OPERATIONS AND SPECTRAL SEQUENCES I
JAMES M. TURNER
Abstract. Using methods developed by W. Singer and J. P. May,
we describe a systematic approach to showing that many spectral
sequences, determined by a filtration on a complex whose homology
has an action of operations, possesses a compatible action of the
same operations. As a consequence, we obtain W. Singer's result
for Steenrod operations on Serre spectral sequence and extend A.
Bahri's action of Dyer-Lashof operations on the second quadrant
Eilenberg-Moore spectral sequence.
1. Introduction
Overview. This is intended to be the first in a series of papers to
address the question of finding a systematic approach for determining
when spectral sequences possess a "good" action of operations, pro-
vided it comes from a filtration on a chain complex whose homology
has an appropriate action of operations. In this paper, we focus on
those filtrations which give us either second quadrant homology spec-
tral sequences or first quadrant cohomology spectral sequences. Since
the latter type can be interpreted as a third quadrant homology spec-
tral sequence, we can capture both of these notions under the rubric of
left plane spectral sequences.
We accomplish our goal, in this situation, by defining the notion of a
Dold algebra. This is essentially a chain complex with product which is
"commutative up to homotopy". The definition and subsequent meth-
ods rely on the algebraic machinery developed by J. P. May in [8].
From this definition, it is easy to see that the homology of a Dold al-
gebra has well defined action of operations. In order to relate this to a
spectral sequence coming from a filtration on a Dold algebra, we define
the notion of a -filtration. We then show that the resulting spectral
sequence has a compatible action of operations, defined in a way that
____________
Date: August 1997.
1991 Mathematics Subject Classification. Primary: 18G40, 55S05, 55U15;
Secondary: 18G30, 55S10, 55S12, 55T10, 55T20.
Key words and phrases. spectral sequences, Dold algebras, Steenrod operation*
*s,
Dyer-Lashof operations, cosimplicial spaces, infinite loop spaces.
1
2 JAMES M. TURNER
captures the type of action originally constructed by W. Singer in [11 ]
and [12 ].
With this in hand, we then focus our attention on bicomplexes and
the spectral sequences arising from the standard filtration on their total
complexes. Again, using the approach developed in [11 ] and [12 ], we
give conditions on the bicomplex so that this standard filtration is a
-filtration. We then analyze the E2-term.
To demonstrate the usefulness of our work, we recover W. Singer's
action of Steenrod operations on the Serre spectral sequence, along with
all the other applications he makes in [11 ] and [12 ]. We also extend
A. Bahri's action of Dyer-Lashof operations on the Eilenberg-Moore
spectral sequence associated to the pullback of infinite loop spaces ([1]),
and generalize it to the homology spectral sequence associated to a
cosimplicial infinite loop space. For further applications see [13 ].
In the sequel [14 ] to this paper, we focus on the notion of truncated
Dold algebras which serve to model, for example, the structure associ-
ated to the total complex for the bicomplex coming from a cosimplicial
iterated loop space. We also examine right plane spectral sequences
where we model phenomena such as occurs in [7].
Organization of this paper. In section 2, we review the needed
material about chain complexes, filtrations, and spectral sequences. In
section 3, we stipulate what we mean by an action of operations on
homology and how it should behave in the spectral sequence coming
from a filtration. We then define Dold algebras and -filtrations and
then show that operations behave well in the associated spectral se-
quence. All this is then applied, in section 4, to bicomplexes where we
give conditions so that the total complex is a -filtered Dold algebra.
We close the section by examining the E2-term. Finally, in section 5,
we give our applications.
Acknowledgments. The author would like to thank Haynes Miller
for suggesting this project, as well as his guidance and advice during
the period at M.I.T. this was being worked on, Jim McClure for his
interest in this project and for the proof of 5.11, and Bill Singer for
sharing an early draft of [12 ]. Thanks also to Julie Riddleberger for
converting this document to LaTeX.
2. Preliminaries on Chain Complexes
For simplicity all modules are over F2. By a chain complex we
mean a sequence of F2-modules and maps
@i+1 @i
. .!.Ci+1 -! Ci -! Ci-1 ! . . .
OPERATIONS AND SPECTRAL SEQUENCES I 3
for all integers i such that
(2.1) @i@i+1 = 0; i 2 Z;
and write (C; @) for shorthand (or just C when the boundary maps
@i are understood). As usual, the homology of C of degree i is defined
as
ker @i
(2.2) Hi(C) = ________:
im @i+1
This definition we use in order to capture the classic notion of homology
and cohomology in one (see, e.g., [15 ]). We denote by Ch the category
of chain complexes.
We now recall the definition (see, for example, [8]) of a key chain
complex, denoted W , and review its properties. Let ss be the group
{1; oe : oe2 = 1} and = F2[ss], the group ring of ss over F2. We then
let
(
j 0;
(2.3) Wj =
0 j < 0;
and define @i by setting
(2.4) @iei = (1 + oe)ei-1; i > 0:
W is in fact a differentially graded coalgebra, that is, there is a map of
chain complexes
(2.5) : W ! W W
making W a (graded) associative coalgebra. We define (2.5) by setting
X
(2.6) em = ei oeiej:
i+j=m
Note that is a map of chain complexes over .
