MULTIPLICATIVE STRUCTURES ON HOMOTOPY SPECTRAL
SEQUENCES II
DANIEL DUGGER
Contents
1. Introduction 1
2. Sign conventions in singular cohomology 1
3. Spectral sequences for filtered spaces 3
4. The Postnikov/Whitehead spectral sequence 8
5. Bockstein spectral sequences 13
6. The homotopyfixedpoint spectral sequence 15
7. Spectral sequences from open coverings 16
References 19
1.Introduction
This short paper is a companion to [D1 ]. Here the main results of that paper
are used to establish multiplicative structures on a few standard spectral sequ*
*ences.
The applications consist of (a) applying [D1 , Theorem 6.1] to obtain a pairing*
* of
spectral sequences, and (b) identifying the pairing on the E1 or E2term with
something familiar, like a pairing of singular cohomology groups. Most of the
arguments are straightforward, but there are subtleties that appear from time to
time.
Originally the aim was just to record a careful treatment of pairings on Post
nikov/Whitehead towers, but in the end other examples have been included because
it made sense to do so. These examples are just the ones that I personally have
needed to use at some point over the years, and so of course it is a very limit*
*ed
selection.
In this paper all the notation and conventions of [D1 ] remain in force. In p*
*articu
lar, the reader is referred to [D1 , Appendix C] for our standing assumptions a*
*bout
the category of spaces and spectra, and for basic results about signs for bound
ary maps. The symbol ^_ denotes the derived functor of ^, and W? denotes an
`augmented tower' as in [D1 , Section 6]. Ho (, ) denotes maps in the homotopy
category of Spectra. If A is a pointed space we will write F(A, X) as an abbrev*
*ia
tion for F( 1 A, X). Finally, the phrase `globally isomorphic' is often used in*
* the
identification of E2terms of spectral sequences. It is explained in Remark 3.5.
2.Sign conventions in singular cohomology
If E is a ring spectrum and X is a space, then the AtiyahHirzebruch spectral
sequence Ep,q2= Hp(X; Eq) ) Ep+q(X) is multiplicative. The naive guess about
1
2 DANIEL DUGGER
what this means is that there is an isomorphism of bigraded rings
p,qEp,q2(X) ~= p,qHp(X; Eq),
where the products on the righthandside are the usual ones
~: Hp(X; Eq) Hs(X; Et) ! Hp+s(X; Eq+t)
induced by the pairings Eq Et ! Eq+t. Unfortunately, this statement is just
not true in general_one has to add an appropriate sign into the definition of ~,
and these signs cannot be made to go away. In order to keep track of such signs
in a simple way, it's useful to reevaluate the `standard' conventions about si*
*ngular
cohomology. I'm grateful to Jim McClure for conversations about these sign issu*
*es.
Let X be a CWcomplex and C*(X) be its associated cellular chain complex.
In most algebraic topology textbooks the corresponding cellular cochain complex
is defined by Cp(X) = Hom (Cp(X), Z) and
(ffiff)(c) = ff(@c), for any ff 2 Cp(X).
The cupproduct of a pcochain ff and a qcochain fi is defined by the formula
(ff [ fi)(c d) = ff(c) . fi(d)
where c is a pchain and d is a qchain. (Note that we have written the above
formula as if it were an external cupproduct, so we technically need to throw *
*in a
diagonal map somewhere_we omit this to simplify the typography). Both of these
formulas obviously violate the Koszul sign rule: we will abandon them and inste*
*ad
define
(2.1) (ffiff)(c) = (1)pff(@c) and (ff [ fi)(c d) = (1)qpff(c) . fi(*
*d).
The first equation may seem to have an unexpected minus sign, but here is the e*
*x
planation. Recall that if A* and B* are chain complexes then there is an associ*
*ated
chain complex Hom (A, B). Our definition of ffi corresponds to the differential*
* on the
chain complex Hom (C*(X), Z[0]), where Z[0] is the complex with Z concentrated
in dimension 0.
The sign conventions from (2.1) appear in [Do ]. We'll of course use these sa*
*me
conventions for cohomology with coefficients, external cup products, and any si*
*milar
construction we encounter.
Exercise 2.1. Check that ffi is a derivation with respect to the cupproduct, a*
*nd
that the dga C*(X; Z) defined via our new formulas is isomorphic to the dga
C*classical(X; Z) defined via the old formulas. In particular, our singular coh*
*omology
ring H*(X; Z) is isomorphic to the classical one.
