TRANSACTIONS OF THE
AMERICANMATHEMATICALSOCIETY
Volume 177, March 1973
A SECONDQUADRANTHOMOTOPYSPECTRAL SEQUENCE
BY
A. K. BOUSFIELD AND D. M. KAN*1)
ABSTRACT. For each cosimplicial
simplicial
set with basepoint,
the
authors construct
a homotopv
soectral
sequence
generalizing
the usual spectral sequence
for a second quadrant
double chain complex.
For such homotopy
spectral
sequences,
a uniqueness
theorem and a general
multiplicative
pairing are established.
This machinery
is used elsewhere
to show the equivalence of various unstable
Adams spectral
sequences
and to construct
for them
certain
composition
pairings
and Whitehead
products.
1. Introduction.
(i)
The purpose
to show that there
of a cosimplicial
of this paper
is such
simplicial
is
a thing as "the"
set with base
(homotopy)
point (resp.
spectral
cosimplicial
sequence
simplicial
group), and
(ii)
to construct
for these
Our prime example
[l] by a different
method.
for that spectral
sequence
We start
abelian
sequences
unstable
smash
a natural
Adams spectral
The above results
with observing
groups
complexes
spectral
is "the
will be applied
and composition
pairings
(in § 2) that the category
is equivalent
with the category
and then devote
the next
pairing.
sequence"
(§3—§8)
in
in [2] to establish
and Whitehead
of cosimplicial
of second
six sections
obtained
quadrant
products.
simplicial
double
to formulating
chain
and
proving
I.
note
Let
X be a cosimplicial
its normalized
homotopy
simplicial
groups,
set with base
Then
there
is (in a sense
sequence
which
n Xs de-
...
n ker ss~l.
will be made precise
in §7) a unique
natural
{E X\ with
F*'fX=
n't Xs
= 0
Received
and let
i.e.
7r'tXs = ntXsnkets0n
spectral
point
by the editors
February
fort>s>0,
otherwise,
29, 1972.
AMS(MOS)subject classifications (1970). Primary 18G30, 18G40, 55J10, 55H05.
Key words
and phrases.
Homotopy
spectral
sequence,
simplicial
set,
cosimplicial
object.
(1) This research was partially supported by NSF GP 8885 and NSF GP 23122.
Copyright © 1973, American Mathematical Society
305
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306
A. K. BOUSFIELD AND D. M. KAN
which,
with
if X is a cosimplicial
the spectral
I . Let
which
simplicial
sequence
abelian
of the associated
B be a cosimplicial
will be made precise
group,
second
simplicial
group.
in §4) a unique
natural
which,
if B is abelian,
associated
double
chain
The remainder
II.
Let
coincides
coincides
quadrant
Then
for t > s > 0
double chain complex.
there
spectral
EA'B = 77,'B5 for t>s>
= 0
[March
is (in a sense
sequence
¡F B} with
0,
otherwise,
for t > s > 0 with the spectral
sequence
of the
complex.
of the paper
is devoted
X and Y be cosimplicial
to proving
simplicial
sets
with base
point.
Then
the
pairing
rt't Xs A n't,Xs' X n[\s + s' A n't,\s^'
— n't+t'(*s + s' hYs + s') =<+/.(X
(where
/ denotes
induces
a pairing
a graded
version
of spectral
of the cosimplicial
r
The definition
sequences,
that
all involve
Our notation
r
for the group
of the spectral
the demonstration
map)
— Es +s'<i+t' (X A Y).
r
result
Alexander-Whitney
sequences
Es'lX A Es''l'Y
II . A similar
AYf +*'
the above
certain
v
case.
sequences,
pairing
universal
and conventions
the proof of their uniqueness
of Ej-terms
examples
for cosimplicial
induces
a pairing
which are introduced
objects
and
of spectral
in §5.
will be as in [l].
2. Preliminaries.
2.1.
The
normalization
of a simplicial
[4], [5] that for a simplicial
complex,
trivial
abelian
in dimension
(i)
group
abelian
map induced
(ii)
< 0, defined
in the following
...
by the "remaining"
NnA = An/(ims0
We recall
A the normalization
N72A = A72 nketd.n 1
with boundary
group.
nketd
N A is the chain
two equivalent
manners:
72
face operator
© •••
from
dQ, or
©imsn_j)
with boundary map given by d = 2n_Q ( - lYd^. N A —>N _ ,A.
Both definitions
over,
ian
the functor
groups
N
vield
and of abelian
Dualizing
the same identification
is an equivalence
chain
between
complexes
which
H N A = n A, 1 > 0.
<?*<?
—
the categories
are trivial
this we get
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of simplicial
in dimension
Moreabel< 0.