Recall that given chain complexes C and C0 their tensor product
C C0 is defined by setting
M
(2.7) (C C0)m = Ci C0j;
i+j=m
X
whose mth boundary map is (@i 1 + 1 @j).
i+j=m
If C and C0 are furthermore chain complexes over we define the
ss-product C C0 to be the chain complex defined by the coequalizer
ss
oe1
(2.8) C C0 C C0 ! C C0:
1oe ss
4 JAMES M. TURNER
An easy calculation shows that W is an acyclic chain complex and
that H0W ~= F2. Let ffl : W ! F2 be the induced augmentation. Then
W is a -free resolution of F2.
When filtering a chain complex C it will be of the form
. . .F sC F s+1C . . .C;
where s 2 Z. Given such a filtration there is an associated spectral
sequence {Er; dr}, bigraded with
F m
(2.9) E0m;t= ______(C)m+t ;
F m-1
with d0 induced by @|Fm C. This shows that
F m
(2.10) E1m;t= Hm+t ______(C) :
F m-1
In general, we can explicitly determine Er as follows: define the bi-
graded module Zr by
(2.11) Zrm;t= {x 2 F mCm+t : @x 2 F m-rC};
it is then a standard exercise (see [15 ]) to show that
Zrm;t
(2.12) Erm;t= ___________________________r-1r-1;
@Zm+r-1;t-r+2 + Zm-1;t+1
and the differential @ on C induces
(2.13) dr : Erm;t! Erm-r;t+r-1:
L
Letting Zrm;*=" t Zrm;tthen Zr+1m;* Zrm;*and Zrm+1;* Zr+1m;*for all r.
Set Z1m;*= Zrm;*. Letting
r
Brm;t= @Zr-1m+r-1;t-r+2+ Zr-1m-1;t+1
[
and B1m;*= Brm;*we have
r
Z1m;*
(2.14) E1m;*= _____
B1m;*
In order to compare the spectral sequence to H*C we define a filtration
{F sH*C} by
i j
C
F sH*C = ker H*C ! H _____ :
F sC
There is then a map
: F mHm+t C ! E1m;t;
OPERATIONS AND SPECTRAL SEQUENCES I 5
which induces m
: _F____(Hm+t C) ! E1 :
F m-1 m;t
In general, we will say that a spectral sequence (Er; dr) is abutting
to H*C if there is a filtration, {F sH*C}, together with a map , in-
ducing . We say such a spectral sequence converges when is an
isomorphism (see [15 ]).
We note that if C and C0 are filtered, then C C0 is filtered by
setting
X
(2.15) F m(C C0) = F iC F jC0:
i+j=m
Finally, recall that a bicomplex is a doubly Z-graded module B
which, for each s, Bs;*and B*;sis a chain complex. We denote by @h
the horizontal differential for B (i.e., @h : Bs;t! Bs-1;t) and by @v the
vertical differential for B (i.e., @v : Bs;t! Bs;t-1). Thus @h@v = @v@h
and @h@h = 0 = @v@v.
Given a bicomplex B, recall that its total complex T (B) is defined
by setting Y
T (B)m = B-i;m+i;
i0
with total differential defined as the formal sum
@T = @h + @v:
We filter T (B) by setting
Y
F -sT (B)m = B-i;m+i;
is
which we call the natural filtration of T (B).
3. Operations, Dold Algebras, and Spectral Sequences
In this section, we focus on chain complexes whose homology pos-
sesses an action of operations. In particular, we give conditions so that
a chain complex has this property. This is done through the notion
of a Dold algebra, the definition of which, and its relationship to op-
erations, is due originally to A. Dold in [6] and generalized by J. P.
May in [8]. Once established, we will be able to give conditions for a
filtration on a Dold algebra so that a compatible well-behaved action
of operations occur on the spectral sequence. This utilizes an approach
pioneered by W. Singer in [11 ] and [12 ].
Definition 3.1. A chain complex C possesses an action of opera-
tions if
6 JAMES M. TURNER
(1) H*C is a graded commutative algebra (not necessarily with unit);
(2) for all integers i there are homomorphisms
Qi : HnC ! Hn+iC
such that
(a) Qi = 0; i < n,
(b) Qnx = x2.
We further call C unstable if, in addition,
(c) Qi = 0; i > 0,
and define, for i 0, the Steenrod operation
Sqi : HnC ! Hn-iC
by setting
Sqi = Q-i:
We next make the
Definition 3.2. A Dold algebra is a chain complex C together with
a map of chain complexes
: W (C C) ! C;
ss
where C C is a -module by having ss act by permutation. We denote
a Dold algebra by (C; ) (or just C when is understood).
Given a Dold algebra (C; ) and m 0 define a map of graded
modules
(3.1) m : C C ! C
of degree m, defined by setting
(3.2) m (x y) = em (x y) :
ss
For m 0, define the set map
(3.3) qm : C ! C
of degree m, by setting, for x 2 Ck,
(3.4) qm (x) = m-k (x x) + m-k+1 (x @x):
An easy exercise shows that
(3.5) @qm = qm @:
Furthermore, one can check that for each k the induced map
(3.6) Qm : HkC ! Hk+m C
OPERATIONS AND SPECTRAL SEQUENCES I 7
is a homomorphism. Since we have a natural map
H*(C) H*(C) ! H*(C C);
given by [u] [v] ! [u v], then 0 induces
(3.7) : H*C H*C ! H*C;
which is a commutative product for H*C. It is now easy to check that
Proposition 3.3. Let (C; ) be a Dold algebra. Then, using (3.6) and
(3.7), C possesses an action of operations. We call this action the
induced action of operations.