2.2. Cohomology with graded coefficients. If A* is a graded ring we next
want to define the singular cohomology ring with graded coefficients H*grd(X; A*
*),
making use of the natural sign conventions. It would be nice to just use the i*
*n
ternal hom for chain complexes Hom (C*(X), A), where A is interpreted as having
zero differential, but unfortunately this might give us infinite products in pl*
*aces
we don't really want them. Instead we'll consider a certain subcomplex. We set
Cp,q(X; A) = Hom (Cp(X); Aq) and Cngrd(X; A) = pq=nCp,q(X; A)_that is, ele
ments of Cp,q(X; A) are regarded as having total degree p  q. For ff 2 Cngrd(X*
*; A)
we define ffiff by the formula
(2.2) (ffiff)(c) = (1)nff(@c).
MULTIPLICATIVE STRUCTURES ON HOMOTOPY SPECTRAL SEQUENCES II 3
The homology of this complex will be denoted H*grd(X; A); it has a natural dire*
*ct
sum decomposition into groups Hp,q(X; A).
The graded cupproduct will be defined on the chain complex C*grd(X; A) as
follows: if ff 2 Cp,q(X; A) and fi 2 Cs,t(X; A) then
(ff [ fi)(c d) = (1)(st)pff(c) . fi(d)
(where c is a pchain and d is a qchain). The sign is again just the one dicta*
*ted
by the usual Koszul convention, and ffi becomes a derivation with respect to th*
*is
product.
If C* D* ! E* is a pairing of graded abelian groups, one also has an extern*
*al
graded cupproduct H*grd(X; C) H*grd(Y ; D) ! H*grd(XxY ; E) defined in a simil*
*ar
fashion. It is this product which arises naturally in pairings of spectral sequ*
*ences.
Exercise 2.3. Construct a bigraded family of isomorphisms jp,q:Hp,q(Z; A) !
Hp(Z; Aq), natural in both Z and A, which makes the diagrams
Hp,q(X; C) Hs,t(Y ;_D)__//Hp+s,q+t(X x Y ; E)
j j j 
fflffl fflffl
Hp(X; Cq) Hs(Y ; Dt)____//_Hp+s(X x Y ; Eq+t)
commute up to the sign (1)sq. Here Hn(Z; Am ) denotes singular cohomology
with coefficients in Am as defined via the formulas in (2.1), and the bottom ma*
*p is
the cupproduct pairing associated to Cq Dt ! Eq+t (again with the signs from
(2.1)).
Exercise 2.4. Repeat the above exercise, but this time show that the isomorphis*
*ms
jp,qcan be chosen to make the squares commute up to the sign (1)pt. Convince
yourself that it is not possible to choose the jp,q's so that the squares commu*
*te on
the nose.
3.Spectral sequences for filtered spaces
In this section we treat the AtiyahHirzebruch spectral sequence, the Serre s*
*pec
tral sequence, and spectral sequences coming from geometric realizations. Some
other references for the former are [K ], [GM , Appendix B], [V ]. For the Se*
*rre
spectral sequence see [K ], [Mc , Chap. 5], [Sp, Chap. 9.4], and [Wh , XIII.8].
3.1. Generalities. Suppose given a sequence of cofibrations ; æ A0 æ A1 æ . . .
and let A denote the colimit. If ; æ B0 æ B1 æ . .æ.B is another sequence of
cofibrations, we may form the product sequence whose nth term is
[
(A x B)n = (Aix Bj).
i+j=n
This is a sequence of cofibrations whose colimit is A x B. Given a fibrant spec*
*trum
E, one can look at the induced tower
. .!.F(A2+, E) ! F(A1+, E) ! F(A0+, E)
and identify the homotopy fibers as F (An=An1, E). This is a limtower rather
than a colimtower, and is not convenient for seeing multiplicative structures;*
* one
doesn't have reasonable pairings F (Ak, E) ^ F(Bn, E) ! F ((A x B)k+n, E), for
instance. Instead we have to use a slightly different tower.
4 DANIEL DUGGER
The cofiber sequences An=An1 ,! A=An1 ! A=An induce rigid homotopy
fiber sequences F (A=An, E) ! F (A=An1, E) ! F (An=An1, E). We define an
augmented colimtower by setting W(A, E)n = F (A=An1, E) and B(A, E)n =
F(An=An1, E). The associated spectral sequence E*(A, E) might be called the
Espectral sequence for the filtered space A.