19731
A SECOND QUADRANT HOMOTOPYSPECTRAL SEQUENCE
2.2.
abelian
The normalization
group
of a cosimplicial
A we define
in dimension
and coboundary
N
groups
and of abelian
between
cochain
A nonabelian
(but not necessarily
need
Finally
N"~lA —>NnA.
are trivial
group;
Clearly,
of cosimplicial
2.2 also makes
abelian)
normalization
sense
the
abelian
in dimension
in that case,
of a cosimplicial
simplicial
sense
abelian
to talk about
normalization
simplicial
trivial
< 0.
when applied
however,
to
the co-
we consider
The double
it makes
double
complex,
not be homomorphisms.
If A is a cosimplicial
hence
For a cosimplicial
nkers""1
which
Definition
boundary
2.4.
group.
A as the cochain
the categories
complexes
remark.
a cosimplicial
maps
n ...
map given by 8 = X"_0 (- l)'dl:
is an equivalence
2.3.
N
< 0, with
NnA = A" Okers0
functor
abelian
its normalization
307
simplicial
its
group,
then
"double"
functor
is an equivalence
abelian
groups
simplicial
abelian
N N ^A = N N A, and
normalization.
between
and of second
group.
Clearly,
the categories
quadrant
double
this
of co-
chain
com-
plexes.
3.
The spectral
spectral
sequence
defined
sequence
as the spectral
N N t A, i.e.,
of a cosimplicial
ÍF AS of a cosimplicial
sequence
of its double
one forms the total
T A=
1
chain
©
this
by putting
abelian
abelian
(TA, dJ)
group.
group
normalization
complex
(2.4)
given
The
A can be
N N
A -
by
NSN,A,
t-s=q
t
>
dTb =8b+ (- lfdb
filters
simplicial
simplicial
FSTA = ©.
tot b £ NsNt\,
/V'/V^A,
and then takes
the associated
spec-
tral sequence.
For our purposes,
3.1.
however,
A more explicit
we need
definition.
homotopy group, i.e. the subgroup
77,AS, and define
df: n(As
"relations",
-> rrt
to c by diagram
1As+r,
chasing,
Denote
i.e.
not always
r > 1, by putting
i.e.
by u As C n As the normalized
n( NSA = H(Nt NSA = NsHtN:): A = Nsn(A C
whenever
db
there
defined
multivalued
= c whenever
are elements
functions
one can get from b
b. £NS+'N
.A
(0 —
< z < r) such that bnu £ b, 8b r— ,1 £ c and Sb.z— ,1 = (- l)s+i + ldb.z (0 < i < r).
One readily
(i)
verifies
naturality:
that these
relations
if b £ntAs
and
if /: A —>B is a cosimplicial
(ii)
additivity: J
have the following
c en'
_,As+r
properties:
ate such
that- d b = c and
map, then dj^b = f c,
if b, b ' £ 77,'AS
and c, c ' e n't
t
,AS+T ate such that d r b =
+r— I
c and d b = c ', then d ib - b') = c - c ',
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308
A. K. BOUSFIELD AND D. M. KAN
2^
[March
(iii)
dx is the function
(- l)z'a^: rr/A5 — a¡\s+1,
(iv)
the domain of definition of d (r > l) is the kernel of d _.:
if b £ 7t'As,
then d b = c for some c if and only if d _ ,b = 0,
(v)
the indeterminacy
of dr (r > l) is the image
of d _,:
if b £n'As,
then
d 0 = b if and only if d _ ,a = b fot some a,
(vi)
the d r are "differentials"
"
: if b £ 77,As
and c £n, t +r-1 ,AS+T ate such
Z
that d r b = c, then d r c = 0.
In view
of these
properties
one now can obtain
the spectral
sequence
\E A}
by putting
e^a = 7t;v,
Es/A
= (n'tAs n keralr_1)/(77;Asnima'r_1),
r> 1,
where
ker ri _ . = \b\ d _ ,b = 0},
and defining
induced
the differentials
by the relations
d : E5,'A
The augmented
spectral
that
case.
all this
i.e.
define
CFq+1\CFqtKC
such
Es,i..A C Es,tA,
fot r > s,
= I >T>sE*'tA.
a natural
..-
generalized
[l, §9],
in
then one can augment
filtration
C F°A = n (A~ 1
with homomorphisms
F«A%,E£t+«A,
that
tained
A as the homomorphisms
has
can be slightly
If A is augmented
sequence,
...
together
one clearly
is given by E^'A
We end with observing
the above
—*E^+r,i+''-
d . Moreover
and hence the E^-term
3.2.
im d _ ^ = \c\ d _ .b - c fot some b\
?>o,
ker e q = Fq+
A,' as follows: Let e q : 77t A- —^ 77.'
t
t+q Aq denote the relation obe a = b whenever
there are elements
a_ , £ N A"
and a. £
by putting
NtZ+z. + ,NlA
(0 —
< i < q)
such that a -1 , £ a, 8a q- ,1 £ b and Sa.z-1 . = (- l)z +1da.2
l
'
(0 < i < q).