We now present a useful context under which Dold algebras arise.
Let E and F be two categories. Given a functor
F : E x E ! F
define its twisting to be the functor
F oe: E x E ! F
given by F oe(C; D) = F (D; C). We then say that the group ss acts on
F if there are natural maps
oe* : F $ F oe: oe*;
such that oe*oe* = 1F and oe*oe* = 1Foe.
Given two functors F; G : E xE ! F on which ss acts, then a natural
map f : F ! G is equivariant if the diagram
f
F -! G
?x ?x
oe*?y??oe* oe*?y??oe*
foe oe
F oe -! G
commutes.
N
Definition 3.4. A triple (E; ; C) is called a complex tensor cat-
egory if E is an abelian tensor category (or symmetric monoidal cat-
egory) with tensor product
O
: E x E ! E
and C : E ! Ch is a fixed functor.
Example Let M be theNcategory of (graded) F2-modules with the
usual tensor product . Then the category of simplicial modulesNsM
becomes a tensor category under the simplicial tensor product where
(V W )s = Vs Ws
8 JAMES M. TURNER
The functor C : sM ! Ch can be chosen to be the normalization
functor, which is an equivalence by the Dold-Kan theorem (see [15 ]).
Unfortunately, C(V W ) and C(V ) C(W ) are only isomorphic after
passing to the homotopy category via the Eilenberg-Zilber theorem.
This theorem was extended by Dold in [6] to extract more information
from this relationship. We now follow Dold's approach for constructing
Dold algebras.
Fix the natural transformation (of graded abelian groups) of degree
q O O
ffq : O(C x C) ! W ( O(C x C));
which is essentially eq 1. Thus ffq@ = (1 @)ffq.
We now make the
Definition 3.5. A homotopyN deviation, associated to a complex
tensor category (E; ; C), is an equivariant chain map
O O
O : W ( O(C x C)) ! C O
where ss acts diagonally on the left, such that:
(1) For each q the natural composite
O ffq
C(-)-q C(-)-q ! ( O(C x C))-2q -!
O O O
(W ( O(C x C)))-q -! (C O )-q
is a monomorphism.
(2) For any i; j the natural composite
O ffq
C(-)-i C(-)-j ! ( O(C x C))-i-j -!
O O O
(W ( O(C x C)))q-i-j -! (C O )q-i-j
is trivial if either q > i or q > j.
LetN(E; ) be a commutative algebroid in a complex tensor category
(E; ; C) possessing a homotopy deviation O. Then we have a chain
map
: W (C(E) C(E)) ! C(E)
ss
which is induced from the composite
O O O C()
[W ( O(C x C))](E; E) -! (C O )(E; E) -! C(E)
OPERATIONS AND SPECTRAL SEQUENCES I 9
since O is equivariant and is commutative. Thus (C(E); ) is a Dold
algebra and
Proposition 3.6. The induced action on (C(E); ) gives C(E) an un-
stable action of operations.
Proof. From above, we just need to check instability, but this follows __
easily from the definitions and (2) of Definition 3.5. |__|
We now turn to spectral sequences. We will assume, for the remain-
der of this paper, that a spectral sequence {Er; dr} will be left plane
spectral sequence, i.e., Ers;*= 0; s > 0. We thus assume that for
any chain complex C, a filtration {F s} must be a left filtration, i.e.,
F sC = 0; s > 0. We now stipulate when operations should behave
well with respect to a left plane spectral sequence.
Definition 3.7. Let C be a chain complex possessing an action of op-
erations and {Er; dr} a left plane spectral sequence abutting to H*C
with induced filtration {F sH*C}. We then call the action of operations
well behaved with respect to {Er; dr} if for any r 2 there exists
homomorphisms
: Er-m;t Er-q;u! Er-m-q;t+u;
Qsv: Er-m;t! Er-m;t+s;
and
Qsh: Er-m;t! Ew-m-t+s;2t;
for some 2r - 2 w r (see [12 ]), such that the following holds for a
fixed x 2 Er-m;t.
(1) Qsvx = 0, for s < t, Qshx = 0, for s < t - m or s > t, and
Qthx = Qtvx.
(2) If x survives to [x] 2 E`; ` r, then Qsvx and Qshx survive to
E` and Qsv[x] = [Qsvx] and Qsh[x] = [Qshx]. Further, if y 2 Er-q;u
survives to E`, then (x y) survives to E` and ([x] [y]) =
[(x y)].
(3) For drx 2 Er-m-r;t+r-1we have that
(a) If y 2 Er-q;usuch that y 6= drx, then
dr(x y) = (drx y) + (x dry);
and
dr(x drx) = Qt-m-1hdrx:
(b) Both Qsvx and Qshdrx survive to EN and
dN Qsvx = Qshdrx
10 JAMES M. TURNER
where
(
2r - 1 + t - s t s t + r - 1;
N =
2r - 1 t - s s t
(c) If t + r - 1 s, then drQsvx = Qsvdrx.
Furthermore, under the map
: F -mHtC ! E1 ;
we have that for x 2 F -mHtC
(4) If y 2 F -qHuC, then (x y) 2 F -m-qHt+uC, and (x y) =
(x y).
(5) For any s, if
(a) t s t+m, then Qsx 2 F -2m-t+sHt+sC and Qsx = Qshx.
(b) t + m s, then Qsx 2 F -mHt+sC and Qsx = Qsvx.