Exercise 3.2. Verify that the tower W(A, E) is weakly equivalent to the tower
F(A*, E) via a canonical zigzag of towers. So the homotopy spectral sequences
can be identified.
Now assume that E had a multiplication E ^ E ! E. Then for any two pointed
spaces X and Y we have the map
F(X, E) ^ F(Y, E) ! F(X ^ Y, E ^ E) ! F(X ^ Y, E).
Using this, the obvious maps of spaces
(A x B)=(A x B)q+t1! A=Aq1 ^ B=Bt1, and
(A x B)q+t=(A x B)q+t1! Aq=Aq1 ^ Bt=Bt1
give pairings W(A, E)^W(B, E) ! W(AxB, E) and B(A, E)^B(B, E) ! B(AxB, E)
which are compatible with the maps in the towers. So [D1 , Thm 6.1] gives us a
pairing of spectral sequences E*(A, E) E*(B, E) ! E*(A x B, E). This is the
`formal' part of the construction.
3.3. The AtiyahHirzebruch spectral sequence. Now we specialize to where
A and B are CWcomplexes which are filtered by their skeleta. In this case we c*
*an
identify the E1 and E2terms, and we will need to be very explicit about how we
do this. For convenience we take A and B to be labelled CWcomplexes, meaning
that they come with a chosen indexing of their cells. Let Iq be the indexing se*
*t for
the qcells in A, and let Cq(A) = oe2IqZ.
Recall that
Ep,q1= ßpF (Aq=Aq1, E) ~=Ho(Sp ^ Aq=Aq1, E).
An element oe 2 Iq specifies a map Sq ! Aq=Aq1 which we will also call oe. Giv*
*en
an element f 2 Ep,q1= Ho (Sp ^ Aq=Aq1, E), restricting to each oe specifies an
element in Ho(Sp ^ Sq, E). We therefore get a cochain in Hom (Cq(A), ßp+qE) =
Cq,p+qgrd(A; E*), and we'll choose this assignment for our isomorphism
Ep,q1~=Cq,p+qgrd(A; E*).
Note that this isomorphism is completely natural with respect to maps of labell*
*ed
CWcomplexes.
We claim that the d1differential corresponds under this isomorphism to the
differential on C*grddefined in Section 2. By naturality (applied a couple of t*
*imes)
it suffices to check this when A is the CWcomplex with Aq1 = *, Aq = Sq, and
Ak = Dq+1 for k q + 1. In this case our d1 is the boundary map in the long ex*
*act
homotopy sequence of F(Sq+1, E) ! F(Dq+1, E) ! F(Sq, E), which takes the form
Ho(Sp^Sq, E) = Ho(Sp, F(Sq, E)) @!Ho (Sp1, F(Sq+1, E)) = Ho(Sp1^Sq+1, E).
We know from [D1 , C.6(d)] that the composite is (1)p1 times the canonical ma*
*p.
Via our identification with cochains, we are looking at a map Cq(A; ßp+qE) !
Cq+1(A; ßp+qE), and the sign (1)p1 is precisely the one for the coboundary ffi
defined in (2.2).
MULTIPLICATIVE STRUCTURES ON HOMOTOPY SPECTRAL SEQUENCES II 5
In a moment we will identify the pairing on E2terms, but before that we make
a brief remark on the case A = B. The diagonal map A ! A x A is homotopic
to a map 0 which preserves the cellular filtration, and so 0 induces a map of
towers W? (A x A, E) ! W? (A, E). Composing this with our above pairing gives
W? (A, E) ^ W? (A, E) ! W? (A, E), and so we get a multiplicative structure on *
*the
spectral sequence E*(A, E).
Theorem 3.4 (Multiplicativity of the AtiyahHirzebruch spectral sequence). There
is a natural pairing of spectral sequences E*(A, E) E*(B, E) ! E*(A x B, E)
together with natural isomorphisms p,qEp,q2(?, E) ~= p,qHq(?, Epq) (for ? =
A, B, A x B) which make the diagrams
Ep,q2(A, E) Es,t2(B,_E)____//_Ep+s,q+t2(A x B, E)
 
 
fflffl fflffl
Hq(A; Epq) Ht(B; Est)___//Hq+t(A x B; Epqst)
commute, where the bottom map is the graded cup product from Section 2.
