. Miller
Title: Professor of Mathematics
1
2
Acknowledgements
Since I get very few opportunities to express my gratitude toward those who
have made a moment like this, i.e. obtaining a doctorate, both memorable and
possible, I'll not refrain from, nor apologize for, being longwinded.
First, since it is only appropriate, let me begin with my colleagues. Foremo*
*st
among them is Haynes Miller. I have learned many things from Haynes, both as a
researcher and as a teacher, but in his role as an advisor I am most appreciati*
*ve.
Many times I have gotten way off track in my objectives (sometimes more than
a graduate student should), but Haynes' advice always got me back on course.
I have never once regretted asking him to act as my advisor.
As far as my work leading up, or directly related, to this document, there
are few, but very important persons to acknowledge. First Bill Dwyer, whose
previous work is integral to this paper, made a crucial observation about abeli*
*an
Hopf algebras which became a lynchpin to establishing the main result. Also
Paul Goerss, whose work also was of great value, enlightened me on many points
concerning simplicial algebras. He also pointed out an error in an earlier vers*
*ion
of this paper. Another person with whom I've enjoyed many conversations is
Phil Hirschhorn. Phil has been crucial in helping me understand the intricacies
of simplicial and cosimplicial spaces. He has also been an indispensable source*
* of
computer information and is responsible for the present appearance of this pape*
*r.
Thanks Phil. Finally, since Mike Hopkins has become a permanent fixture here,
I have had many confusing conversations with him. But, as usually happens, the
truth of his deep insights become wonderful revelations days, sometimes weeks,
later. For sharing these and his delightful humor, I thank him. I also thank him
for his many tutorials on group cohomology.
During my formative years, as a graduate student, I had made many comrades
in my efforts to understand the subtleties of teaching and homotopy theory. My
most memorable moment at the very beginning was learning Quillen's book
with Matthew Ando. It was my greatest influence to algebraic topology. Also,
around this time, my duties as a tutor placed me in an office with Rick Scott
and Farshid Hajir. For two years, I had a wonderful time learning from them
methods of teaching, number theory, toric varieties, philosophy, dealing with r*
*ed
tape, dealing with faculty, the joys of Indian food, etc. Laura Anderson was
another person who helped make my life at MIT enjoyable. I missed her the
year she was away. Finally, I want to thank John Palmieri and Hal Sadofsky for
being kind mentors and good role models during my early years.
Two and a half years ago, I experienced, for about two semesters, a loss of
desire to pursue mathematics. While two of my most productive efforts occurred
during this time, they happened during brief intervals. One person who was quite
pivotal in reawakening this desire was Brooke Shipley. Since then she has been a
powerful ally in pursuing several mathematical projects, as well as a good frie*
*nd.
I look forward to collaborating with her in the future.
At this point, I want to mention my email buddy, Michele Intermont. A year
ago I had the good fortune of meeting her at a conference in Toronto. Since
then I have had many delightful correspondence with her, which, I hope, will
continue.
3
Another aspect of the mathematics department which I've enjoyed dealing
with is the personnel that make it run. First among them is Phyllis Ruby, for
whom each graduate student owes a debt of gratitude for insuring that their
journey toward a doctorate is a smooth one. Of equal importance is the crew
of the Math. Headquarters: Carla, Dennis, Karen, and Sonya, who insure that
the entire department runs like a well oiled machine. Of course I can't ignore
the staff of the Undergraduate Math. Office: Anna, Joanne, and John. Their
hospitality and efficiency create an atmosphere within which undergraduates,
graduates, and professors can interact in a positive and productive way.
One of the most important places for a graduate student is the library. While
I have never been happy with the set up there, I have been very happy with the
staff. In all areas, they have been of great help while being both courteous and
pleasant to talk to.
Before I move on I should mention Judy Romvos. While I've only made a
brief acquaintance with her, without her this document would not have existed.
Outside of life in academia, there have been several close friends who have
helped me keep my sanity over the years. First, Gaby Canalizo has been someone
for whom I could discuss many things that I would not dare with anyone else.
She has also been pivotal in keeping me in touch with the Christian community.
Another person, without whom I would have had many boring days, is Bandita
Joarder. There have been very few people in my life that truly deserve the title
friend, Dita is one of them. One other person who also deserves this title is
John Yarosh. A friend since high school, John has made it possible for me to
experience a lot of things, in my life, I would have otherwise missed.
Since I have now actually neared the end, it is time that I pay homage to the
most important people in my life, my family. During the past twelve years, I ha*
*ve
experienced the most trying and the most exciting times of my life. Through it
all they have each given me their solid support and their valuable assistance. *
*In
particular, my parents have been pivotal in helping me achieve all my goals I've
pursued. To all of them my love and my thanks.
Last, but definitely not least, I thank God through the Lord Jesus Christ,
without whom none of this would have been possible.
4
Contents
Abstract 1
Acknowledgements 3
Introduction 7
Conventions 9
Chapter I. Preliminaries 11
1. Simplicial F2modules 11
2. EilenbergZilber Theorem. 13
3. Group Actions on Tensor Products. 14
Chapter II. Simplicial Abelian Hopf Algebras 17
1. Simplicial Algebras and Dalgebras. 17
2. Simplicial Coalgebras and Acoalgebras. 21
3. Simplicial Hopf Algebras and Hopf Dalgebras. 25
Chapter III. Proof of the Main Theorem 31
1. The Reduction 31
2. Proof of Theorem 2.3.15. 34
3. A Detection Scheme 36
4. Dwyer's detection map and the cohomology of groups 43
5. Proof of the Detection Scheme 48
Bibliography 57
5
6
Introduction
The goal of this paper is to determine all the natural relations that occur *
*in
the homotopy groups of a simplicial abelian Hopf algebra over F2, the field of *
*two
elements. Here Hopf algebra means a unitary algebra and a counitary coalgebra
for which certain diagrams commute (see (2.3.1)). An abelian Hopf algebra then
is one which is commutative as an algebra and cocommutative as a coalgebra.
It is wellknown that over F2, the homotopy groups of a simplicial commuta
tive algebra possesses, in addition to an algebra structure, a compatible action
of a certain operator ring. These operations are viewed as higherorder versions
of divided squares. Dually, the homotopy groups of a simplicial cocommutative
coalgebra possesses an operational action which extends the coalgebra structure.
In fact these are just the Steenrod operations viewed as the dual of higherord*
*er
squaring operations. In each case, the higherorder structure exists because of
the (co)commutativity. Thus the homotopy groups of a simplicial abelian Hopf
algebra possesses both of these structures and the additional properties will p*
*ro
duce relations between them.
These relations contribute to the understanding of the cohomology of iterated
loop spaces with F2coefficients. In particular, the cohomology of a cosimplici*
*al
iterated loop space is a simplicial abelian Hopf algebra. The E2term of the
generalized EilenbergMoore spectral sequence (see, for example, [3]) associated
to this cosimplicial space, is the homotopy groups of this particular simplicial
algebra. Thus the relations assist in making computations. Further, theses
operations play a role in understanding the action of the Steenrod and Dyer
Lashoff operations on the abutment of the spectral sequence (see [10], [18], [1*
*9],
and [20]).
This paper is organized as follows. Chapter 1 is a review of relevant simpli*
*cial
homotopy and symmetric group actions. Chapter 2 sets up the background for
and makes the statement of the Main Theorem. In particular, section 2.1 reviews
simplicial commutative algebras and the properties of their homotopy groups, as
presented in [9]. Section 2.2 does a similar summary for simplicial cocommutati*
*ve
coalgebras following [12]. Finally, section 2.3 reviews Hopf algebras, establis*
*hes
an abelian version of the Hopf condition, and states the Main Theorem which
portrays the natural relations that occur in the homotopy of a simplicial abeli*
*an
Hopf algebra.
Chapter 3 is devoted to proving the Main Theorem. We begin, section 3.1,
by stating the Reduction. This is a theorem which computes the homotopy
groups of a functor on simplicial commutative algebras. We immediately reduce
the proof of this Reduction to computing the effect of a natural map, between
two functors on simplicial vector spaces, in homotopy. This natural map arises
from the abelian Hopf condition, established in section 2.3. In section 3.2, we
use the Reduction to prove the Main Theorem. In section 3.3, we begin the
proof of the computation for the natural map of section 3.1. We first fit this
map into two commuting diagrams. This reduces our efforts further by allowing
us to divide the computations between two new natural maps, each possessing
properties amenable to calculations. In particular, in section 3.4, we recall a
method developed in [9] which allows us to convert our simplicial calculations *
*to
7
ones in the cohomology of groups. Finally, in section 3.5, we make these group
cohomological calculations, completing the proof of the Reduction.
8
Conventions
All groups throughout are finite.
Let R be a ring, G a group, and V a left Rmodule. Then V is a Gmodule if
V is a left R[G]module. On the category of Gmodules there are two functors.
The first functor
()G :(Gmodules) ! (Rmodules)
called the Ginvariant functor, is defined by
V G = {x 2 V : gx = x for all g 2}G:
The second functor
()G :(Gmodules)! (Rmodules)
called the Gcoinvariant functor, is defined by
ffi
VG = V {(1  g)x : x 2 V; g 2 G}:
Further, given a subgroup H G, the inclusion induces a natural transforma
tion, called restriction,
r(G; H): V G ! V H:
Also, if g1; : :;:gm are coset representatives of G=H, where m = (H : G), then
the action of the element g1 + . .+.gm 2 R[G] on V H induces a natural trans
formation, called transfer,
t(h; G): V H ! V G:
The two transformations are related by
t(H; G)r(G; H)x = (H : G)x
for any x 2 V G.
For a fixed group G, we denote by i the inclusion
V G ! V
and by ae the projection
V ! VG
Next we call V a graded Rmodule if V = {Vn}n0 where each Vn is an R
module. If W is another graded Rmodule we define the graded tensor product
V W by M
(V W )n = Vi Wj
i+j=n
for all n 0.
On the category of graded Rmodules we have a functor
: Rgradedmodules! Rgradedmodules
9
called the doubling, defined by
(
(V )n = 0 n odd :
Vn_2 n even
For an element x 2 V we denote its associated element in V by __x.
Finally, for n 2 Z and k 2 N define nk as the coefficient of xk in the Tay*
*lor
expansion of (1 + x)n. These numbers satisfy the general Pascal relation
n n n + 1
k  1 + k = k :
Further, we have
n n + k  1
k = k :
Also we define for i; j 0
(i; j) = i +jj :
For the rest of this work R = F2, the field of two elements.
10
CHAPTER I
Preliminaries
1. Simplicial F2modules
Define a simplicial F2module V to be a graded F2module together with maps
of modules
dj:Vn ! Vn1
called face maps, and
sj:Vn ! Vn+1
called degeneracies, for 0 j n, satisfying standard identities (See [14]). A
map f :V ! W of simplicial F2modules is a map of graded modules which com
mutes with the face and degeneracy maps. We denote the category of simplicial
F2modules by sF2.
Next given two simplicial F2modules V and W we define the simplicial tensor
product V W by
(V W )n = Vn Wn
such that for x y 2 (V W )n then
dj(x y) = djx djy sj(x y) = sjx sjy
0 j n.
Now, define the normalization functor
N :sF2 ! cF2chainomplexes
as follows: For V in sF2 define, for each n 0, the submodule DnV Vn by
DnV = ims0 + . .+.imsn1:
From this define ffi
NnV = Vn DnV:
Further, define
@ :NnV ! Nn1V
by
@ = d0 + . .+.dn:
11
12
As shown in [14], (NV; @) is a welldefined chain complex.
Moreover, N has a left adjoint
S : F2chaincomplexes! sF2:
As shown in [6], the adjoint pair (S; N) determine an equivalence of categories.
We are now in a position to define, for V in sF2, the homotopy groups ss*V by
ssnV = Hn(NV; @):
This defines a functor
ss*: sF2 ! F graded :
2modules
Moreover, this functor is corepresentable as follows:
For V and W in sF2 define its homotopy set of maps to be
ffi
[V; W ] = homsF2(V; W ) ~
where f ~ g if f is homotopic to g (see [15]) for f; g 2 hom sF2(V; W ). Now,
define, for n 0, K(n) in sF2 by SC(n) where C(n) is the chain complex such
that
(
C(n)q = F2 q = n
0 otherwise.
From the equivalence of categories we have
(
(1.1.1) ssqK(n) = F2 q = n
0 otherwise.
Moreover, the correspondence
(1.1.2) [K(n); V ] ! ssnV
given by
[f] ! f*()
where 2 ssnK(n) is the generator, is a bijection (see [15]).
Remark. An equivalent definition of K(n) is given as follows:
Let [n] be the standard nsimplex and _ [n] the simplicialfsetfgeneratediby
djn where n 2 [n]n. Then K(n) = freeF2module on [n] _[n].
2. EILENBERGZILBER THEOREM. 13
2. EilenbergZilber Theorem.
We begin by summarizing the EilenbergZilber theorem as given in [14] and
[9].
Theorem 1.2.1. Let V and W be two simplicial F2modules. Then there
exists a unique natural chain map
D :N(V ) N(W ) ! N(V W )
which is the identity in dimension 0.
Moreover, there exists a natural chain map
E :N(V W ) ! N(V ) N(W )
such that
ED = 1 DE ' 1:
In [9] it was noticed that since D is necessarily the shuffle map (see [14])*
* thus
D possesses a symmetry. This symmetry was exploited by Dwyer to construct
higher order versions of D which we now describe.
Definition 1.2.2.For each k 0, let
OEk: N(V ) N(W ) ! N(V W )
be the chain map such that for x 2 N(V ) and y 2 N(W )
(
OEk(x y) = x y x = k = y
0 otherwise
OEk is called an admissible map.
Let T denote the switching map for either
N(V ) N(W ) ! N(W ) N(V )
or
N(V W ) ! N(W V ):
Theorem 1.2.3. Let V and W be simplicial F2modules. For each k 0
there exists a natural chain map
Dk: [N(V ) N(W )]m ! N(V W )mk
defined for m 2k and satisfying
1. D0 + T D0T + OE0 = D
2. Dk+1 + T Dk+1T + OEk+1 = @Dk + Dk@
Remark. Dwyer showed in [9] that each Dk is unique in a certain sense.
14
3. Group Actions on Tensor Products.
Let V be an F2module and define
V m = V . . .V :
mtimes
Then m , the symmetric group on m letters, acts on V m by permutation.
Thus for any subgroup G m , V m is a Gmodule. With this we define the
Gsymmetric invariant functor
(1.3.1) SG :mF2 ! mF2
by
SG (V ) = (V m )G
and the Gsymmetric coinvariant functor
(1.3.2) SG :mF2 ! mF2
by
SG (V ) = (V m )G :
If G = m then_we denote (1.3.1)by Sm and (1.3.2)by Sm .
Now, let N 2 F2[G] be defined by
__ X
(1.3.3) N = g
g2G
__
Then the action of N on V m defines a map which factors
__N
V m [ ____________Vwm
[ aeo
(1.3.4) o [] aei
ae
SG V
but since, for any x 2 V m , o(gx) = o(x) for any g 2 G then we have a further
factorization.
o
V m [ ___________wSG V
[ aeo
(1.3.5) ae[] aeN
ae
SG V
defining the norm map N.
Because of its importance later, we analyze the norm map N in the case
G = 2. First, we define the diagonal map
d: V ! V 2
3. GROUP ACTIONS ON TENSOR PRODUCTS. 15
by d__x= x x. This is not a homomorphism, nonetheless we have a commuting
diagram
S2V