If {Er; dr} is induced by a left filtration {F sC} we call the action of
operations well behaved with respect to {F sC} if it is well behaved
with respect to {Er; dr}.
We now turn to Dold algebras and determine when a filtration is
well behaved with respect to the induced action of operations. First,
note that if C and D are filtered chain complexes over , then we filter
C D by letting F m(C D) be the image of F m(C D). Next we
ss ss
filter W by setting
(
Wq 0 q j;
(F jW )q =
0 otherwise :
Nonetheless, we use this to define F m W (C C) , for any filtered
ss
chain complex C, as before. We then call a Dold algebra (C; ) filtered
if there is a filtration {F mC} such that, for each m; induces
m
: F m W (C C) ! F C:
ss
Unfortunately, this is insufficient for our needs. To accomplish our goal
we define the -filtration of W (C C) by setting
ss
(3.8) F m W (C C)
ss
X
= F iW F m-i(C C) + W F 2m+1(C C):
i<-m-1 ss ss
We then call a filtered Dold algebra (C; ) -filtered if for all m
m
(F m W (C C) ) F C:
ss
OPERATIONS AND SPECTRAL SEQUENCES I 11
We now proceed to prove
Theorem 3.8. Let (C; ) be a -filtered Dold algebra. Then the in-
duced action of operations on C is well behaved with respect to this
filtration.
We first prove
Proposition 3.9. Let (C; ) be a -filtered Dold algebra. Let {Er; dr}
be the associated left plane spectral sequence. Suppose r 2 and x 2
Zr-m;t. Then, if
(1) t - m s t then there is an integer w, with r w 2r - 2,
such that qsx 2 Zw-m-t+s;2tand the correspondence
x ! qsx
passes to a homomorphism
Qsh: Er-m;t! Ew-m-t+s;2t
(2) t s we have qsx 2 Zr-m;t+sand the correspondence
x ! qsx
passes to a homomorphism
Qsv: Er-m;t! Er-m;t+s:
Proof. (1) Since x 2 Zr-m;t, then x x 2 F -2m(C C). We have
es-t+m (x x) 2 F -m-t+s W (C C) , thus s-t+m (x x) is in
ss ss
F -m-t+sC, since C is a filtered Dold algebra. Also, @x 2 F -m-rC,
so s-t+m+1 (x @x) 2 F -m-r-t+s+1C F -m+t+sC; r 1. Thus
qsx 2 F -m-t+sC and by (3.5)
@qsx = s+m-t+1 (@x @x) 2 F -m-2r-t+s+1C
F -m-t+s-w C;
for any 2r - 2 w r, hence qsx 2 Zw-m-t+s;2t. By a similar analysis,
w can be finessed so that qsx 2 Bw-m-t+s;2t. See [12 ] for further details.
12 JAMES M. TURNER
Next, given x; y 2 Zr-m;twe show qs(x + y) - qsx - qsy represents
0 2 Er-m-t+s;2t. Let k = t - m,
qs(x + y) - qsx - qsy = es-k (1 + oe)(x y) + es-k+1 (x @y)
ss ss
+ oees-k+1 (@x y)
ss
= @es-k+1 (x y) + es-k+1 (x @y)
ss ss
+ oees-k+1 (@x y)
ss
= @ es-k+1 (x y)
ss
+ @es-k+1 (@x y)
ss
= @ es-k+1 (x y)
ss
+ es-k+1 (1 + oe)(@x y) :
ss
One can check, using the fact that C is a -filtered Dold algebra, that
the last part of the equation is an element of
@Zr-1-m-t+s+r-1;*+ Zr-1-m-t+s-1;* (r 2);
as required.
(2) This is essentially the same as (1) except that for s t and
x 2 Zr-m;twe have e.g., es-t+m (x x) 2 F -m W (C C) , and
ss ss
so qsx 2 Zr-m;t+s, as before, and the rest of the proof follows the same_
path. |__|
Proof of 3.8.One can easily check that since 0 : C C ! C is a map
of chain complexes, it induces
: Er-m;t Er-q;u! Er-m-q;t+u
from the definitions. Also the existence of Qshand Qsvfollows immedi-
ately from Proposition 3.9. We now confirm the axioms
(1) This is immediate from Proposition 3.9.
(2) This is also immediate from Proposition 3.9 and the above.
(3)(a) This follows from the definitions and the equation
@0(x y) = 0(@x y) + 0(x @y):
OPERATIONS AND SPECTRAL SEQUENCES I 13
(c) Let u 2 Er-m;t, and x 2 Zr-m;trepresent u. Then if x 2 F -mC
and @x 2 F -m-rC. Since t s t + r - 1, then qs@x = s-t+m+1 (@x
@x) is in F -m-t+s-2r+1C = F -m-N C F -m-rC, using the fact that
C is a filtered Dold algebra. By (3.5) and the fact that C is a -
filtered Dold algebra qsx 2 ZN-m;t+s, and so Qsvu survives to EN . Next
observe that Qshdru is represented by qs@x which lies in ZN-m-N;* since
@qs@x = qs@@x = 0. Hence Qshdru survives to EN and (3.5) tells us
that dN Qsvx = Qshdrx.
Cases (b) and (d) are similar.