As in 3.1, these
relations
have
the properties:
(i) naturality,
(ii) additivity,
(iii)
eQ is /¿e function
(iv)
i/be domain
(v)
the indeterminacy
(vi)
the image
d^: ntA~
of definition
of e
of e
contains
of e
—» 77,A = 77^
,
(q > O) is the kernel
of e
.,
(q > O) z's the image of d ,
only "infinite
cycles",
i.e.
it a £ rt A~
and
b £ 77,
Aq ate such that e q a = b, then d r b - 0 for all r.
Z+<7
In view of these
properties
we can now put
F'A
= 77A-
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n ker e
., q > 0,
1973]
A SECOND QUADRANT HOMOTOPY SPECTRAL SEQUENCE
309
eq
and observe
that each
q > 0, whose
4.
relation
kernel
is exactly
The spectral
sequence
late our first
result
groups
is a unique
there
which,
e q then induces
of a cosimplicial
roughly
speaking,
(natural)
simplicial
spectral
To be precise
4.1.
of § 3.
Theorem.
Let
77'Bs = Nsn B C 77 B5.
FfA
i
—>F*'i+9A,
oo
there
group.
We now formu-
that for cosimplicial
sequence
abelian
generalizing
groups
simplicial
the one of §3,
coincides
with (part of) the
we have
B be an augmented
Then
simplicial
states
spectral
i.e. which for cosimplicial
sequence
a homomorphism
r
F*+ A.
cosimplicial
are unique
simplicial
group and let
relations
dA<Qs^n't+r_fis",
r>l,t>s>0,
eq:np-l^n'uW,
q>0,t>0,
which
(i)
sist
have
properties
on the additivity
(i) through
of d
tot
(vi) of 3.1 and 3.2 (except
that we do not in-
t = s = 0, r> 1; but we instead
770'ß° — Z7r'_jB' to be "crossed-additive"
require
d :
for r >.2, i.e. if b. b ' £ tzJB0 and c,
c £ 77 ,Br are such that db = c and djb = c , then d (b - b ') = c - (b - b ')c '
where the left action
morphism
(ii)
of ker d. C ?70B
on 77
ker d. —» <70Br and the left action
for abelian
B, coincide
,Br
is induced
of z70Br on 77
and
by the natural
,Br),
with the relations
d
e
of an (augmented)
cosimplicial
homo-
and
of § 3.
This allows us to define
4.2.
The spectral
The spectral
sequence
sequence
sequence
obtained
iErBS
of a cosimplicial
simplicial
group
simplicial
group.
B is the spectral
by putting
f^íb
= ít;bs,
/>5>o,
= 0,
otherwise,
Es/B = (F^B
n ketdr_l)/(E\'tB
n imz^j),
ESttB= fl F*.'B
00
with,
as differentials,
the relations
*
the homomorphisms
d : Es,tB
—» f? +r,z+r-lß
¡ncjucecj Dy
d r of 4.1.
Furthermore
sequence
r>s
if B is augmented,
by putting
lations
e
exactly
F^ + 1B.
F*B = ^B"
of 4.1 then induce
then,
1 O ker e
homomorphisms
as in 3.2, we can augment
this spectral
., q > 0, and observing
that the re-
F*B
—>F9,i+9B
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whose
kernel
is
7
310
A. K. BOUSFIELD AND D. M. KAN
Clearly
if B is abelian
this
(augmented)
t - s > 0 with the one of § 3, justifying
spectral
[March
sequence
coincides
for
the use of the same notation.
Also, [2, §5], readily implies
4.3.
Corollary.
The above
(augmented)
spectral
sequence
is naturally
equiv-
alent with the one of [2, § ll].
4.4.
Remark.
The above
spectral
sequence
is "fringed"
in dimension
the sense of [l, § 4], i.e. EB = H(Ef_ jB, df_ A except possibly
0 in
when t - s = 0,
when one may have E B C H(E _ .B, d _ .) instead.
The proof of Theorem 4.1 will be given in §6.
5.
Universal
cosimplicial
r
amples
(§6)
examples
objects
'
for the d
D,{r,s,t) . and
and
and will also
5.1.
interpreted
(ii)
the existence
D,
,.
we discuss
of pairings
certain
ex-
point
where
each
point
(for
4.1
(§ 10).
For r > 1 and s > 0, D,
set with base
D"
. will be
. is obtained
A[n] D, p. 14] by
the (s - l)-skeleton
as "taking
section
are the main tool in the proof of Theorem
example
simplex
collapsing
and e . In this
E,(q,t) ,. which will turn out to be universal
to prove
simplicial
from the standard
(i)
e . They
be used
The universal
the cosimplicial
for the d
the disjoint
to a base
union
with a base
s = 0 this
point"),
should
be
and
taking the (s + r - l)-skeleton,
and where the cofaces
and codegeneracies
are induced
by the standard
maps [5, p. 14]
cr.