oeaeaeo
i
ae 
ae d u
(1.3.6) V [_____wV 2
[ 
[[] ae

u
S2V
oe is not a homomorphism, but is one. From this we define the exterior square
functor E2 by E2V = coker. We now have the following commutative diagram
(1.3.7)
V '
oe ''')
N u ss
0 _______Vw ______S2Vw'________________S2Vw____wcokerN _____w0
' ') [[]
[
E2V
from which we have that E2V = imN = kerss. Note that ssoe is a linear isomor
phism.
As an application of (1.3.7)we have
Proposition 1.3.8.For any ! 2 S2V there exists ff 2 E2V and x 2 V ,
uniquely determined by !, such that
! = (ff) + oe(__x):
Proof. Let eoe:cokerN ! S2V be the composite oe . (ssoe)1 so that sseoe= 1.
Then the selfmap
1 + eoess :S2V ! S2V
satisfies ss(1 + eoess) = 0. Thus since is injective, there exists ff 2 E2V su*
*ch that
(ff) = ! + eoess(!):
Finally, let x 2 V be the element which satisfies
__x= (ssoe)1(ss!):
Conclusion follows. ___
16
CHAPTER II
Simplicial Abelian Hopf Algebras
1. Simplicial Algebras and Dalgebras.
Recall that a (graded) algebra is a triple (; m; j) consisting of a (graded)
vector space and maps of (graded) vector spaces
(2.1.1) m: ! ;
called multiplication, and
(2.1.2) j :F2 ! ;
called the unit, such that the two diagrams
m1!
? ?
(2.1.3) 1m ?y ?ym
!m
and
1j
F2 ' ' F2fl _____w
 flfl 
(2.1.4) j1  flfl m
u 1 flflfflu
______________wm
commute.
We further call our algebra commutative if (2.1.1)factors as
'_____________wm
(2.1.5) a')e [ [[]
S2
Notation. For brevity, we denote an algebra (; m; j) by . Also, for x; y 2
we denote the image of x y 2 under m by x . y.
17
18 II. SIMPLICIAL ABELIAN HOPF ALGEBRAS
Next, given algebras and 0, a linear map f : ! 0is a map of algebras
if the diagrams
ff
_____w0 0
m   0
(2.1.6)  m
u u
_________w0f
and
j 
aeo
ae 
(2.1.7) F2 [ f
[] 
j0 u
0
commute.
Note that if and 0are commutative then (2.1.6)can be replaced by
S2(f)
S2 _____wS20
 