(4) and (5). Let u 2 F -mH*C. Then we can represent u by
x 2 F -mC such that @x = 0. Thus qsx = s-t(x x), which is in
F -2m-t+sC, for s t + m, and in F -mC for s t + m, since C is
-filtered Dold algebra. Thus this result (and the result for ) follows__
from the definitions. |__|
4. Bicomplex Tensor Categories
In this section, we present a context which gives rise to bicomplexes
whose total complexes are -filtered Dold algebras. We will assume
that all our bicomplexes B . .are left bicomplexes, that is, Bs; .= 0
for s > 0.NWe denoteNby BCh the category of (left) bicomplexes. We
also let and ssbe the obvious generalization to BCh of tensor
product and ss-tensor product that occurs on Ch.
Definition 4.1. (1) A (left) bicomplex tensor category is a triple
N N
(E ; ^ ; B) where (E ; ^ ) is a tensor category, with E abelian, and
B : E ! BCh a fixed functor. N
(2) A bicomplex extension of a complex tensor category (E; ; C)
N
is a bicomplex tensor category (E ; ^ ; B) where E is a subcategory
of grE, the category of graded objects over E, and for s 2 Z we
have
a. For E; E0 2 E then, in E,
(E ^E0)s = Fs(E E0)
where E E0 is the object of bigrE, the category of bigraded
objects over E, such that (E E0)m;t = Em E0tand the func-
tor Fs : bigrE ! E is additive exact together with a natural
monomorphism fs : Fs ! diags, where diags : bigrE ! E is
given by diagsE = Es;s.
b. There is a naturally monic quasi-isomorphism
's : B . ;s! C O prs
where prs : grE ! E is given by prs(A) = As.
14 JAMES M. TURNER
Next we define bicomplexes W hand W vby W*h;0= W = W0v;*and
W*h;s= 0 = Wsv;*; s 6= 0 with the differentials induced by the one on
W . We now make the
N
Definition 4.2. Let (E ; ^ ; B) be a bicomplex tensor category
(1) A Dold complex is a pair (E; ) consisting of an object E 2 E
and a map of bicomplexes
: W v B(E ^E) ! B(E):
ss
(2) A complex homotopy deviation is an equivariant map
O O^
fl : W h ( O(B x B)) ! B O
such that, for each s, the induced map
O O^
W ( O(B x B)) . ;s! (B O ) . ;s
satisfies conditions 1. and 2. of 3.5
(3) If ENis a bicomplex extension of a complex tensor category
(E; ; C), with homotopy deviation O, then a map of bicomplexes
O O^
fl : W h ( O(B x B)) ! B O
is an extension of O if, for each s, the diagram
N fl N
W ( O(B . ;sx B . ;s)) -! (B O ^ ) . ;s
? ?
1('sx's)?y 's?y
N O N
W ( O(C O prs x C O prs)) -! C O O (prs x prs)
commutes, where 's is induced from 's and fs, using 2. of 4.1.
N
Lemma 4.3. Let (EN; ^ ; B) be a bicomplex extension of the complex
tensor category (E; ; C). Then any extension fl of a homotopy devi-
ation O is a complex homotopy deviation.
__
Proof. This is just a diagram chase using the definitions. |__|
Now, given a functor B : E ! BCh define TB : E ! Ch by TB (E) =
T [B(E)].
Next, given A; B left bicomplexes there is a natural inclusion
(4.1) i : T (A) T (B) ! T (A B):
OPERATIONS AND SPECTRAL SEQUENCES I 15
Indeed, for each m; n, i is given by the composite
iY j iY j Y
(4.2) A-j;m+j B-k;m+k ! (A-j;m+j B-k;m+k )
j k j;k
Y
(A B)-`;m+n+`:
`
N
Now let (E ;N^ ; B) be the bicomplex extension of the complex tensor
category (E; ; C) and fl an extension of a homotopy deviation O.
Define the twisting of fl to be the natural map
O O
o (fl) : W ( O(TB x TB )) ! T O (W v (B O ))
which for (E; E0) 2 E x E is defined by the composite
W TB (E) TB (E0) i-! W W T (B(E) B(E0))
OE1 h 0 1T(fl) 0
-! W T (W B(E) B(E )) -! W TB (E E )
OE2 v 0
(4.3) -! T (W B(E E ));
where OE1; OE2 are defined in an obvious way.
Lemma 4.4. For fl an extension of a homotopy deviation O, the twist-
ing o (fl) is equivariant.
Proof. This is immediate from the definition and properties of the var-_
ious maps. |__|
For the rest of this section, we fix a (left) bicomplex extension
N N
(E ; ^ ; B) of the complex tensor category (E; ; C) and fl an exten-
sion of a homotopy deviation O. Let (E; ) be a Dold complex in this
bicomplex tensor category. Then define
(4.4) (fl) : W TB (E) TB (E) ! TB (E)
ss
as induced by the composite
O o(fl) O^
(W ( O(TB x TB )))(E; E) -! (T O (W v (B O )))(E; E)
v T( )
(4.5) - ! T W (B(E) B(E)) -! TB (E);
ss
which is ss-equivariant by Lemma 4.4.
Theorem 4.5. TB (E); (fl) is a -filtered Dold algebra with respect
to the natural filtration.