In the diagonal case, there is a natural isomorphism of rings p,qEp,q2(A, E)*
* ~=
p,qHq(A; Epq), where the latter is again given the graded cup product.
Remark 3.5. Rather than repeat the above statement for every multiplicative
spectral sequence we come across, we'll just say that the E2term is globally i*
*so
morphic to the graded cup product (that is, they are naturally isomorphic as
pairings of bigraded abelian groups).
Proof.We have done everything except identify the product. The pairing on E1
terms is the map
ßpF (Aq=Aq1, E) ^ ßsF (Bt=Bt1, E) ! ßp+sF (Aq=Aq1 ^ Bt=Bt1, E).
Recall that this sends ff: Sp ^ Aq=Aq1 ! E and fi :Ss ^ Bt=Bt1 ! E to the
composite
fffi :Sp ^ Ss ^ Aq=Aq1^ Bt=Bt1! Sp ^ Aq=Aq1^ Ss ^ Bt=Bt1! E ^ E ! E.
Choosing a qcell oe of A yields a map Sq ! Aq=Aq1, and a tcell ` of B gives a
map St ! Bt=Bt1. Under our identification with cochains, the `value' of fffi on
the cell oe ^ ` is the restriction of fffi to Sp ^ Ss ^ Sq ^ St.
If, on the other hand, we compute ff(oe).fi(`) in the ring ß*E, we get the co*
*mposite
[Sp ^ Sq] ^ [Ss ^ St] ! [Sp ^ Aq=Aq1] ^ [Ss ^ Bt=Bt1] ! E ^ E ! E.
By inspection, this differs from (fffi)(oe ^ `) by the sign (1)sq, which is th*
*e same
sign that was used in defining the graded cup product from section 2.2 (remember
that under our isomorphism ff lies in Cq,p+q(A; E*) and fi lies in Ct,s+t(B; E**
*)).
Remark 3.6. In the square from the statement of the theorem, the bottom map is
(1)t(p+q)times the `ungraded' cup product on cohomology induced by the pairing
Epq Est ! Epqst. This follows from Exercise 2.3. The signs are easy to
remember, because they follow the usual conventions: The index `t' is commuted
across the index `(p + q)', and as a result the sign (1)t(p+q)is picked up (*
*the
minus sign can of course be left off the exponent). Note that for most of the f*
*amiliar
cohomology theories, like Ktheory or complex cobordism, the signs end up being
irrelevant because the coefficient groups are concentrated in even dimensions.
6 DANIEL DUGGER
3.7. The Serre spectral sequence. Let p: X ! B be a fibration with fiber
F , where B is a pointed, connected CWcomplex. Let B0 B1 . . .be the
skeletal filtration of B, and define Xi = p1Bi. We'll assume that the inclusio*
*ns
Xi,! Xi+1are cofibrations between cofibrant objects, and consider the augmented
tower Wn = F(X=Xn1, HZ), Bn = F(Xn=Xn1, HZ). The associated homotopy
spectral sequence E*(X) is the Serre spectral sequence for the fibration.
It is easy to see that there is a natural identification
` h i
Xn=Xn1 ~= p1enff=p1@(enff) ,
ff
where the wedge ranges over the ncells enffof B. The interior of a cell en is *
*just
the interior of Dn, so we can take a closed disk around the origin with radius *
*1_2_
call this smaller disk U. Then ~H*(p1en=p1@(en)) may be canonically identified
with ~H*(p1U=p1@U), and we are better off than before because @U is actually a
sphere (rather than just the image of one). The diagram
p1(0)__~___//_p1U
fflffl __99______
~ _~_________
fflffl______fflfflfflffl
U x p1(0)_____////_U
has a lifting as shown, and this lifting will be a weak equivalence. It restric*
*ts to a
weak equivalence @U x p1(0) ! p1(@U) (because this is a map of fibrations over
@U, and it is a weak equivalence on all fibers). Therefore we have the diagram
*oo_____p1(@U) //_____//_p1(U)
 OO OO
 ~ ~
  
  
*oo___@U x p1(0) //__//U x p1(0)
and this necessarily induces a weak equivalence on the pushouts. In this way we
get an identification
~Hk(p1U=p1(@U)) ~=~Hk([U=@U]^p1(0)+ ) ~=~Hk(Sn^p1(0)+ ) ~=Hkn (p1(0)).