Aln -I] A AW,
For
t > s > 0, D,
of [l, §9] of the (t - s)-fold
Note
that
this
definition
<5-la>
Let
Z be the "free
functor [5, p. 124].
also
sense
makes
by the simplices
5.2.
Properties
r
nondegenerate
(ii)
Proof.
point
the reduced
homotopy
This
(in the sense
if r = <*>and that
= D(~.s+,.i + r)on"
point
functor
[l],
i.e.
X the simplicial
of X with the base
the functor
(free)
which
abelian
point (and its degeneracies)
group
put
Then we have
of ZD.(°a,s.t) .. (i) 7r,ZDf_
Z
(aa.s.t)
t-simplex
72-72'
ZD*
group
set with base
to the identity.
0 < 2 < 72.
, by application
suspension
abelian
to a simplicial
generated
base
from D,
D<~,S.yD(W)
assigns
equal
AU + 1] A A[n],
, then is obtained
of Df^
. * Z and is generated
°
by
the
'
.,
. = 0 otherwise.
follows
X there
(integral)
readily
is a natural
homology,
type of a wedge
from the fact
isomorphism
that
for any simplicial
rr^ZX « H^X,
and the observation
where
that each
of ¡"-spheres.
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set
f/t
D. x
with
denotes
. has the
1973]
A SECOND QUADRANTHOMOTOPYSPECTRAL SEQUENCE
5.3.
Properties
degenerate
t-simplex
(ii)
77'
of ZD.
.. (i) Z7ZDf
of Df
.ZDf+r
311
. * Z and is generated
by the non-
s ty
. * Z and is generated
by the collapsing
map A[t + r] —»
rjs+r
ir.s.t)'
(iii)
n'ZDf
(iv)
the differential
.1=0
otherwise,
d
Fs<t - 77' 7Ï15
fcr
Proof.
is a simple
Milnor's
to a simplicial
by the simplices
identity,
- 77'
er
parts
follow
"free
group on"
^í
readily
ZDs+r
+ r-l¿U(7,S,í)'
trom 5.2 and (5.1a)
set with base
To do this
point
of X with the base
then we can for the
we need
functor
[3J, i.e.
X the simplicial
point
FD,
the notion
f l¡fu + f 2*u where
the functor
(free)
properties
j y f2, fl2:
consisting
to the
to 5.2 and 5.3.
For K e 5\* and n >
1 let
—
of all
FK —>Fièvre)
zz £ n FK
such
that
are the homomorphisms
/. 7*u =
which respec-
x in the zth K).
to show that
K —> FK
induces
a homomorphism
(ii) for
anyJ K, L £ S* , rfn F(KVL) * nn FK®n
'
. n tt'FD)
\ T1 S t I )
71
77
n
n K—> "n FK,
and
FL. If we rput nn'FDk( r.s.t)
. then we can formulate
\ T1 S r I J
5.5. Properties of FD(r ,„
fl«<s(z's generated
(ii)
as-
of
the subgroup
the inclusion
tt FD~
71
which
put equal
similar
tively send x £K to x., x 2, x yX2 £ F(KVK) (with xy denoting
(i)
part (iv)
group generated
(and its degeneracies)
. formulate
5.4. The additive subgroup
of 77'FD,
,,.
0
r
n
( r.s.t)
rf F K C 77 F K denote
It is easy
while
calculation.
If F denotes
signs
three
,, z's an isomorphism
JL F-y+r'i+''-1
- 77ZZU(r1s,z)^'
The first
and
in E ZD,
for t > 0. (i) fr/FDf,#5#()= w/FDff>,it) * Z
by the nondegenerate
_ jFDf+r
t-simplex
. * Z arzz^ z's generated
of Dj?
.;
by the collapsing
map Alt + r] —*
(r.s.t)
Proof.
wedge
This
of /-spheres
5.6.
^fjFD^o
is a unique
(ii)
¿ft
easily
the fact
n Q. for r > 2.
0) module.
function
using
that
D,
( r,s,
, is equivalent
t)
^
to a
it + r - l)-spheres.
of FD,
h: "^FD^
¿ z's fl crossed
X €D(r,0,0)
("i)
and
Properties
as a left
(i)
follows
A straightforward
computation
q q) —» ?7r'_ ,FD'
q
homomorphism,
= £,%h where
As in 4.1 (i) we view
cf: FD(r
Q
i.e.
n'r _ ¡FD^
then shows
d,8* = £*^r "^ere
i?: ^(0,0)
that there
szzcè z/W
hix + y) = bx + xihy),
—» FD(r
Q Q z's the homomorphism
*»*> x2 eFD(r,0,0)'
—» ZD(r 0 0) z's the abelianization.