 0
(2.1.8)  
 
u u
_______w0f
We denote the category of commutative algebras (respectively commutative
graded algebras) by A (respectively A*).
Given a graded algebra , let Is() s 0 denote the ideal of elements x
in such that x s.
Definition 2.1.9.A algebra is a commutative graded algebra together
with a map
fl2: I2 !
such that
1. I1 is exterior under the product of ,
2. for x; y 2 I2
fl2(____x).=yfl2(__x) + fl2(__y) + x . y;
3. for x; y 2 such that x . y 2 I2
8
><0 x; y 2 I1
fl2(____x).=y> (x . x) . fl2(__y)x = 0
: fl2(__x) . (y.yy) = 0:
We now make the following, as given in [12].
1. SIMPLICIAL ALGEBRAS AND DALGEBRAS. 19
Definition 2.1.10.A Dalgebra is a algebra together with maps
ffii:n ! n+i
for all 2 i n such that
1. ffii is a homomorphism, for i < n, and ffin = fl2,
2. for x; y 2 such that x . y 2 n then
8
><(x . x) . ffii(y)x = 0
ffii(x . y) = > ffii(x) . (y.yy) = 0
: 0 otherwise,
3. for x 2 n and j < 2i then
X i  j + s  1
ffijffiix = ffij+isffisx:
j+1_2si+j_3 i  s
A map of Dalgebras is a map in A* that commutes with the ffii. We denote the
category of Dalgebras by AD.
We now define a simplicial algebra (; m; j) so that (2.1.1)(2.1.4)are satis*
*fied
with the caveat that (2.1.1)and (2.1.2)are now maps of simplicial modules (F2
is replaced with its constant simplicial alias). Further (2.1.5)is satisfied f*
*or
simplicial commutative algebras with the factorization occurring in sF2. We
denote the category of simplicial commutative algebras by sA.
The following was proved in [9] and [13].
Theorem 2.1.11. Let be a simplicial commutative algebra. Then ss* is
naturally a Dalgebra i.e. we have a functor
ss*: sA ! AD:
Remark. The operations ffii in (2.1.10)were first discovered in [4]. Their
properties were subsequently derived in [2] and [9]. In the latter, they were
called higher divided squares.
We conclude this section by indicating why Theorem 2.1.11 completely deter
mines the homotopy of a simplicial commutative algebra.
In light of (2.1.5), a computation of the homotopy of S2V , for a simplicial
module V , in terms of ss*V would give a complete picture of the primary operat*
*or
algebra for the homotopy of a simplicial commutative algebra. Such a description
is known to exist by [8]. We now proceed to make this description explicit.
Fix a simplicial module V . For each 0 i n define
(2.1.12) i:NnV ! Nn+iS2V
by
(2.1.13) i(a) = aeDni(a a) + aeDni1(a @a)
where the Ds are from Theorem 1.2.3.
A computation gives us that
@i= i@:
20 II. SIMPLICIAL ABELIAN HOPF ALGEBRAS
Thus, for 2 i n, i induces a natural map
_
(2.1.14) ffii:ssnV ! ssn+iS2V:
Also, the chain map
aeD :NsV NtV ! Ns+tS2V
induces a homomorphism
__m:ss
(2.1.15) sV sstV ! sss+tS2V:
Combining the results of [4], [2], and [9] we are led to
Proposition 2.1.16.Let V be a fixed simplicial module. Define W to be the
graded module with basis
ffii(x)for x 2 ssnV and 2 i n,
x . y for x 2 sssV and y 2 sstV .
Define a submodule B in W with basis
(
ffii(x + y) + ffii(x) + ffii(y)+02 i < n for x; y 2 ssnV
x . y i = n;
x . y + y .fxor x 2 sssV and y 2 sstV ,
x . (y + z) + x . y + xf.ozr x 2 sssV and y; z 2 sstV ,
x . xfor x 2 ssnV and n > 0.
Then the map W ! ss*S2V given by
x . y! __m(x y)
_
ffiix! ffiix
is natural and induces a linear isomorphism
W=B ' ss*S2V:
Note 2.1.17.Given (graded) algebras and 0 then 0 is a (graded)
algebra under the product
0
( 0) ( 0) 1T1!( ) (0 0) mm! 0:
Further, if and 0are algebras then we define
fl2: I2 ! 0
by demanding that the diagrams
I2(1j0)
I2() ' I2( F2) _________wI2( 0)
 
 
fl2 fl2
 
 
u u
' F2 ________________w10j0
2. SIMPLICIAL COALGEBRAS AND ACOALGEBRAS. 21
and
I2(j1)
I2(0) ' I2(F2 0) _________I2(w 0)
 
 
fl2 fl2
 
 
u u
0' F2 0 ________________wj10
commute and then extending using 3. of Definition 2.1.9. Similarly, we define a
Dalgebra structure on 0, when and 0 are Dalgebras, by demanding
that
ffii:( 0)n ! ( 0)n+i
for 2 i n, fits in the commuting diagrams
0
n ' ( F2)n __________w(1j0)n
 
 
ffii ffii
 
u u
n+i ' ( F2)n+i ________(w10)n+ij0
and
j1
0n' (F2 0)n __________(w 0)n
 
 
ffii ffii
 
u u
0n+i' (F2 0)n+i ________w(j10)n+i
and then extending using 2. of (2.1.10).
2. Simplicial Coalgebras and Acoalgebras.
Recall that a (graded) coalgebra is a triple (; ; ffl) consisting of a (grad*
*ed)
module and maps of (graded) modules
(2.2.1) : ! ;
called comultiplication, and
(2.2.2) ffl: ! F2;
called the counit, such that the diagrams
!
? ?
(2.2.3) ?y ?y1
1!
22 II. SIMPLICIAL ABELIAN HOPF ALGEBRAS
and
______________wflfl
 flfl1 
(2.2.4)  flfl 1ffl
u flflfflu
_____wF2ffl'1 ' F2
commute.
We further call our coalgebra cocommutative if (2.2.1)factors as
_____________w'
'') [[]
(2.2.5) [ i
S2
Notation. For brevity, we denote a coalgebra (; ; ffl) by .
For two (graded) coalgebras and 0, a map f : ! 0of (graded) modules
is a map of (graded) coalgebras if the two diagrams
f
__________0w

 0
(2.2.6)  
u u
_____w0f0f
and
[
 []ffl

(2.2.7) f  F2
 aeo
uaeffl0
0
commute.
Note that for and 0cocommutative, (2.2.6)can be replaced by
f! 0
? ?
(2.2.8) ?y ?y 0
S2 ! S20
S2(f)
We denote the category of cocommutative coalgebras (resp. cocommutative graded
coalgebras) by CA (resp. CA*).
Next, given a cocommutative graded coalgebra we define the coalgebra map
(2.2.9) v : !
2. SIMPLICIAL COALGEBRAS AND ACOALGEBRAS. 23
called the verschiebung, as follows: Fix x 2 . Then x 2 S2. By Proposi
tion 1.3.8 there exists unique ff 2 E2 and fi 2 such that
__
x = (ff) + oe(fi):
__
From this we let v(x) = fi.
Definition 2.2.10.An Acoalgebra is a cocommutative graded coalgebra
together with homomorphisms
Sqi:n ! ni
for i 0 such that for x 2 n we haven
1. xSqi = 0 for02i0> n and xSq _2= v(x),
2. if x = x0 x then
X X
(xSqi) = (x0Sqs) (x00Sqt);
s+t=i
3. for j < 2i we have
X i  s  1
xSqjSqi = j  2s xSqi+jsSqs:
2sj
We define a map of Acoalgebras to be a map in CA* which commutes with
the Sqi. Denote by K* the category of Acoalgebras.
Note. A clearly denotes the Steenrod algebra.
We now define a simplicial coalgebra to be a triple (; ; ffl) where is a
simplicial module and satisfies (2.2.1)(2.2.4)with the exception that all maps
are maps of simplicial modules. Further, a simplicial cocommutative coalgebra
also satisfies (2.2.5). (2.2.6)(2.2.8)also define maps with the requirement th*
*at
they be maps of simplicial modules.
We denote the category of simplicial cocommutative coalgebras by sCA. A
consequence of [7] (see also [12]) is the following
Theorem 2.2.11. Let be a simplicial cocommutative coalgebra. Then ss*
is naturally an Acoalgebra. That is, we have a functor
ss*: sCA ! K*:
We close this section by indicating why Theorem 2.2.11 completely determines
the homotopy of a simplicial cocommutative coalgebra.
As in the algebra case, (2.2.5)indicates that it is sufficient to determine
ss*S2V , for a simplicial module V , in terms of ss*V . This description exists
by [8]. We now proceed to make this explicit.
Fix a simplicial module V . Consider the composite
(2.2.12) NnV i! Nn+iS2V N*!Nn+iS2V
of chain maps. Here i is from (2.1.12)and N is the norm map (1.3.7). This
induces a natural map
(2.2.13) oei:ssnV ! ssn+iS2V
24 II. SIMPLICIAL ABELIAN HOPF ALGEBRAS
for each 0 i n. Also, the composite
(2.2.14) NsV NtV D!Ns+t(V V ) ss*!Ns+tS2V N*!Ns+tS2V
induces the homomorphism
(2.2.15) o :sssV sstV ! sss+tS2V
The following is given in [12].
Proposition 2.2.16.Let V be a simplicial module. Let T be the graded
vector space with basis
oei(x)for x 2 ssnV and 0 i n,
[x; y]for x 2 ssnV and y 2 ssm V , n; m 0.
Let R be the submodule of T with basis
[x; y] + [y; x]for x 2 ssnV , y 2 ssm V , n; m 0,
[x; y + z] + [x; y] + [x;fz]or x 2 ssnV , y; z 2 ssm V , n; m 0,
(
oei(x + y) + oei(x) + oei(y) + 0 0 i < n
[x; y] i = n for x; y 2 ssnV ,
[x; x] for x 2 ssnV , n 0.
Then the map T ! ss*S2V defined by
oei(x) ! oei(x)
[x; y] ! o(x y)
induces a natural linear isomorphism
T=R ' ss*S2V:
Moreover, if we let
e: ss*S2V ! ss*V ss*V
be induced by the composition of chain maps
NS2V i*!N(V V ) E!NV NV
(see Theorem 1.2.1) then for x 2 ssnV y 2 ssm V n; m 0
e([x; y]) = x y + y x
and for x 2 ssnV 0 i n
(
e(oei(x)) = 0 0 i < n
x x i = n:
We take a moment to note a corollary given in [12].
3. SIMPLICIAL HOPF ALGEBRAS AND HOPF DALGEBRAS. 25
Corollary 2.2.17. The effect of the homomorphism
N*: ss*S2V ! ss*S2V
is given by
x . y ! [x; y]
for x 2 ssnV , y 2 ssm V , n; m 0, and
ffii(x) ! oei(x)
for x 2 ssnV , 2 i n. Moreover, under the homomorphism (1.3.6)
*: ss*V ! ss*S2V
we have
im* = kerN*:
Finally, given in sCA, then for x 2 ssn Proposition 2.2.16 tells us that
(2.2.18) *x = [x0; x00] + oei(xSqi)
which defines the action of the Steenrod operations. From this and Corol
lary 2.2.17 we conclude 1. of (2.2.10). Also, we define the coproduct
(2.2.19) : ss* ! ss* ss*
by e * from Proposition 2.2.16.
3.Simplicial Hopf Algebras and Hopf Dalgebras.
Recall that a (graded) Hopf algebra (in the sense of [16]) is a (graded) mod*
*ule
H which is both a (graded) algebra and a (graded) coalgebra for which the two
diagrams
H H _____Hw H H H