Of course the first isomorphism depended on the lifting ~, and so is not canoni*
*cal.
We refer to [Mc , Chap. 5] for a detailed discussion of local coefficient sys*
*tems and
their use in this particular context. But once the right definitions are in pla*
*ce the
argument we gave for the AtiyahHirzebruch spectral sequence in the last section
adapts verbatim to naturally identify the (E1, d1)complex as
Ep,q1~=ßpF (Xq=Xq1, HZ) ~=Cq(B; Hp+q(F ))
where H*(F ) denotes the appropriate system of coefficients. The differential o*
*n the
cochain complex is still the one from section 2, appropriate for cellular cohom*
*ology
with graded coefficients.
Now suppose X0! B0is another fibration satisfying the same basic assumptions
as X ! B. We give B x B0 the product cellular filtration, and then pull it back
to get a corresponding filtration of X x X0. This coincides with the product of
the filtrations on X and X0, and so we get a pairing of spectral sequences by t*
*he
discussion in section 3.1. The identification of the pairing with the graded c*
*up
product again follows exactly as for the AtiyahHirzebruch spectral sequence.
MULTIPLICATIVE STRUCTURES ON HOMOTOPY SPECTRAL SEQUENCES II 7
Theorem 3.8 (Multiplicativity of the Serre spectral sequence). There is a natur*
*al
pairing of Serre spectral sequences E*(X) E*(X0) ! E*(X x X0) such that the
pairing of E2terms is globally isomorphic to the graded cup product on singular
cohomology with local coefficients.
3.9. Spectral sequences for simplicial objects. Filtered spaces also arise in t*
*he
context of geometric realizations. Let X* be a Reedy cofibrant simplicial space*
*, in
which case the skeletal filtration of the realization X is a sequence of cofi*
*brations.
There is a resulting tower of rigid homotopy cofiber sequences with Wq(X*, E) =
F(X= skq1X, E) and Bq(X*, E) = F(skqX= skq1X, E).
If Y* is another Reedy cofibrant simplicial space, we can equip X x Y  wi*
*th
the product filtration. We also have the product simplicial space X*xY*, equipp*
*ed
with its skeletal filtration. There is a natural map j :X x Y  ! X x Y , *
*and
this is actually an isomorphism (using adjointness arguments one reduces to the
case where X and Y are the simplicial sets m and n, where it is (T2) from [D1*
* ,
Appendix C]). Unfortunately j does not preserve the filtrations, as can be seen*
* by
taking X and Y both to be the simplicial set 1 (regarded as a discrete simplic*
*ial
space). The product filtration on  1 x  1 is smaller than the skeletal filt*
*ration
coming from  1 x 1.
The formal machinery of section 3.1 gives a pairing from E*(X, E) and
E*(Y , E) to the spectral sequence for the product filtration on XxY _let*
*'s call
this E*(XxY , E). Often one would like to have a pairing into E*(X xY , E*
*), but
this doesn't seem to follow from our basic results. Here are two ways around th*
*is.
One can replace E*(X, E) with the homotopy spectral sequence for the cosimpli
cial spectrum [n] 7! F(Xn, E), and similarly for E*(Y , E) and E*(X xY , E)*
*. The
paper [BK ] proves that if M* ^ N* ! Q* is a (levelwise) pairing of cosimplici*
*al
spaces, then there is an associated pairing of spectral sequences_this gives us*
* what
we wanted. Having not checked the details in [BK ], I can say nothing more about
this approach; their results clearly depend on more than the formal theorems of
[D1 ], but I couldn't tell from their paper exactly what the important ingredie*
*nt is.
Here is another approach which sometimes works. While j :X xY  ! XxY 
does not preserve the filtrations, j1 is filtrationpreserving (by functoriali*
*ty and
adjointness arguments it suffices to check this when X and Y are the simplicial
sets n and m ). So j1 induces a map of spectral sequences E*(X x Y , E) !
E*(X x Y , E). We have the following:
Proposition 3.10. If X* and Y* are simplicial sets, the natural map of spectral
sequences E*(X x Y , E) ! E*(X x Y , E) is an isomorphism on E2terms.
Proof.This follows from the work in section 3.3, since it identifies both E1te*
*rms
as cellular chain complexes computing H*(X x Y , E*), but for different CW
decompositions.