We end with considering
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.
sending
312
A. K. BOUSFIELD AND D. M. KAN
5.7.
augmented
tained
The universal
example
cosimplicial
simplicial
from the standard
(i)
collapsing
E,
to a base
.. For q > 0 and t - 0, E.
set with base
simplex
[March
point
where
each
„. will be the
E"
. is ob-
A\_n + l] by
point
the image
of A[n] under the map <50: A[n] —*
A[t2 + l], and
(ii)
taking the »-skeleton,
and where
the operators
15?' and s'
Aln] -^Abi + 1],
For
t > 0, E,
pension
ate induced
by the standard
A[t2+ 2] ^Aln
. is then obtained
from
E,
+ l],
maps
0 < z< n.
„, by application
of the r-fold
sus-
functor.
One readily
5.8.
proves
Properties
r
degenerate
of ZE, (<j,z)..
t-simplex
(ii) 77
the following
of E?
ZE?
analogues
(i)
v '
of 5.3 and 5.5.
7r,ZE7
generated
z
(q.t) ,n * Z ««a7 z's 6
by' the non-
.,
. * Z and is generated
by the collapsing
map Alt + q + l] —*
Eq(q.t)'
(iii)
77n'ZE, (q,t) ,. = 0 otherwise,
(iv)
the homomorphism
e
and
(3.2)
is an isomorphism
Fq=n,ZE7l.e^Eq>t+q=7T'
Z
5.9.
Properties
degenerate
6.
time
Proof
that
(q.tYs,
ZE«
oo
(i)
ET
the
of Theorem
t+q
n.PET
6.1.
indeed
Definition
ntF^(r,s,t)'
of the
.
(q,t)'
, « Z and is generated
by the non-
and
by the collapsing
We now prove
Theorem
To do this
we define
D,(r.s.t) ,. and
all the desired
map A[f + q + l]
4.1 and show at the same
E,(q,t) .. of §ü 5 are universal
theorem.
4.1 in terms
have
4.1.
D,(.r.s.t) ,, and
e q of this
Theorem
tions
of
,.
¡s
77t+qEE?(.q.t)* Z and is generated
Eq
d r and
(q,t)
of EE,
t-simplex
(ii)
t
examples
r
for the relations
the relations
E.(q.t) ,. and then
d r and
show
that
e q of
these
rela-
properties.
of the d r in terms
of the
denote by / e nr- 1^(7,0,0)
D,(r.s.t)
..
Choose
the element
a generator
b
i £
d°A - dll if r - 1 and the
element hi (see 5.6) if r > 2, and for t > 0 denote by' '/ £ rf,t +r-1 ,EDf+r
. the
(r,s,t)
(in view of 5.5) unique
ZD,
element
such
. is the abelianization.
elements
b £ rr Bs
t
and
c £ rr
is a homomorphism f: ED.
Similarly
that
d^g^i
= g^j
For any cosimplicial
t+r-
,Bs+r
1
we then
put
r
where
g: FD.
simplicial
. —>
group
B and
d b = c if and only
r
'
. —» B such that f i = b and f%j = c.
we have
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*
if there
•
1973]
A SECOND QUADRANT HOMOTOPY SPECTRAL SEQUENCE
6.2.
Definition
77 PET
that
of the
n and denote
e g^i = g^j.
ments
in terms
and
b £ n[t
homomorphism f: FE.
+q
definitions
verifies
cosimplicial
Choose
a generator
simplicial
do not depend
that the relations
they are uniquely
3.2 (iv) are equivalent
to the ones
..
i £
element
group
such
B and ele-
Bq we then rput e q a = b if' and only' if' there
3.1(i), (ii) (modified as in 4.1(i)),
similar
E,
. the (in view of 5.9) unique
is a
. —►
B such that f^i = a and ftj = b.
both these
One readily
versely,
of the
FE?
For any augmented
a £ 77t BClearly
e
by / £ n
313
above
of the generator
have properties
i.
4.1 (ii),
and (iii) and 3.2 (i), (ii) and (iii) and that, con-
determined
by these
to the following
used
on the choice
defined
in proving
properties.
lemma
that
which
Properties
can be proved
a simplicial
group
3.1 (iv) and
by calculations
satisfies
the extension
condition dI6.3.
Lemma.
momorphism
r
Let
B be an augmented
,
'f: FD, (r-l,s,z)
.. —» B (resp.r
to a homomorphism FD.
. —>B (resp. FE,
The proof of Properties
erty 3.1 (vi) follows
^(r.s,z)=
directly
6.6.
Lemma.
Proof.
E,
by naturality
from this
Let
A ho-
. —»B) if and only if f j = 0.
And finally
from the fact that (ignoring
and Lemma
same
group.
6-6 below,
Prop-
the augmentation)
while
Property
3.2(vi)
lemma.
77n'FE^(q,t) ,, = 0 'for n - k < t.
d be the smallest
. has the homotopy
integer
type of a wedge
ton-Milnor
theorem
such
77n FE, \q,t) , = TH q+t ,E, Kq.t) ..
that
simplicial
. —» B) then can be extended
3.1 (v) and 3.2 (v) is not hard.