 
 1T1
 u
(2.3.1) m  H H H H

 mm
 
u u
H ____________wH H
and
j
F2[_____wH
[ 
(2.3.2) 1 [] ffl
u
F2
commute. A map of Hopf algebras is simply a map of algebras and a map of
coalgebras. We further define a Hopf algebra to be abelian if it is commutative
as an algebra and cocommutative as a coalgebra. Given an abelian Hopf algebra
26 II. SIMPLICIAL ABELIAN HOPF ALGEBRAS
the diagram (2.3.1)possesses a modification. To describe it, we need some
preliminaries.
Lemma 2.3.3.Let V be a (graded) module. Then there exist maps OE0, OE00of
modules such that the following diagram commutes
S2V S2V ii!V V V V ae!S2(V V )
? ? ?
OE0?y 1T1 ?y ?yOE00
S2(V V ) ! V V V V ! S2V S2V
i aeae
Proof. Define bT:V 4 ! V 4 by bT(a b c d) = c d a b. Then on
V 4 , we have the identity
(1 T 1)(T T ) = bT(1 T 1):
From this, the two composites
S2V S2V ii!V 4 1T1!V 4
and
V 4 1T1!V 4 aeae!S2V S2V
factors to give us the maps OE0and OE00respectively. ___
Lemma 2.3.4.For a module V there exists a map OE of modules such that the
following cube commutes
ii
S2V S2V' ________________wV 4 '
 ' 'ae  ' ae
 ') 1T1  '')
0 
OE S2S2V _______________S2(VwSV2)(i)
   
 OE  
u  u 
S2(V V )'_______________wVi4 ' OE00
'  ' aeae 
2 '')  '') 
S (ae) u u
S2S2V _______________wS2Vi S2V
Proof. The identity (1 T 1)(T T ) = bT(1 T 1) from the proof of
Lemma 2.3.3 tells us that the composite
0 S2(ae)
S2V S2V OE!S2(V V ) ! S2S2V
factors to give us the desired map OE. The commutativity of the cube now follows
from Lemma 2.3.3, the surjectivity of ae, and the injectivity of i. ___
3. SIMPLICIAL HOPF ALGEBRAS AND HOPF DALGEBRAS. 27
Proposition 2.3.5.For an abelian Hopf algebra H, the following diagram
commutes
S2( ) OE
S2H ______wS2S2H _____wS2S2H

 S2()
 