16 JAMES M. TURNER
Proof. Let B = B(E). Since the filtration is naturally defined, it is
sufficient to show that the twisting induces
-m v
o (fl) : F -m W TB (E) TB (E) ! F T (W B(E ^E))
for all m. Let x y 2 F -qTB (E)s F -rTB (E)t. Then we may assume
x 2 B-i;s+i; i q and y 2 B-j;t+j; j r. Let e 2 Wp, which we may
assume is ep, by equivariance, for which
X
ep = ea oeaeb;
a+b=p
thus, since fl is an extension of O, we can write
X
o (fl)(ep x y) = ea O(oeaeb x y):
a+b=p
By definition, O(oeaeb x y) 2 B(E E)-i-j+b;s+t+i+j, which is
trivial for b > i or b > j by (2) of Definition 3.5, and so is trivial for
q + r + 2b > 2i + 2j. Thus, since i + j - b q + r - p we have that
O(oeaeb x y) is in both F -r-q+pTB (E E) and F -`TB (E ^E), where
r + q
` = least integer greater than_____:
2
Now, if p < m-1, then 2p < m+p-1 < r+q and so 2r+wq-2p > r+q,
i.e., ep x y 2 F -m W TB (E) TB (E) , otherwise p m - 1, for
which ep x y 2 F -m W TB (E) TB (E) by definition, and by
the analysis above we have
o (fl)(ep x y) 2 F -mT (W v B(E E));
since we may assume q + r = m + p, and so q + r 2m - 1, so that __
` m. Conclusion follows. |__|
We now analyze the E2-term of the spectral sequence associated to
a Dold complex. Recall that for any left bicomplex B the E2-term has
the form
(4.6) E2-m;t= Hh-mHvt(B)
where Hv*denotes homology with respect to @h. If E is a Dold complex
and B = B(E), then for each m, B-m;* is a Dold algebra, and therefore
there are chain homomorphisms
(4.7) : Hvq(B) Hvt(B) ! Hvq+t(B);
and for each s,
(4.8) Qs : Hvm(B) ! Hvm+s(B);
OPERATIONS AND SPECTRAL SEQUENCES I 17
which is the induced action of operations on B-m;* for each m. Next
for each t; Ht(B) is an unstable Dold algebra by Proposition 3.6. Thus
(4.7) induces
(4.9) : Hh-mHvt(B) Hh-qHvu(B) ! Hh-m-qHvt+u(B);
(4.8) induces
(4.10) Qs : Hh-mHvt(B) ! Hh-mHvt+s(B)
and (4.7) and h induce
(4.11) Sqs : Hh-mHvt(B) ! Hh-m-sHv2t(B):
Lemma 4.6. For x 2 Hh-mHvt(B) we have
_i i s
SqsQ2 x = Q Sq x
provided Qkv satisfies the Cartan formula on products in Hv*(B).
Proof. Since Qi : Hvt(B) ! Hvt+i(B) is a map of chain complexes, then
representing x by u in Hvt(B) we have that Sqsx is represented by
O(es-m u u) in Hv2t(B). By the assumption we compute, using
naturality, that
X
QiO(es-m u u) = O(es-m Qk u Q`u)
k+`=i
_i _i
= O(es-m Q2 u Q2 u)
X
(4.12) + O es-m (1 + oe) Qku Qi-ku :
2k*< d s i < j
sjdi = id i = j; j + 1
>:
di-1sj i > j + 1
sjsj = si-1sj i > j:
Suppose C = M the category of (possibly graded) F2-modules. Define
N on cM by
(M L)s = Ms Ls
with diagonal cofaces and codegeneracies. We also define the functor
C : cM ! Ch
Xn
by setting C(M)n = M-n and @ : C(M)n+1 ! C(M)n by @ = dj.
j=0
We then define the cohomotopy groups of M by
ssnM = H-n C(M)
for n 0.
20 JAMES M. TURNER
Definition 5.1. A cosimplicial Eilenberg Zilber map {Dk} is a
sequence of natural maps
O O
Dk : O(C x C) ! C O ;
each of degree k such that
(1) Dk = 0; kN< 0; N
(2) D0 = id: O(C x C) 0 ! C O 0;
(3) k 0; @Dk + Dk@ = Dk-1 + oeDk-1oe.
In addition, we call {Dk} special if for each pair M; L in cM,
(4) Dk : C-j(M) C-i(L) ! C-j-i+k(M) x C-j-i+k(L) is trivial if
either k > i or k > j;
(5) Dn : C-n (M) C-n (L) ! C-n (M) x C-n (L) is the identity.
Proposition 5.2. There exists a cosimplicial Eilenberg-Zilber map
{Dk}. In addition, we can choose {Dk} to be special.
__
Proof. This is due to Dold in [6] and an argument of Dwyer in [7]. |__|
We note that for x 2 C-s(M) and y 2 C-t(L) with s + t > 0, then
D0(x y) = ds+t. .d.s+1x ds-1 . .d.0y
in C(M L)-s-t (see [3]). Also, by naturality, if we define
N-sM = Ms \ kers0 \ . .\.kerss-1
as the normalized chain complex, then {Dk} defines
Dk : N-sM N-tL ! N-s-t+k(M L)
for each k. We nowNdefine a cosimplicialNdeviation to be a homotopy
deviation O : WN ( O(C x C)) ! C O for the complex tensor
category (cM; ; C). Given a cosimplicial Eilenberg-Zilber map {Dk}
and M; N in cM define
(5.1) O(eq x y) = Dq(x y);
and
(5.2) O(oeeq x y) = oeDq(y x):
OPERATIONS AND SPECTRAL SEQUENCES I 21
Then one can check from the definitions that
@O(eq x y) = Dq-1(x y) + oeDq-1(y x) + Dq@(x y)
= O(eq-1 x y) + O(oeeq-1 x y)
+O eq @(x y)
= O((1 + oe)eq-1 x y) + O eq @(x y)
= O(@eq x y) + O eq @(x y)
= O@(eq x y):
Thus O is a map of chain complexes and therefore a cosimplicial devi-
ation provided {Dk} is special. Conversely, given a cosimplicial devi-
ation O (5.1) and (5.2) defines a special cosimplicial Eilenberg-Zilber
map. We sum this up in
Proposition 5.3. (5.1) and (5.2) determines a one-to-one correspon-
dence 8 9
ae oe < special cosimplicial=
cosimplicial
Eilenberg-Zilber :
deviations ! : ;
maps
Next, let X be a bicosimplicial object over M, i.e., for each s, X.;s
and Xs;.are cosimplicial objects over M and the vertical operators
commute with the horizontal operators. Let {Dkh} be a cosimplicial
Eilenberg-Zilber maps in the horizontal direction and {Dkv} a cosim-
plicial Eilenberg-Zilber map in the vertical direction.N Let bicM be
the category of bicosimplicialNmodules and let ^ and B the obvious
generalizations of and C to this category.