^(A-+r.z-s-l/E(s.z—s-l)
follows
cosimplicial
'f: FE, (4-1.f),
[3], a functor
such
that
T from free abelian
hence, by [4, p. 2331? ^„'^E,
n < id + l)iq
of iq + ¿)-spheres,
Moreover
this
there
groups
functor
+ t).
As each
is, by the Hil-
to abelian
is of degree
6
groups
< d and
—
. = 0 for k > dq. The desired result now follows
readily.
7.
The spectral
Our next result
also
is that,
zs a unique
precise,
(natural)
of a cosimplicial
for cosimplicial
spectral
simplicial
simplicial
sequence
sets
set with base
with base
point,
generalizing
the one of § 3.
cosimplicial
simplicial
point.
there
To be
we have
7.1.
point
sequence
Theorem.
and let
Let
X be an augmented
n Xs = Nsn X C 77 Xs.
dr:«'tXS-*"'t
Then
there
+ r-lXS + T>
are unique
'>!»
'>*>0,
which
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relations
set with base
314
A. K. BOUSFIELD AND D. M. KAN
(i)
sist
d
have
properties
on the additivity
(i) through
of d
(vi) of 3.1 and 3.2 (except
(ii)
for X an augmented
we can define
7.2.
The spectral
spectral
point.
cosimplicial
simplicial
sequence
such
group,
coincide
with
simplicial
as in 4.2.
set with base
point,
set
then its
of the spectral
orem 4.1, the above
otherwise,
And, as in 4.2, an augmentation
of X induces
an aug-
sequence.
// X z's a?2 augmented
justifying
simplicial
t>s>0,
= 0,
can be defined
cosimplicial
ÍE X} with
Es1'tX = 77,tXs,
§4,
abelian
of an (augmented)
If X is a cosimplicial
sequence
mentation
we require
as in 4.1 (i)), and
d r and e q of' § 3.
Thus
with base
that we do not in-
tot t = l, s = 0, r > 1; but instead
for r> 2 to be "crossed-additive",
the relations
[March
cosimplicial
spectral
sequence
simplicial
group,
coincides
then,
in view of The-
for t —s > 0 with the one of
the use of the same notation.
Again [2, § 51 readily implies
7.3.
Corollary.
The above
(augmented)
spectral
sequence
is naturally
equiv-
alent with the one of \2, § 4L
7.4.
Remark.
The
above
spectral
sequence
is "fringed"
in dimension
1
with the one of [2, v 4], i.e. E X = H(Et_ jX, d _ A except possibly when t - s =
1,' when one may' have E r X C H(E r— ,X,
d r— ,)
instead.
1
1
8.
Equivalence
G and the classifying
of the results
functor
8.1.
Proposition.
Theorems
8.2.
Proposition.
Let
boundary
Using
4.1 and 7.1 imply each
B be an augmented
and let d: n Z+l,WBS * n t Bs denote
"other"
of § 4 and § 7.
the loop group
functor
W oJ we will show
other.
cosimplicial
the boundary s isomorphism
c
simplicial
ofi [5].
group
Then the
isomorphisms
d0t = (- lYd:vt+lWB*mw]P
induce
an augmented
spectral
sequence
isomorphism,
i.e.
isomorphisms
<9„ :
E-s.'
+ l WB * ES'(B,
dn : Fq ,WB * FqB
which commute with the d r and e q .
r
rOzZ+1
Z
8.3.
base
point
Proposition.
and let
Let
d: n
X be an augmented
.Xs « tt GXS denote
cosimplicial
the boundary
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simplicial
set with
isomorphism
of [5].
1973]
A SECOND QUADRANT HOMOTOPY SPECTRAL SEQUENCE
Then the
"other"
boundary
315
isomorphisms
d0t = i-lYd:nt+1X**ntGX*
induce
an augmented
spectral
sequence
isomorphism,
ß . ps,t+ lv Ä ns,tr\
°or
r
A
r °
which
commute
These
G and
with
the
d r and
propositions
follow
readily
"other"
boundary
_
which
cosimplicial
isomorphisms
d
01
. „
properties
t+1
w\*
// B z's an augmented
relations
on the
77 Bs and
"other"
boundary
~ „ \s
t
the
on the
"other"
n
boundary
Remark
with
D,(r.s.t) . and
slight
problem
the analogue
places
9.
pairing
set
Theorem
E,(q,t) ,, instead
because
of 6.3
Xs need
However,
(the abelian
case).
0/
(of § 3).
and
e
(vi),
relations
are
then
the
on the
7.1.
point
d
and
e
Xs « 77 GXS induce
(vi),
relations
3.1 and 3.2 (i) through (vi).
Instead
, and
of reducing
using
these
it to The-
the arguments
FE, (q.t) ,,.
the extension
both
and
3.1 and 3.2 (i) through
7.1 directly
of FD, (r.s.t)
not satisfy
be true.
<9 •
r
and e
with base
dQ : 77
on the proof of Theorem
isomorphisms
induce
satisfying
which also satisfy
orem 4.1, one can try to prove
induce
3.1 and 3.2 (i) through (vi).