u u
H _____________________wS2H
Proof. The diagram (2.3.1)can be expanded to give
H H _______________________________________________HwNHNNH H
 A N N N N flflflffl
 A A N NNP flflflflflii u1T1
 aeA A S2H S2H H H H H
 AC ae 
 S2( ) u aeae 
 S2H ________S2S2Hw 
m  OE 
 ' u i 
 ' ' S2S2H _____wS2H S2HA mm
 S2() A 
 ' u A 
 '' flflfS2HlfNfNlN N A 
 flflflflfl N NiN N AAC 
u''*flflflfl N N NNP u
H fl___________________________________________________Hw H
which commutes by (2.1.5), (2.2.5), Lemma 2.3.4, the surjectivity of ae, and the
injectivity of i. ___
We now pause to give a useful reinterpretation of Proposition 2.3.5.
Let be a commutative algebra. Then S2 is a commutative algebra with
product
2()
(2.3.6) S2S2 OE!S2S2 S! S2
and unit
2(j)
(2.3.7) F2 ' S2(F2) S! S2
Corollary 2.3.8. For an abelian Hopf algebra H, the coproduct
:H ! S2H
is a map of commutative algebras.
Also, if is a algebra then by (2.1.17) is a algebra. Moreover, from
its definition we have
(2.3.9) fl2T = T fl2:
Thus S2 is also a algebra.
We denote by H (resp. H*) the category of abelian Hopf algebras (resp. abeli*
*an
graded Hopf algebras).
28 II. SIMPLICIAL ABELIAN HOPF ALGEBRAS
Definition 2.3.10.A Hopf algebra is a pair (H; fl2) consisting of an abelian
graded Hopf algebra H together with a map
fl2: I2 ! H
satisfying 1. through 3. of Definition 2.1.9 along with the additional conditio*
*n 4.
for x 2 I2
___
fl2__x= fl2(x ):
A map of Hopf algebras is just a map in H* which is also a map of algebras.
We pause here to record a basic relation on a Hopf algebra H. Our objective
is to give a description of the composite
I2 fl2!H v!H:
To do so we define a map
(2.3.11) h: I2 ! H
which fits in the following expansion of (1.3.7)
H '
'~'=
oeu ')
0 _______wH ______S2Hwi________________S2HwN'___cokerNwss___w0fli
 ' ') [] fffl
 [ [ u fl
 E2H _____ E2H fl
 " fl
 " fl
u""^
H u______I2_y_________________Hwh
Here ff is the natural map determined by Proposition 1.3.8 and the dotted arrow
exists in positive degrees by 1. of Definition 2.3.10.
Proposition 2.3.12.For a Hopf algebra H the diagram
fl2
I2 [_____wH
[ 
h [[] v
u
H
commutes.
To prove this, we note that since i: S2H ! H H is a map of algebras then
:H ! S2H is a map of algebras, by Corollary 2.3.8 and Definition 2.3.10.
In light of this and Proposition 1.3.8 we are reduced to proving
3. SIMPLICIAL HOPF ALGEBRAS AND HOPF DALGEBRAS. 29
__
Lemma 2.3.13.Let be a algebra and ! 2 S2. Write ! = (ff) + oe(fi)
as in Proposition 1.3.8. Then
ss(fl2!) = ssoe((__ff))
where __ff2 S2 satisfies (__ff) = ff.
Sketch of proof. Since fl2 is quadratic, we have
__ __
fl2! = fl2(ff) + fl2oe(fi) + (ff) . oe(fi):
__
Using the algebra map i: S2_! we can compute ss((ff) . oe(fi)) = 0.
Also, since ! 2 I2, fl2oe(fi) = 0. We are thus left with computing fl2(ff). Cho*
*ose
z 2 such that it maps to ff under ! E2 and let __ffbe its image in
S2. Then in
i(ff) = (1 + T )z
so that a computation using (2.1.5), (2.1.17), and (2.3.9)gives us
ifl2(ff) = fl2i(ff)= (1 + T )fl2z + z . T z
= i(y) + ioe((__ff))
for some y 2 E2 (in fact y is the image of fl2z). ___
Definition 2.3.14.A Hopf Dalgebra is a Hopf algebra H together with
maps
ffii:Hn ! Hn+i
for all 2 i n, satisfying conditions 1.3. of Definition 2.1.10, and with maps
Sqi:Hn ! Hni
for all i 0, satisfying conditions 1.3. of Definition 2.2.10, such that the f*
*ol
lowing relations are satisfied for a fixed x 2 Hn
1. for each 2 i n
ffiix = ffiix
and for any y 2 H, j 0
X
(x . y)Sqj = (xSqs) . (ySqt)
s+t=j
2. for each 2 j < n and i 0
8 P
>>> (i  j; j  2i + 2s  1)ffiji+s(xSqs)i > j
> vfl2x + ffis(xSqs) i = j
>>>P 2s>j
: (i  2s; j  2i + 2s  1)ffiji+s(xSqs)i < j
8 s
>>>0 i > n
>> P (xSqs) . (xSqis)
>>>2s: + P (i  2s; n  2i + 2s  1)ffini+s(xSqs)i < n:
s
30 II. SIMPLICIAL ABELIAN HOPF ALGEBRAS
A map of Hopf Dalgebras is simply a map of Dalgebras and a map of A
coalgebras. We denote the category of Hopf Dalgebras by HD.
We now define a simplicial Hopf algebra to be both a simplicial algebra and a
simplicial coalgebra which satisfies (2.3.1)and (2.3.2). Clearly, Proposition 2*
*.3.5
applies to a simplicial abelian Hopf algebra. We denote by sH the category of
simplicial abelian Hopf algebras.
We now come to the main theorem of this work, whose proof is postponed to
Chapter 3.
Theorem 2.3.15. Let H be a simplicial abelian Hopf algebra. Then ss*H is
naturally a Hopf Dalgebra. That is we have a functor
ss*: sH ! HD:
On the proof : Consider the simplicial map (2.2.5)
:H ! S2H:
In light of Corollary 2.3.8, if x; y 2 ss*H then we have the equations
*(x . y) = ( *x) . ( *y)
and
*(ffijx) = ffij( *x):
Thus by (2.2.18), we are reduced to understanding ss*S2H as a Dalgebra. This
is the main focus of Chapter 3.
CHAPTER III
Proof of the Main Theorem
1. The Reduction
As we noted at the end of Chapter 2, the key to proving the main theorem
(Theorem 2.3.15) is a complete understanding of the Dalgebra ss*S2, where
is a simplicial commutative algebra. This is achieved in the following
Theorem 3.1.1. Let be a simplicial commutative algebra. Then for the
associated simplicial commutative algebra S2 the following relations hold in
the Dalgebra ss*S2
a.For x 2 ssn, 0 i n, 2 j n + i
X s  i  1
ffijoei(x) = oei+jsffis(x)
j<2s 2s  j  1
b.For x 2 ssn, y 2 ssm , 2 j n + m
8
> oej(x . y) + [ffijx;iyf.my]= 0
: oej(x . y) otherwise
c.For x 2 ssn, y 2 ssm , 0 i n, 0 j m
oei(x) . oej(y) = oei+j(x . y)
d.For x 2 ssn, y; z 2 ss*, 0 i n
(
oei(x) . [y; z] = [x . y; x .iz]f i = n
0 otherwise
e.For x; y; z; w 2 ss*
[x; y] . [z; w] = [x . z; y . w] + [x . w; y . z]:
31
32 III. PROOF OF THE MAIN THEOREM
To prove this theorem, we note that the algebra structure on S2 is completely
determined from the one on through the map of (2.3.4)
OE: S2S2 ! S2S2
by (2.3.6). Thus, we are reduced to computing this map in homotopy when is
an arbitrary simplicial module.
First, if we combine Proposition 2.1.16 and Proposition 2.2.16 then for V a
simplicial module we have
Proposition 3.1.2.The following are generators for ss*S2S2V :
a. oeiffij(x) for x 2 ssnV , 0 i n, 0 j n + i
b. ffii[x; y] for x 2 ssnV , y 2 ssm V , 0 i n + m
c. oei(x) . oej(y)for x 2 ssnV , y 2 ssm V , 0 i n, 0 j m
d. oei(x) . [y;fz]or x 2 ssnV , y; z 2 ss*V , 0 i n
e. [x; y] . [z;fw]or x; y; z; w 2 ss*V
Proposition 3.1.3.The following are generators for ss*S2S2V :
a. oeiffij(x) for x 2 ssnV , z j n, 0 i n + j
b. oei(x . y) for x 2 ssnV , y 2 ssm V , 0 i n + m
c. [ffii(x); y .fz]or x 2 ssnV , y; z 2 ss*V , 2 i n
d. [ffii(x); ffij(y)]for x 2 ssnV , y 2 ssm V , 2 i n, 2 j m
e. [x . y; z . w]for x; y; z; w 2 ss*V
We now arrive at the following which clearly implies Theorem 3.1.1.
Proposition 3.1.4.Let V be a simplicial module. Then the effect of the
map
OE: S2S2V ! S2S2V
in homotopy is given by the following
a.For x 2 ssnV , 0 i n, 2 j n + i
X s  i  1
OE*ffijoei(x) = oei+jsffis(x)
j<2s 2s  j  1
b.For x 2 ssnV , y 2 ssm V , 2 i n + m
8
> oei(x . y) + [ffiix;fyo.ry]m = 0
: oei(x . y) otherwise
c.For x 2 ssnV , y 2 ssm V , 0 i n, 0 j m
OE*(oei(x) . oej(y)) = oei+j(x . y)
1. THE REDUCTION 33
d.For x 2 ssnV , y; z 2 ss*V , 0 i n
(
OE*(oei(x) . [y; z]) = [x . y; x .iz]= n
0 otherwise
e.For x; y; z; w 2 ss*V
OE*([x; y] . [z; w]) = [x . z; y . w] + [x . w; y . z]
We end this section by taking a closer look at the map OE. Let V be a module.
Then we have
Generators of S2S2V :
[x; y] . [z; w]
oe(x) . [y; z]
oe(x) . oe(y)
for any x; y; z; w 2 V .
Generators of S2S2V :
[x . y; z . w]
oe(x . y)
for any x; y; z; w 2 V .
Here oe is the map of (1.3.6).
The effect of
OE: S2S2V ! S2S2V
is given by
[x; y] . [z;!w][x . z; y . w] + [x . w; y . z]
oe(x) . [y;!z][x . y; x . z]
oe(x) . oe(y)! oe(x . y):
We can use this to compute the kernel and cokernel of OE. First, we have a map
a: V 4 ! S2S2V
given by
a b c d ! [a; b] . [c; d] + [a; c] . [b; d] + [a; d] . [b; c]:
It is easy to see that
OEa = 0:
Further, we have a factorization
a
V 4 ____________wS2S2V'
'') [[]
[ b
E4V
34 III. PROOF OF THE MAIN THEOREM
Here E4V is the 4thexterior power of V i.e. the cokernel of the composite
(V ) V 2 d1!V 4 ! S4V
where d is from (1.3.6).
Claim. The induced map
__
b :E4V ! kerOE
is a linear isomorphism.
Proof. By naturality of b and simplicity of the functor_E4,_b is injective,
since it is nontrivial. To see surjectivity, we note that bis onto when dimV 4.
Thus, since E4 is a polynomial functor of degree 4 the result follows. ___
Now, an easy calculation shows
(S2V )* = S2V *
and
(S2V )* = S2V *:
From this and Lemma 2.3.4 we have
OE* = OE:
Further (E4V )* = E4V *so that the claim gives us an exact sequence
0 ! E4V ! S2S2V OE!S2S2V ! E4V ! 0
which is natural as functors of modules. This defines a map
F2 ! Ext2F(E4; E4)
where F is the category of endofunctors on the category of modules. L. Schwartz
has shown (private communication) that this map is an injection.
2. Proof of Theorem 2.3.15.
First, by Theorem 2.1.11 and Theorem 2.2.11 ss*H is both a Dalgebra and
an Acoalgebra. Moreover, is a map of simplicial commutative algebras
by (2.1.17), (2.1.5), (2.1.8), and Lemma 2.3.3. By Theorem 1.2.1 and Theo
rem 2.1.11 we conclude ss*H is a Hopf algebra.
We now proceed to establish 1. and 2. of Definition 2.3.14. For the remainder
of this section we fix x 2 ssnH and write
X X
*x = [x0k; x00k] + oes(xSqs)
k s
as in (2.2.18).
2. PROOF OF Theorem 2.3.15. 35
1.The first part is an easy consequence of the fact that is a map of sim
plicial commutative algebras. For the second part let y 2 ssm H and write
X X
*y = [y0`; y00`] + oet(xSqt):
` t
By Theorem 3.1.1 we have
X
( *x) . ( *y) oes(xSqs) . oet(ySqt)
s;t
X X
oei(xSqs . ySqt)
i0 s+t=i
X X
oei xSqs . ySqt)
i0 s+t=i
where, here and throughout, "" means "equal modulo [ , ]'s". By (2.2.18)we
have X
*(x . y) oei((x . y)Sqi):
i0
The conclusion follows from Corollary 2.3.7. ___
2.Fix 2 j < n. By (2.2.18)we have
X
*ffij(x) oei((ffijx)Sqi):
i
Next, Theorem 3.1.1 gives us
X X
ffij *x oej(x0k. x00k) + ffijoes(xSqs)
k s
X X X `  s  1
oej(x0k. x00k) + 2`  j  1oej+s`ffi`(xSqs)
k s j<2`
X X X j  i  1
oej(x0k. x00k) + j  2i + 2s  1oeiffiji+s(xSqs)
k s 2ij<2s
X X i X j  i  1 j
oej(x0k. x00k) + oei j  2i + 2s  1ffiji+s(xSqs) :
k i 2ij<2s
When i < j we immediately get the third equation. When i > j the expression
m m + r  1
r = r
gives us the first equation. When i = j we just need to verify
X
vfl2x = x0k. x00k
36 III. PROOF OF THE MAIN THEOREM
which is just a consequence of Proposition 2.3.12. Finally, combining Theo
rem 3.1.1 and Definition 2.1.9 we get
X X X
ffin *x oen(x0k. x00k) + ffinoes(xSqs) + oes(xSqs) . oet(xSqt)
X k X s X s
2 = <(1; 3)(2; 4)>:
R
Moreover, it is wellknown that 2 2 ' D8; the dihedral group of order 8.
We thus have the identity
(3.3.2) S2S2V ' SD8 V:
3. A DETECTION SCHEME 37
Lemma 3.3.3.There exists a natural idempotent map
ff: SD8 V ! SD8 V
such that the diagram
NS2V D
S2S2V ! S 8 V
? ?
OE?y ?yff
S2S2V ! SD8 V
S2NV
commutes. Explicitly
ff = 1 + r(4; D8)t(D8; 4):
The proof will follow from the next lemma.
Lemma 3.3.4.There exists a natural map
ff00:S2(V 2 ) ! (S2V )2
such that the diagram
NV 2 2 2
S2(V 2 ) ! S (V )
? ?
OE00?y ?yff00
(S2V )2 ! (S2V )2
(NV)2
commutes. Here OE00is the map of Lemma 2.3.3. Indeed, we can take
ff00= fflt(2; 2 x 2)
where the transfer is associated to the diagonal 2 ! 2 x 2 and ffl is the
isomorphism induced by 1 T 1: V 4 ! V 4 .
Proof of Lemma 3.3.4. First, we have commuting diagrams
1+(1;3)(2;4)
V 4 ________________wV 4u
 
 
aeV 2 iV 2
 
u 
S2(V 2 )____________wS2(VN2 )
V 2
and (1+(1;2)).(1+(3;4))
V 4 _________________Vw4u
 
2  2
aeV iV
 
u 
(S2V )2 _____________(S2Vw)2N:2
V
38 III. PROOF OF THE MAIN THEOREM
An easy computation shows that the diagram
1+(1;3)(2:4)
V 4 ________________wV 4
 