N N
Lemma 5.4. (bicM; ^ ; B) is a bicomplex extension of (cM; ; C).
Proposition 5.5. Suppose X is a bicosimplicial commutativeNF2-al-
gebra. Then {Dkv} makes X a Dold complex in (bicM; ^ ; B). Fur-
thermore, the obvious extension of the homotopy deviation associated
to {Dkh} makes TB (X) a -filtered Dold algebra and, at the E2-term,
the operation
h v
Qs : Hh-mHvtB(X) ! H-m Ht+s B(X)
is unstable and satisfies the Adem and Cartan relations, as also does
h v
Sqs : Hh-mHvtB(X) ! H-m-s H2t B(X) :
Proof. This follows immediately from Proposition 3.6 and Theorem 4.5, __
Theorem 4.7, and the results of [6]. |__|
22 JAMES M. TURNER
Corollary 5.6. (see [11 ] and [12 ]) Suppose Z is a bisimplicial cocom-
mutative F2-coalgebra and N an F2-module. Then the third quadrant
spectral sequence {Er; dr} converging to H*(diag Z; N) possesses, for
each r, a differential graded algebra structure and an action of Steenrod
operations
Sqsh: Er-m;-t! Er-m+t-s;-2t;
and
Sqsv: Er-m;-t! Er-m;-t-s;
satisfying all the usual relations and compatible at E1 with the induced
filtration on H*(diag Z; N).
__
Proof. Apply Proposition 5.5 to Hom(Z; N). |__|
This has various topological applications gotten by starting with a
bisimplicial set Y and letting Z be the free bisimplicial module on
Y . The diagonal Y ! Y x Y induces a cocommutative coalgebra
structure on Z. In this context, Corollary 5.6 applies to the Serre
spectral sequence and the bar spectral sequence. For details on this
and other applications, see [11 ] and [12 ].
Next, let A be a simplicial object over M. Let NA be the Moore
complex of A, i.e.,
NsA = As \ kerd1 \ . .\.kerds;
and @ = d0. For example, the chain complex W is a chain homotopy
equivalent to N[F2Ess] where F2Ess is the free simplicial module on Ess
(see [8]).
Consider now the category J of infinite loop spaces (where space =
simplicial set). By [9], the infinite loop structure on an object X in J
is determined by, among other things, a map : Ess x (X x X) ! X.
ss
By the Eilenberg-Zilber theorem (see [15 ]), there is a natural chain
homotopy equivalence
W N(F2X) N(F2X) ! N F2(Ess x X x X) ;
which induces
W N(F2X) N(F2X) ! N F2 Ess x (X x X) :
ss ss
Now, define an E21 - algebra in sM to be a pair (M; ) consisting
of a simplicial module M and a map of simplicial modules
: F2(Ess) (M M) ! M
ss
It is clear that for X 2 J then F2[X] is an E21- algebra.
OPERATIONS AND SPECTRAL SEQUENCES I 23
Lemma 5.7. For each M an E21- algebra; N(M) is a Dold algebra.
As a consequence, for X in J ; N(F2X) is a Dold algebra.
N N
Now consider (csM; ^ ; B), where ^ is the obvious generalization
of tensor product from cM and sM, and
B : csM ! BCh
is defined by B(M) = C [N(M)], where C : cCh ! BCh is the pro-
longation of C.
N N
Lemma 5.8. (csM; ^ ; B) is a bicomplex extension of (cM; ; C).
Furthermore, any cosimplicial deviation induces an extension through
the simplicial Eilenberg-Zilber map.
Corollary 5.9. Given a cosimplicial object Z over J then F2(Z) is a
Dold complex in csM and TB (F2Z) is a -filtered Dold algebra.
__
Proof. This follows immediately from 5.8 and 4.5. |__|
We now have enough to prove
Theorem 5.10. Let X. be a cosimplicial infinite loop space and
{Er; dr} the second quadrant spectral sequence associated to the total
complex of B(F2X). Then H*(X.; F2) is a cosimplicial graded commu-
tative F2-algebra with a compatible action of Dyer-Lashof operations
(see [5]), and so ss*H*(X.; F2) is a bigraded commutative algebra pos-
sessing an action of the Dyer-Lashof operation
Qs : ssm Ht(X.; F2) ! ssm+s H2t(X.; F2);
satisfying Adem, Cartan, instability, etc., for all s, and the relations
of Lemma 4.6. Under the identification E2-m;t= ssm Ht(X.; F2), this
product and action of operations is well behaved, in the sense of Defi-
nition 3.7, with the spectral sequence {Er; dr}.