77 .X"
isomorphisms
then the
group and d
d„ : 77 .WBS ~ ^.B5
and
group,
3.1 and 3.2 (i) through
cosimplicial
.Xs
on the 77'GXS and n GX~
8.4.
satisfying
abelian
d
simplicial
WB~ l which also satisfy
// X z's an augmented
are relations
then
77 B
isomorphisms
77' WBS and n
(iii)
cosimplicial
of the functors
are not hard to verify.
simplicial
77' WAS « 77' As which commute with the relations
(ii)
\s¡FqGX
ríÜA
Oz-rz+lA
from the basic
observations
If A is an augmented
. pq
isomorphisms
eq.
W [5] and the following
(i)
ß
'
i.e.
One then
condition,
problems
of § 6
runs into a
nor need
disappear
if one re-
X by WGX.
Pairings
of spectral
(or equivalently
existence
sequences
(2.4)
of a similar
for augmented
for augmented
second
quadrant
pairing
cosimplicial
We end this
for augmented
sets
with recalling
cosimplicial
double
simplicial
paper
chain
cosimplicial
with base
simplicial
complexes)
point.
the usual
abelian
groups
and showing
simplicial
This
groups
section
the
and
will
deal with
9.1.
The abelian
group case.
For any two simplicial
B we denote by 77(AA 77 ' B —»77
abelian
groups
A and
' (A ® B) the pairing induced by the Eilen-
berg-Zilber map U, p. 217] N^ A ® N^ B -^ /V^A ® B), and recall that it is
(i) bilinear,
(ii)
associative,
1
(iii)
commutative
(with sign
(- l)n
1
), i.e.
t Au A v) = (- l)"
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iv A u) for
316
A. K. BOUSFIELD AND D. M. KAN
u £ n A, v £ ir i B and f. A®ß
For two augmented
can define
—»B ® A the twisting
cosimplicial
simplicial
map.
abelian
groups
„'A* An'(lBs'A\„'t+tl(\®B)s
as the map induced
*
nA'^'tn.B'^'tn
■ t
/ is the (graded)
*■
Alexander-Whitney
f(u, v) = ((- l)tsV
A straightforward
calculation
+ s'
Theorem.
The pairing
is augmented,
linear,
I
t+ t
,(A ® B)s + S'
map [4, p. 217J given
+ *' . .- ds+ 1u, ds~l...
then yields
Es'lA A£s'-!'B
which
B one then
by the composition
77A* A 77 ,B*' ¿
9.3.
A and
a pairing
(9-2)
where
[March
(9.2)
the following
induces
a pairing
and commutative
d°v).
"well-known"
result.
of spectral
L\Es + s'-t+i'(A ® B),
associative
by
sequences
1< r < «,
(with sign
(- i)tI—■s'i' ""^ z),
i.e.
(i)
(ii)
the pairing
on E . is the pairing
for u £Esr'lA
dr(u Av)=
(iii)
the pairing
ing on Ex
(iv)
on E
is induced
the pairing
(9.2),
and y £ E* '■l 'ß
.is
(l < r < ~),
(du A !/)+(induced
by the ones
D'-5(zz A¿rw),
by the one on E
on the
on EM is compatible
E
(l < r < oo) and the pair-
(l < r < °°),
with the augmentations,
i.e.,
it u £
E?A
® B) and e q+q , (u A v) = e qu A e q ,v,
t and v £ Fq\
t i B, then u A v £ Fq+q'(A
t+t'
(v)
the pairing
is bilinear,
(vi)
the pairing
is associative,
(vii)
the pairing
is commutative
9.4.
A slight
refinement.
(with
Let
and all simplices
of the form (ds+s
v £ Bs
> 0.
and
s, s
Then
A ®
sign
(- i)(f_5')(i
_s
B C A ® B consist
)) for r > 2.
of the augmentation
■■ ■ ds + lu ® ds~ l ■ ■■ d°v)
the pairing
(9.2)
can clearly
where
be factored
u £ As,
through
pairing
tt'A*
Aff.B*'Á'
tt't + t ,(A®'Br +*'
Z
f
and hence
9.5.
one has
Theorem.
The pairing
of 9.4 induces
a pairing
e*-'a ae^i'b^e;+s''í+('(A®'B),
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of spectral
i<r<°o,
sequences
a
1973]
A SECOND QUADRANTHOMOTOPYSPECTRAL SEQUENCE
which
is augmented,
10.
Pairings
10.1.
point
with base
Y we denote
such
diagram
and associative
(the other cases).
The sets
X and
pairing
bilinear
that,
First
(commutativity
X and
does not make sense).
we consider
point case.