 
(2;3) (2;3)(1+(1;3))
 
u (1+(1;2))(1+(3;4)u)
V 4 ________________wV 4
commutes. Consider now the map
iV 2 4
S2(V 2 ) ! V :
In the group ring F2[4], we have the identity
(2:3)(1 + (1; 3))(1; 3)(2;=4)(1 + (1; 2))(2; 3)(1; 3)(2; 4)
= (1 + (1; 2))(1; 2)(3; 4)(2; 3)
= ((1; 2)(3; 4) + (3; 4))(2; 3):
This shows that the image of the above map is invariant under the action of
<(1; 2); (3:4)>. We thus have a commuting diagram
iV 2 4
S2(V 2 )! V
? ?
ff00?y ?y(2;3)(1+(1;3))
(S2V )2 ! V 4
i2V
defining ff00.
Combining these four diagrams and Lemma 2.3.3 gives us a cube
V 4 _______________Vw4''*
 ''))  ' '
  *
 S2(V 2 )__________S2(Vw2 )
 
   
u  u 
V 4 _____________Vw4''* 
''))  ' ' 
u * u
(S2V )2 ___________(S2Vw)2
from which our desired commutative diagram results. The identification of ff00
follows from our construction and the definition of transfer. ___
Proof of Lemma 3.3.3. Consider the composite
2(iV) ff00
S2S2V S! S2(V 2 ) ! (S2V )2 :
From Lemma 3.3.4 and a computation we have
ff00(1; 2)(3; 4) = ffl(1; 2)(3; 4)t(2; 2 x 2) = (1; 3)(2; 4)ff00:
3. A DETECTION SCHEME 39
Thus (1; 3)(2; 4)ff00S2(iV ) = ff00S2(iV ). Hence we have a diagram
2(iV)
S2S2(V ) S! S2(V 2 )
? ?
ff?y ?yff00
S2S2(V ) ! (S2V )2 :
iS2V
By Lemma 2.3.4 and Lemma 3.3.4 our desired diagram commutes. From this
and the identity (2; 3)(1; 3) = (1; 3)(1; 2) we arrive at the commuting diagram
S2S2(V ) ! V 4
? ?
ff?y ?y(1;3)+(2;3):
S2S2(V ) ! V 4
Clearly 1; (2; 3); (1; 3) are coset representatives for D8 in 4. Also ((2; 3) +
(1; 3))2 = (1; 3)(1; 2) + (2; 3)(1; 2) from the above and the identity (1; 3)(2*
*; 3) =
(2; 3)(1; 2). Hence ff2 = ff. ___
Corollary 3.3.5. The following cube commutes
S2S2V '_______________S2S2Vw'
 '')  ' ')
 ff
OE S2(V 2 )___________wS2(V 2 )
   
 00  
u OE u 
S2S2V _____________S2S2Vw ff00
' '  ' ' 
')  ') 
u u
(S2V )2 _____________w(S2V )2
Proof. This easily follows from Lemma 2.3.4, Lemma 3.3.3, Lemma 3.3.4,
and naturality. ___
Note. The effect of the map
ff: SD8 V ! SD8 V
on elements is
[x; y]; [z;!w][x; z]; [y; w] + [x; w]; [y; z]
oe(x); [y; z]! [x; y]; [x; z]
[oe(x); oe(y)]! oe[x; y]
oe[x; y]! oe[x; y]
from which we easily verify idempotence. We further note that the module of
natural maps
()D8 ! ()D8
40 III. PROOF OF THE MAIN THEOREM
on the category of 4modules has as basis the set {1; ff}. In light of this,
Lemma 3.3.3 should not be surprising.
Now, by Proposition 2.2.16 we have
Proposition 3.3.6.The following are generators of ss*S2S2V :
a. oejoei(x) for x 2 ssnV , 0 i n, 0 j n + i
b. oei[x; y] for x 2 ssnV , y 2 ssm V , 0 i n + m
c. [oei(x); oej(y)]for x 2 ssnV , y 2 ssm V , 0 i n, 0 j m
d. oei(x); [y; z]for x 2 ssnV , y; z 2 ss*V , 0 i n
e. [x; y]; [z; w]for x; y; z; w 2 ss*V .
By Corollary 2.2.17, the effect of the map
(NS2V)*: ss*S2S2V ! ss*S2S2V
is given by
ffijoei(x)! oejoei(x)
ffij[x;!y]oej[x; y]
oei(x) . oej(y)! [oei(x); oej(y)]
oei(x) . [y;!z]oei(x); [y; z]
[x; y] . [z;!w][x; y]; [z; w] :
Also, the effect of the map
(S2NV )*: ss*S2S2V ! ss*S2S2V
is given by
oeiffij(x)! oeioej(x)
oei(x . y)! oei[x; y]
[ffii(x); ffij(y)]! [oei(x); oej(y)]
[ffii(x); y!. z]oei(x); [y; z]
[x . y; z .!w][x; y]; [z; w] :
Further, by Proposition 2.2.16, the effect of the map
(S2iV )*: ss*S2S2V ! ss*S2(V 2 )
3. A DETECTION SCHEME 41
is given by
(
ffijoei(x)! ffij(x x)i = x
0 otherwise
ffij[x;!y]ffij(x y + y x)
(
oei(x) . oej(y)! (x x) . (y y) i = x j = y
0 otherwise
(
oei(x) . [y;!z](x x) . (y z + z y) i = x
0 otherwise
[x; y] . [z;!w](x y + y x) . (z w + w z)
Also, the effect of the map
(iS2V)*: ss*S2S2V ! ss*(S2V )2
is given by
(
oeiffij(x)! ffij(x) ffij(x)i = x + j
0 otherwise
(
oei(x .!y)(x . y) (x . y)i = x + y
0 otherwise
[ffii(x); ffij(y)]! ffii(x) ffij(y) + ffij(x) ffii(x)
[ffii(x); y!.fz]fii(x) (y . z) + (y . z) ffii(x)
[x . y; z .!w](x . y) (z . w) + (z . w) (x . y):
From this we conclude that the map
(S2NV )* (iS2V)*: ss*S2S2V ! ss*S2S2V ss*(S2V )2
is injective. We are thus reduced, by Corollary 3.3.5, to computing, in homotop*
*y,
the maps induced by ff and OE00. For this we have
Proposition 3.3.7.Let V be a simplicial module. Then the effect of
ff*: ss*S2S2V ! ss*S2S2V
is given by
a.For x 2 ssnV , 0 i n, 0 j n + 1
X s  i  1
ff*oejoei(x) = oei+jsoes(x)
j<2s 2s  j  1
b.For x 2 ssnV , y 2 ssm V , 0 i n + m
ff*oei[x; y] = oei[x; y]
c.For x 2 ssnV , y 2 ssm V , 0 i n, 0 j m
ff*[oei(x); oej(y)] = oei+j[x; y]
42 III. PROOF OF THE MAIN THEOREM
d.For x 2 ssnV , y; z 2 ss*V , 0 i n
(
[x; y]; [x; z]i = n
ff* oei(x); [y; z] =
0 otherwise
e.For x; y; z; w 2 ss*V
ff* [x; y]; [z; w] = [x; z]; [y; w] + [x; w]; [y; z] :
Proposition 3.3.8.Let V be a simplicial module. Then the effect of
OE00*:ss*S2(V 2 ) ! ss*(S2V )2
is given by
a.For x 2 ssnV , y 2 ssnV , y 2 ssm V , 2 j n + m
8
> x . x ffijyn = 0
: 0 otherwise
b.For x; y; z; w 2 ss*V
OE00*[(x y) . (z w)] = (x . z) (y . w):
We will actually prove a much more general result then Proposition 3.3.8. To
state it we first need the following set up.
Let V and W be modules. Then the map
1 T 1: V W V W ! V V W W
induces __
OE00:S2(V W ) ! S2V S2W:
Following the proof of Lemma 3.3.4 verbatim gives us
Lemma 3.3.9.There exists a map
__ff00:S2(V W ) ! S2V S2W
such that the diagram
S2(V W ) NVW!S2(V W )
_OE00??y ??y_ff00
S2V S2W ! S2V S2W
NVNW
commutes. Indeed we can take
__ff00= fflt(
2; 2 x 2)
as in Lemma 3.3.4.
4. DWYER'S DETECTION MAP AND THE COHOMOLOGY OF GROUPS 43
Proposition 3.3.10.Let V and W be simplicial modules. Then the effect of
__00
OE:ss*S2(V W ) ! ss*S2V S2W
is given by
a.For x 2 ssnV , y 2 ssm V , 2 j n + m
8
__00 > x . x ffijyn = 0
: 0 otherwise
b.For x; z 2 ss*V , y; w 2 ss*W
__00
OE*[(x y) . (z w)] = (x . z) (y . w):
Clearly Proposition 3.3.10 implies Proposition 3.3.8. Finally, Proposition 3*
*.1.4
follows from Lemma 3.3.3, Lemma 3.3.4, Proposition 3.3.7, and Proposition 3.3.8.
The proof of Proposition 3.3.7 and Proposition 3.3.10 will be given in x5.
4. Dwyer's detection map and the cohomology of groups
In this section, we gather the tools necessary for proving Propositions 3.3.7
and 3.3.10. The key is the following theorem found in [9].
Theorem 3.4.1. Given a simplicial module V and a subgroup G m there
exists a natural homomorphism
M
G :ssiSG V ! Hk(G; ssi+kV m )
0k
such that for a subgroup H G
r(G; H)G = H r(G; H) t(H; G)H = G t(H; G):
We summarize the proof of this theorem. Let G m and CG the category
of Gmodules. Let
F :CG ! B
be an additive functor to some abelian category. This induces a functor
F :chCG ! chB
of bounded above chain complexes over these categories. For a fixed C in chCG ,
there exists an injective resolution C ! I i.e. an object I in chCG which is
degreewise injective, together with a quasiisomorphism from C. Such an object
is unique up to chain homotopy. Define, as in [17], the total right derived fun*
*ctor
of F to be
RF (C) = F (I)
which comes equipped with a natural map
F (C) ! RF (C):
At this point, we should remark that given C in chCG we can construct an
injective resolution C ! I as follows: for each k 2 Z we have an injective
44 III. PROOF OF THE MAIN THEOREM
resolution Ck ! Ik;*in CG by homological algebra. The chain maps for C
extend to give us a bichain complex I**. Upon letting I = Tot I**, the total
chain complex, we immediately get a quasiisomorphism
C ! I
which serves as an injective resolution. The advantage of this construction is
that it gives us a spectral sequence
(3.4.2) E2i;j= RiF (HjC) =) Ri+jF (C)
where RkF (C) = HkRF (C) and RkF () is the kth derived functor of F on
CG . This spectral sequence is constructed in ch. 17 of [5]. As an application,*
* if
C is a Gchain complex with trivial differential then (3.4.2)collapses to give *
*us
M
(3.4.3) RkF (Ckm ) ' Rm F (C)
k0
in B. To define our desired map G let F be the Gfixed point functor i.e. for
M in CG
(3.4.4) F (M) = MG = H0(G; M):
Now, let V be a simplicial module such that NV is bounded above. Then the
EilenbergZilber map provides us with a Gequivariant chain equivalence
(NV )m ! N(V m ):
Moreover, since we are over a field, there is a chain equivalence
NV ! ss*V
where ss*V has trivial differential, which induces a Gequivariant chain equiva
lence
(NV )m ! (ss*V )m :
By (3.4.2), we have quasiisomorphisms
RF (N(V m )) RF ((NV )m ) ! RF ((ss*V )m ):
By (3.4.3)and (3.4.4)we obtain
M
RnF (N(V m )) ' Hk(G; ssn+kV m ):
k0
Now, H*F (N(V m )) ' H*NF (V m ) ' ss*SG V by functoriality. Combining
the above, we have a natural homomorphism
M
ssiSG V ' HiF (N(V m )) ! RiF (N(V m )) ' Hk(G; ssi+kV m )
k0
which is what we call G . The case of a general simplicial module V follows
from a limit argument.
The relations involving restriction and transfer follow immediately from the
naturality and equivariance of all maps involved.
4. DWYER'S DETECTION MAP AND THE COHOMOLOGY OF GROUPS 45
The usefulness of the map of Theorem 3.4.1 is now made precise by the fol
lowing
Proposition 3.4.5.For any simplicial module V , the natural homomorphism
of Theorem 3.4.1 is injective for the group 2.
Proof. We first prove the result for V = K(n). From chapter 1 x1 we have
8
><0 s < n
Ns(K(n) K(n)) = > nonzero n s 2n
: 0 2n < s:
Let C be the 2chain complex such that
8
> n < s 2n
Cs = > F2 n = s
: 0 otherwise.
If we write 2 = {1; T }, then the differential @ on C is given by
@xs+1= (1 + T )xs n < s < 2n
@xn+1 = y:
Write ssn K(n) = F2 and define a map
f :C ! N(K(n) K(n))
by
xs ! D2ns(a a)
y ! OEn(a a):
By Theorem 1.2.3 this is a map of 2chain complexes. Moreover, it is a quasi
isomorphism. Let F be the functor H0(2; ). We wish to compute
HnF (C) ! Rn F (C):
To do so define the complex bCby
(
Cbs= F2[2] s 2n
0 otherwise
with differential b@given by b@bxs+1= (1 + T )bxs. This is clearly a free 2cha*
*in
complex and the map
C ! bC
given by
xs! bxs
y! (1 + T )bxn
is clearly a quasiisomorphism. Thus RF (C) = F (Cb) and an easy calculation
gives that
HsF (C) ! RsF (C)
is an injection for all s.
46 III. PROOF OF THE MAIN THEOREM
To obtain the general case, we first take V so that NV is bounded above.
Then we have a weak equivalence
M
K(nff) ! V:
ff
Thus it suffices to show that if 2 is injective for W1 and W2 then it is injec*
*tive
for W1 W2. First, we have a decomposition of
N((W1 W2) (W1 W2))
as
N(W1 W1) N(W2 W2) N((W1 W2) (W2 W1)):
Since the last summand is 2free and since 2 respects this decomposition,
injectivity follows. A limit argument completes the proof. ___
We now pause to record a useful property of total derived functors.
Lemma 3.4.6.Let G; H be finite groups and B an abelian category. Let
F1: CG ! CH and F2: CH ! B be additive functors such that F1 preserves
injectives. Then R(F2 O F1) is chain homotopic to RF2 O RF1. Moreover, the
natural map
F2 O F1 ! R(F2 O F1)
is chain homotopic to the composite
F O F1 ! (RF2) O F1 ! RF2 O RF1:
As an application, we give a Corollary to Proposition 3.4.4.
Corollary 3.4.7.RFor any simplicial module V , G is injective for G =
2 x 2 and G = 2 2.
Proof. Let F1 = H0(2; ) and F2 = H0(2x 2; ). A Kunneth theorem
argument shows that
F2 ! RF2
is equivalent to
F1 F1 ! RF1 RF1:
Thus injectivity follows from Proposition 3.4.5. Now F1 and F2 can be viewed
as functors
F2: C2R 2 ! C2
and
F1: C2 ! (modules):
R
Since F2 preserves injectives then using the fact that F1 O F2 = H0(2 2; )
our desired result follows from Proposition 3.4.5, Lemma 3.4.6, and the result
for 2 x 2. See [9] for details. ___
We now proceed to recall some useful tools in group cohomology. See [11] or
chapter 12 of [5] for details.
4. DWYER'S DETECTION MAP AND THE COHOMOLOGY OF GROUPS 47
LyndonSerreHochschild Spectral Sequence. Consider the extension
of finite groups
K ae G i Q:
Let M be a Gmodule. Then we have a first quadrant spectral sequence
(3.4.8) E*;*2= H*(Q; H*(K; M)) =) H*(G; M):
Here H*(K; M) is a Qmodule since we have the functor
H0(K;): CG ! CQ :
To make this spectral sequence useful we have
Lemma 3.4.9.Given a diagram
K v_____G_w______wQw
  