It follows that when X* and Y* are simplicial sets we get our desired pairing
E*(X, E) E*(Y , E) ! E*(X x Y , E) from the E2terms onward. This obser
vation will be used in section 7.
Exercise 3.11. Is Proposition 3.10 true for simplicial spaces? I haven't worked
out the answer to this.
8 DANIEL DUGGER
4.The Postnikov/Whitehead spectral sequence
For each spectrum E and each n 2 Z, let PnE denote the nth Postnikov section
of E; this is a spectrum obtained from E by attaching cells to kill off all hom*
*otopy
groups from dimension n + 1 and up. The construction can be set up so that if E
is fibrant then all the PnE are also fibrant, and there are natural maps E ! PnE
and PnE ! Pn1E making the obvious triangle commute. So we have a tower of
fibrant spectra
. .!.P2E ! P1E ! P0E ! . . .
and the homotopy cofiber of Pn+1E ! PnE is an EilenbergMacLane spectrum of
type n+2H(ßn+1E).
If A is a cofibrant, pointed space, we can map 1 A into this tower and there*
*by
get a tower of function spectra
. .!.F(A, P2E) ! F(A, P1E) ! F(A, P0E) ! . . .
The homotopy cofiber of F (A, Pn+1E) ! F (A, PnE) is weakly equivalent to
F_(A, n+2H(ßn+1E)), and the resulting homotopy spectral sequence has
Ep,q1~=ßpF_(A, q+2H(ßq+1E)) ~=Hqp+2(A; ßq+1E).
The spectral sequence abuts to ßp1F_(A, E) = E~1p(A). This turns out to be
another construction of the AtiyahHirzebruch spectral sequence_see [GM , Ap
pendix] for some information about how the two spectral sequences are related.
Assume that E, F , and G are fibrant spectra, and that there is a pairing E ^
F ! G. There do not exist reasonable pairings PnE ^ Pm F ! Pn+m G, and so
Postnikov towers are not convenient for seeing multiplicative structures on spe*
*ctral
sequences. This is related to the Postnikov tower being a limtower rather than*
* a
colimtower. Instead we will use the `reverse' of the Postnikov tower, sometim*
*es
called the Whitehead tower. If WnE denotes the homotopy fiber of E ! Pn1E,
then there are natural maps WnE ! Wn1E and so we get a new tower. The
homotopy cofiber of Wn+1E ! WnE is weakly equivalent to nH(ßnE). We will
modify these towers in an attempt to produce a pairing W*E ^ W*F ! W*G.
To explain the idea, let's forget about cofibrancy/fibrancy issues for just a*
* mo
ment. Consider the following maps:
Wm E ^_WnF _____//E ^ F
___ 
~ ______ 
fflffl___p fflfflj
Wm+n G ________//_G_____//_Pm+n1 G.
The horizontal row is a homotopy fiber sequence. The spectrum Wm E ^ WnF is
(m + n  1)connected, and so the composite Wm E ^ WnF ! Pm+n1 G is null
homotopic. Choosing a nullhomotopy lets us construct a lifting ~: Wm E ^WnF !
Wm+n G. If we had two different liftings ~ and ~0, their difference would lift *
*to a
map Wm E ^ WnF ! Pm+n1 G and so would be nullhomotopic (again, because
the domain is (m + n  1)connected). So the lift ~ is unique up to homotopy.
The situation, then, is that we can produce pairings Wm E ^ WnF ! Wm+n G,
but so far they don't necessarily commute with the structure maps in the towers.
They certainly commute up to homotopy_this follows from the `uniqueness' con
siderations in the above paragraph_ but we need them to commute on the nose.
MULTIPLICATIVE STRUCTURES ON HOMOTOPY SPECTRAL SEQUENCES II 9
By using obstruction theory we will be able to alter these maps so that the rel*
*evant
diagrams do indeed commute. The argument proceeds in a few steps.
Lemma 4.1. For each fibrant spectrum E there is a natural tower of rigid homoto*
*py
cofiber sequences (fW*E, eC*E) such that every fWnE and eCnE is cofibrantfibra*
*nt,
together with a natural zigzag of weak equivalences from fW*E to W*E.