For any two simplicial
by n X A tt( , Y —> n
whenever
317
sets
with base
,(X A Y) the unique
Y ate simplicial
abelian
groups,
natural
the following
commutes:
7TÍ+ Í,(X
A V)
ntX A nt, Y
where
p(x, y) = x ® y for all
(i)
linear
(ii)
in the first
nt+tAX®
Y)
(x, y) £ X A V. Clearly
this
(second)
variable
whenever
pairing
is
t > 0 (t ' > O),
associative,
(iii)
commutative
(with sign
For two augmented
this pairing
induces,
(- l)lt
cosimplicial
as in (9.2),
(10.2)
)•
simplicial
sets
^'X* A tt;,Ys' ÍAn't+t,(X
which,
as in 9.4,
can be factored
(10.3)
X A
10.4.
Y C X A Y has
Theorem.
point
X and
Y
through
¡\Y)S +S'
a pairing
n'tXs A vYs'^7r;+,,(X
where
with base
a pairing
the obvious
The pairings
>\'Y)s + s'
meaning,
(10.2)
and one has,
and (10.3)
induce
as in 9.3 and 9.5,
pairings
of spectral
sequences
Esr~'X A E^'»''Y
-^Esr+s'-t+t'(X
Es,tX A, EAz'y.^.Es
which
are augmented,
commutative
Proof.
the first
(with
ond pairing.
bilinear
and associative.
(-
-s
sign
l)'i —s''i
The commutativity
pairing
+ s',z+í'(X
follow
And the latter,
in turn,
l <r<oo,
A.Y))
l<r<oo,
Moreover
the first
pairing
is
' )•
is easy.
immediately
AY),
The existence
and other
from the existence
is a ready
properties
and properties
consequence
of
of the sec-
of Theorem
9.5 and
the observations:
(i)
it suffices
"
and X=E(9>/y
(8.4)
to consider
only the cases
J
X = D,
Y= E(9,((),
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(r.s.t)
,,, Y = D.
(r,s
(
,t
,.
)
A. K. BOUSFIELD AND D. M. KAN
318
(ii) in the first case the obvious map (X A' \)k ~* Z(X A' Y)fe =
(ZX
® ZY)
77.
.
t+t
+T-1
induces
, when
(iii)
an isomorphism
k = s + s
in the second
of n
¡ when
k = s + s ' and of
+ r,
case
when k = - 1 and of1 n,t+q
this
obvious
map induces
an isomorphism
of 77
,
. +q , when k = q
+ q '.
11
+t
We end with
10.5.
The group case.
77(B A 77i, C —>nt
For any two simplicial
1FÍB A C) the composite
77(BA nt,cAnt
where
groups
B and
C we denote
by
pairing
+t,(B A C) -^nt +t,F(B A C)
B A C —-»FÍB A C) is the obvious
map.
Clearly
the pairing
nas properties
10.1 (i), (ii) and (iii).
For two augmented
duces,
as in (10.2)
cosimplicial
and (10.3),
simplicial
groups
B and
C this
pairing
in-
pairines
(10.6)
n[Bs A n't,CsAn't +tlF(B A C)s +S',
(10.7)
n'tBs An't<Cs A+rr'^^FiB
A'C)S +S',
and one has, as in 10.4,
10.8.
Theorem.
The pairings
(10.6)
and (10.7)
induce
pairings
of spectral
sequences
Es/B A Ef't'C
-E;+s'''t,,F(B
Fs,tB rXFA'-t'C ^.Es + s''t + t'F(B
which
are augmented,
commutative
(with
The proof
bilinear
sign
and associative.
i- l)(i_s'('
~s
to that
of Theorem
is similar
AC),
l < , < 00,
A'C),
K r < 00,
Moreover,
the first
pairing
is
' ).
10.4.
BIBLIOGRAPHY
1.
A. K. Bousfield
coefficients
and D. M. Kan,
in a ring, Topology
2.-,
Pairings
Tzze homotopy
spectral
sequence
of a space
with
11 (1972), 79-106.
and products
in the homotopy
spectral
sequence,
Trans.
Amer.
Math. Soc. 177 (1973), 319-343.
3. E. B. Curtis,
4.
A. Dold
Simplicial
and D. Puppe,
homotopy theory, Advances
Homologie
nicht-additiver
in Math. 6 (1971),
Funktoren.
107-209.
Anwendungen,
Ann.
Inst. Fourier (Grenoble) 11 (1961), 201-312. MR 27 #186.
5.
J. P. May, Simplicial
no. 11, Van Nostrand,
objects
Princeton,
in algebraic
topology,
Van
Nostrand
Math.
Studies,
N. J., 1967. MR 36 #5942.
6. E. H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR. 35 #1007.
DEPARTMENT OF MATHEMATICS, BRANDEIS UNIVERSITY, WALTHAM,MASSACHUSETTS
02154
DEPARTMENT OF MATHEMATICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY, CAMBRIDGE, MASSACHUSETTS02139 (Current address of D. M. Kan)
Current
at Chicago
address
Circle,
(A. K. Bousfield):
Chicago,
Illinois
Department
of Marhematics,
60680
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University
of Illinois