  
  
u u u
K0 v_____G0w______Q0ww
whose rows are extensions then the induced map
H*(Q0; H*(K0; M)) ! H*(Q; H*(K; M))
is a map of spectral sequences for a Qmodule M. Moreover, the induced map
on E1 is compatible with
H*(G0; M) ! H*(G; M):
Further, if the vertical maps are injective, then the map
H*(Q; H*(K; M)) ! H*(Q0; H*(K0; M))
induced from the associated transfers, becomes a map of spectral sequences.
Again, the induced map on E1 is compatible with the associated transfer,
H*(G; M) ! H*(G0; M):
Double Coset Formula. Let H; K be subgroups of a finite group G. A
double coset representation of G with respect to H and K is a subset S G
such that [
G = HoeK
oe2S
and is minimal among all such subsets. Next, if x 2 G and J G define the
conjugation map
cx: J ! xJx1
by cx(u) = xux1.
48 III. PROOF OF THE MAIN THEOREM
Proposition 3.4.10.Let S be a double coset representation of G with respect
to H and K and let M be a Gmodule. Then for ff 2 H*(K; M)
X
r(G; H)t(K; G)(ff)= t(H \ xKx1; H)r(xKx1; H \ xKx1)cx(ff)
X x2S
= t(H \ xKx1; H)cxr(K; x1Hx \ K)(ff)
x2S
holds in H*(H; M).
5.Proof of the Detection Scheme
In this section, we prove Proposition 3.3.7 and Proposition 3.3.10 using the
methods of the previous section. First, we need some basic results to facilitate
our computations.
Let K(n) be the EilenbergMacLane module so that ss*K(n) = F2 where
a = n 0. Then by Proposition 2.2.16
(
ss*S2K(n) ' F2* = n + i 0 i n
0 otherwise:
Also H*(2; F2) = F2[w] where w is dual to the generator H1(2; F2) ' F2. We
then have
Proposition 3.5.1.Under the homomorphism
2 : ss*S2K(n) ! H*(2; F2)
of Theorem 3.4.1
2 oei(a) = w2ni
for all 0 i n.
Proof. This follows easily from Proposition 3.4.5. ___
Now, take K(m) so that ss*K(m) ' F2 where b = m 0.
2
Proposition 3.5.2.Let M be the 2submodule of ss* K(n) x K(m)
generated by a b. Then
(
Hi(2; M) = 0 i > 0
F2<2 [a; b]> i = 0
Proof. For i > 0 this just follows from the fact that M is a free 2module.
For i = 0 we note that under the projections
2
S2 K(n) x K(m) ! S K(n)
2
S2 K(n) x K(m) ! S K(m)
[a; b] projects to 0 in homotopy. Hence by naturality and Proposition 3.4.5 the
result follows. ___
5. PROOF OF THE DETECTION SCHEME 49
Proposition 3.5.3.Consider the extension
2 x 2 ae D8 i 2:
Then for a simplicial module V
H*(D8; ss*V 4 ) ' H*(2; H*(2; ss*V 2 )2 ):
Moreover, we have a factorization
D8
ss*SD8 V_________________________H*(D8;wss*V 4 )
   