Proof. The action of Dyer-Lashof operations follows from [5] and nat-
urality. By Proposition 3.6 and Proposition 5.3 the action of Steenrod
operations occurs and is independent of choice of cosimplicial Eilenberg-
Zilber map as shown in [6]. The relationship to the spectral sequence_
follows from Corollary 5.9, Theorem 4.5, and Theorem 4.7. |__|
Next, recall that given a cosimplicial space X. there is an associated
total space (see [4]) Tot(X.) which for q 0,
Tot(X.)q = Hom (.x [q]; X.);
where . is the standard cosimplicial space with m = [m]. From
[2] there is a filtration F -sH* Tot (X.); F2 , together with natural
24 JAMES M. TURNER
maps
: F -sH* Tot (X.); F2 ! F -sH*TB (F2X.):
Proposition 5.11. If X. is a cosimplicial infinite loop space, then
Tot(X.) is an infinite loop space, and for x 2 F -mHt Tot(X.); F2
if
(1) t s t + m, then Qsx 2 F -2m-t+sHt+s Tot(X.); F2 and
(Qsx) = Qsh(x).
(2) t + m s, then Qsx 2 F -mHt+s Tot(X.); F2 and (Qsx) =
Qsv(x).
Proof (sketch).The following proof is due to J. McClure in [10 ]. Let
D2 : s S ! s S
be the functor D2X = Ess x (X x X). By [5] it is enough to compute
ss
H* D2 Tot(X.) = H* Tot (D2X.) from the spectral sequence. From
section 5 of [3] there is a cosimplicial space A(s;t)for each t s 0
which is universal for elements of the E1 -term of the homotopy spectral
sequence associated to a cosimplicial space. From their results we have
(
F2 l = s; m = t;
(5.3) E2-l;m(A(s;t)) =
0 otherwise
and from [2] this spectral sequence converges to show that Tot(A(s;t)) =
St-s up to 2-completion. Thus Hn Tot (D2As;t) = F2 for n 2(t - s)
(see [5]). From the universality of A(s;t)we are reduced to showing that
8
>:
0 otherwise
but this is a straightforward computation utilizing the definition of __
A(s;t). |__|
As an application, let
M -! E
? ?
? ?
y py
f
X -! B
be a fibre square of infinite loop spaces and maps. Then the geometric
cobar construction B is a cosimplicial infinite loop space such that
Bs = X x B x . .x.B x E
OPERATIONS AND SPECTRAL SEQUENCES I 25
with s copies of B in this product. The resulting homology spectral
sequence is the Eilenberg-Moore spectral sequence (see e.g. [2])
E2-m;t= CotormH*B(H*X; H*E)t =) Ht-m M
As a consequence of 5.10 we have operations
Qi : CotormH*B(H*X; H*E)t ! CotormH*B(H*X; H*E)t+i
and
Sqi : CotormH*B(H*X; H*E)t ! Cotorm+iH*B(H*X; H*E)2t
which is well behaved in the spectral sequence and is compatable with
the action of Dyer-Lashof operations on H*M, as described by 5.11.
This generalizes the results of [1].
References
[1]A. Bahri, Operations in the second quadrant Eilenberg-Moore spectral se-
quence J. Pure and Appl. Alg. 27(1983), 207-222
[2]A. K. Bousfield, On the homology spectral sequence of a cosimplicial space,
Amer. J. of Math. 109(1987), 361-394.
[3]A. K. Bousfield and D. M. Kan, A second quadrant homotopy spectral se-
quence, Trans. A.M.S. 177(1973), 305-318
[4]___________, Homotopy Limits, Completions, and Localizations, Lecture
Notes in Mathematics 304, Springer-Verlag, 1972.
[5]F. Cohen, T. Lada, and J. May, The Homology of Iterated Loop Spaces, Lecture
Notes in Mathematics 533, Springer-Verlag.
[6]A. Dold, U"ber die Steenrodschen kohomologie operationen, Ann. of Math.
73(1961), 258-294.
[7]W. Dwyer, Higher divided squares in second quadrant spectral sequences,
Trans. A.M.S. 260(1980), 437-447.
[8]J. P. May, A general algebraic approach to Steenrod operations, The Steenrod
Algebra and its Applications, Lecture Notes in Mathematics 168, Springer-
Verlag (1970), 153-231.
[9]___________, The Geometry of Iterated Loop Spaces, Lecture Notes in Math-
ematics 271, Springer-Verlag, 1972.
[10]J. McClure, Private communication, November 1993.
[11]W. Singer, Steenrod squares in spectral sequences I, II, Trans. A.M.S.
175(1973), 327-336, 337-353.
[12]___________, Steenrod squares in spectral sequences: the cohomology of Hopf
algebra extensions and of classifying spaces, preprint, Fordham University
(1997)
[13]J. Turner, Looping Bousfield-Kan Towers, in preparation
[14]___________, Operations and spectral sequences II, III, in preparation
[15]C. Weibel, An Introduction to Homological Algebra, Cambridge Studies in Ad-
vanced Mathematics 38, Cambridge University Press, 1995.
26 JAMES M. TURNER
Department of Mathematics, College of the Holy Cross, One Col-
lege Street, Worcester, MA 01610-2395
E-mail address: jmturner@math.holycross.edu
*