   
   
  

ss*S2S2V 
 
  
2  
u  
H*(2; ss*(S2V )2 )_______________wH*(2;HH*(2;*ss*V(2 )2 ) 2
2;(2 ) )
Proof. Define functors
F1: CD8 ! C2
and
F2: C2 ! (modules)
by F1 = H0(2x 2; ) and F2 = H0(2; ). Then F1 preserves injectives and
F2 O F1 = H0(D8; ). So by Lemma 3.4.6,
R(F2 O F1) ' RF2 O RF1:
Thus it suffices to compute H*(RF2 O RF1) for NV 4 . Since we have an equiv
ariant equivalence
NV 4 ! (ss*V )4
and since RF1 is 2equivalent to RF2 RF2 we conclude that we have a 2
equivalence
RF1(NV 4 ) ! H*(2; ss*V 2 )2
so by (3.4.3)
H*(RF2 O RF1(NV 4 )) ' H*(2; H*(2; ss*V 2 )2 ):
The identification of D8 follows from the 2nd part of Lemma 3.4.6. ___
Note. The identification in Proposition 3.5.3 can also be worded to say that
the spectral sequence (3.4.8)collapses at the E2term. We also note that this
identification gives us a choice of representatives for the generators for the *
*co
homology of D8, but we will see that in most cases the spectral sequence (3.4.8)
has only one nontrivial column or row at E2, forcing our hand.
Before proving Proposition 3.3.7, we note that by Lemma 3.3.3 and Proposi
tion 3.4.1 we have
(3.5.4) ff*D8 = D8ff*
Also, combining Lemma 3.3.3 and Proposition 3.4.10, we have
50 III. PROOF OF THE MAIN THEOREM
Proposition 3.5.5.Let D8 be the subgroup (2; 3)D8(2; 3) \ D8. Then
the map
ff*: H*(D8; ss*V 4 ) ! H*(D8; ss*V 4 )
satisfies the identity
ff* = t(; D8)c(2;3)r(D8; ):
Now, we proceed to prove Proposition 3.3.7. To do so we exploit naturality
using (1.1.2)and reduce to universal examples. To this end we fix the following
throughout
ss*K(m)= F2 a= m
ss*K(n)= F2 b= n
ss*K(p)= F2 d= p
ss*K(q)= F2 e= q
where m; n; p; q 0.
Proof of Proposition 3.3.7 part a: First, since = 2x2, H*(; F2) '
F2[v1; v2] where v1; v2 2 H1(; F2) is dual to the elements of H1(; F2) associ
ated to the generators of . We now summarize a result in [9].
Proposition 3.5.6.There exist elements x; y 2 H1(D8; F2) and z 2 H2(D8; F2)
such that
1. H*(D8; F2)' F2[x; y; z]=(xy)
2. r(D8; )x = v2
r(D8; )y = 0
r(D8; )z = v1(v1 + v2)
X m  `  1
3. t(; D8)vm1= ` xm2`z`
02` ' 2 x 2, then N is a direct sum of two trivial
Bmodules. Thus by the Kunneth theorem
H*(B; N1) ' H*(B; F2) H*(B; F2) ' F2[i01; i02] F2[i001; i002]:
Here 2 acts by exchanging summands, which is a free 2action. Hence (3.4.8)
tells us that
H*(D8; N1) ' H0(2; H*(B; N1)) ' F2[i1; i2]
where i1 corresponds to i01 i001and i2 corresponds to i02 i002, i1 = 1 = i2.
Next, N2 is a free Bmodule so by (3.4.8)
H*(D8; N2) ' H*(2; H0(B; N2)) ' F2[]
with  = 1.
Now, we have an extension
2 ae i 2
so that
H*(; N1) ' H0(2; H*(2; N1)) ' F2[j] j = 1
since N1 is a direct sum of two trivial 2modules with respect to the inner
2action and so proceed as above. Now, N2 factors into N02 N002as modules
where N02is generated by a b a b and N002is generated by a b b a.
Thus
H*(; N2) ' F2[1] F2[2] i = 1 i = 1; 2
by a computation as above.
5. PROOF OF THE DETECTION SCHEME 53
Proposition 3.5.11. 1. Under the map r(D8; ): H*(D8; N) ! H*(; N)
i1! j
i2! j
! 1 2
2.Under the map c(2;3):H*(; N) ! H*(; N)
j! 1
2 ! 2:
3.Under the map t(; D8): H*(; N) ! H*(D8; N)
jr ! 0
r1 ! ir
r2 ! r
for all r > 0.
Proof. 1. Consider the diagram of extensions
2 v______w ______2ww

  
ffi   
  
u u  
B v_____D8_w____w2w
where ffi is the diagonal map. This induces
H0(2; H*(B; N1)) ! H0(2; H*(2; N1))
and
H*(2; H0(B; N2)) ! H*(2; H0(2; N2)):
These are the restriction maps
H*(D8; Ni) ! H*(; Ni)
for i = 1; 2, by our above computations and Lemma 3.4.9. The first restriction
is an easy computation. For the second restriction we have H0(B; N2) ' F2 and
H0(2; N2) ' F2 F2 so that the induced map F2 ! F2 F2 is the diagonal
map.
2. This is an easy consequence of the fact that
c(2;3)N1 = N02
c(2;3)N002= N002:
3. First, N2 is a free Bmodule so that r(D8; B) is trivial on H*(D8; N2) in
positive degrees. Next, N1 is a direct sum of two trivial Bmodules thus
H*(B; N2) ' F2[fl1; fl2] F2['1; '2]
54 III. PROOF OF THE MAIN THEOREM
where fli = 1 = 'i i = 1; 2. From the diagram of extensions
Bv_____B_w_____1ww
 
 
 
 
  u u
B v_____D8_w____w2w
and Lemma 3.4.9, the restriction map r(D8; B) on H*(D8; N1) is equal to the
inclusion
H0(2; H*(B; N1)) ! H*(B; N1):
Thus
r(D8; B)is1it2= fls1flt2 's1't2:
We now pause to bring in the transfer
Claim.
r(D8; B)t(; D8) = 0
Proof. By Proposition 3.4.10
r(D8; B)t(; D8) = t(I; B)r(; I)
where
I = \ B:
Since I is a factor of B; r(B; I) is onto, but t(I; B)r(B; I) = 0 so that t(I; *
*B) =
0. ___
From this claim and our computations, we conclude that
t(; D8)ri= cir
ci2 F2, i = 1; 2. From Proposition 3.4.10, we have
r(D8; )t(; D8) = 1 + c(1;2):
Since
(1; 2)N1= N1
(1; 2)N01= N001
we get that under r(D8; )t(; D8)
ri! r1 r2:
So ci = 1 for i = 1; 2. Finally, t(; D8)jr = 0 since jr is in the image of
r(D8; ). ___
Now, the relevance of the module N comes from
Proposition 3.5.12. 1. For 0 i m, 0 j n
D8[oei(a); oej(b)] = imi1inj22 H*(D8; N1)
2.For 0 i m + n
D8oei[a; b] = n+mi 2 H*(D8; N2):
5. PROOF OF THE DETECTION SCHEME 55
Proof. These follow from Propositions 3.5.1, 3.5.2, and 3.5.3. ___
Combining Corollary 3.4.7, (3.5.4), Propositions 3.5.5, 3.5.11, and 3.5.12 g*
*ives
us our desired result.
Proof of Proposition 3.3.7 part d: Again it is sufficient to prove the result
for V = K(m) x K(n) x K(p). Let N be the 4submodule of ss*V 4 generated
by a a b d. As such it is a summand of the 4module ss*V 4 .
Proposition 3.5.13.For all 0 i m
mi
D8 oei(a); [b; d] 2 H (D8; N):
Proof. Again, this is a computation utilizing Propositions 3.5.1, 3.5.2, and
3.5.3. ___
Now, since N is a free module, then by (3.5.4)and Proposition 3.5.5 the re
sult follows from a computation utilizing Proposition 2.2.16 and Corollary 3.3.*
*5.
Proof of Proposition 3.3.7 part e: Let V = K(m) x K(m) x K(p) x K(q)
and N the 4submodule of ss*V 4 generated by a b d e.
Proposition 3.5.14.
*
D8 [a; b]; [d; e] 2 H (D8; N):
Proof. Combine Proposition 3.5.2 and 3.5.3. ___
N is 4free so another computation using Proposition 2.2.16 and Corol
lary 3.3.5 gives us our result.
This completes the proof of Proposition 3.3.7.
Proof of Proposition 3.3.10. a. It is sufficient to prove the result for
V = K(n) and W = K(n). Suppose n; m > 0. Then
(NV NW )*: ss*S2V S2W ! ss*S2V S2W
is injective. Thus it suffices to compute __ff00*. By Theorem 3.4.1 and Lemma 3*
*.3.9
our conclusion follows from t(2; 2 x 2) = 0 since r(2 x 2; 2) is onto
H*(2; ss*(V W )2 ). Suppose n = 0. Define
i1: (S2V ) W ! S2(V W )
as the unique simplicial map such that
(xy) b ! (x b)(y b):
Also define
i2: (S2V ) W ! (S2V ) (S2W )
56 III. PROOF OF THE MAIN THEOREM
as 1 (see (1.3.6)). Then the diagram
S2(V W )
i1 AC 
A A 
A 
(S2V ) W _OE00
flfl 
flflffl
i2 u
S2V S2W
commutes. A computation gives the result. The case of m = 0 is the same.
b. This is an easy computation using the diagram
(V W )2 ! S2(V W )
? ?
1T1 ?y ?y_OE00:
(V 2 ) (W 2 ) ! S2V S2W
___
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57