Proof.First take W*E and apply a cofibrantreplacement functor Q to all the lev*
*els:
this produces QW*E, a tower of cofibrant spectra. Then perform the telescope
construction from [D1 , B.4] to get a tower of cofibrations between cofibrant o*
*bjects
T W*E and a weak equivalence T W*E ! QW*E. Let CnE denote the cofiber of
T Wn+1E ! T WnE. Finally, let F be the fibrantreplacement functor for Spectra
such that F (*) = * given in [D1 , C.3(c)]. Applying F to the rigid tower (T W *
*E, CE)
gives a new rigid tower which has the desired properties.
At this point we have towers where everything is cofibrantfibrant, so the ar
gument we have already explained will construct maps fWmE ^ fWnF ! fWm+nG
which commute up to homotopy with the maps in the towers. By considering the
diagram
fWmE ^ fWnF ____//_fWm+nG
 
 
fflffl fflffl
eCmE ^ eCnF eCm+nG
one can see that there is a unique homotopy class eCmE ^ eCnF ! eCm+nG which
makes the square commute. This is because Cem+nG is an EilenbergMacLane
spectrum of type n+m H(ßn+m G), and fWmE ^ fWnF ! CemE ^ eCnF induces
an isomorphism on the corresponding cohomology group (since both the domain
and codomain are (m + n  1)connected). So at this point we have produced a
homotopypairing (fWE, eCE) ^_(fWF, eCF ) ! (fWG, eCG) (see [D1 , 6.3]). We wi*
*ll
prove:
Proposition 4.2. The homotopypairing (fWE, eCE) ^_(fWF, eCF ) ! (fWG, eCG) is
locally realizable.
The following lemma encapsulates the basic facts we will need. The proof will
be left to the reader.
Lemma 4.3 (Obstruction theory). Suppose that X ! Y is a fibration of spectra
which induces isomorphisms on ßk for k n. Let A æ B be a cofibration which
induces isomorphism on ßk for k < n. Then any diagram
Af_____//flfflX>>__
_______
 ______
fflfflfflfflfflffl___
B _____//Y
has a lifting as shown.
Proof of Proposition 4.2.First we truncate the towers, and we might as well ass*
*ume
we are dealing with truncations ø0 k(fWE, eCE) and ø0 l(fWF, eCF ) because the
argument will be the same no matter what the lower bounds are. For the rest of
10 DANIEL DUGGER
the argument we will only be dealing with these finite towers, and will omit th*
*e ø's
from the notation.
We replace (fWE, eCE) and (fWF, eCF ) by the equivalent towers (T W E, CE)
and (T W F, CF ) constructed in the proof of Lemma 4.1, because these consist of
cofibrations between cofibrant spectra. It is easy to see that one can also fin*
*d a tower
cW*G consisting of fibrant spectra and fibrations, together with a weak equival*
*ence
fW*G ! cW*G (remember that all our towers are finite). We will construct a
pairing of towers T W*E ^ T W*F ! cW*G which realizes the homotopypairing.
For T W0E ^ T W0F ! cW0G we choose any map in the correct homotopy class.
Next consider the diagram
cW1G


fflffl
T W0E ^ T W1F _____//T W0E ^ T W0F___//_cW0G.
The vertical map is a fibration inducing isomorphisms on ß1 and higher, and the
spectrum T W0E ^ T W1F is 0connected; so there is a lifting ~(0,1). Next, look*
* at
the diagram
T W1E ^ T W1F _____//T W0E ^ T W1F____//cW1G
 
 
fflffl fflffl
T W1E ^ T W0F _____//T W0E ^ T W0F___//_cW0G.
This diagram commutes (because the missing vertical arrow in the middle may be
filled in). The right vertical map is a fibration which induces isomorphisms on*
* ß1
and higher. The left vertical map induces isomorphisms on ß0 and lower (because
both domain and range are 0connected), and is a cofibration. So there is a lif*
*ting
~(1,0):T W1E ^ T W0F ! cW1G.
This process may be continued to inductively define ~(0,2), ~(1,1), and ~(2,0*
*),
and then onward from level three. In this way, we construct the required pairing
of towers. This pairing agrees with the original homotopypairing because of the
`uniqueness' of the liftings ~ in our original discussion; the details are left*
* for the
reader.
At this point we have a pairing of towers, but we need a pairing of augmented
rigid towers. We can't just take~cofibers in cW*G because the maps are fibratio*
*ns, not
cofibrations. So let Q(cW*G) i cW*G be the cofibrant approximation guaranteed
in [D1 , Lemma B.2]. We have a diagram
Q(cW*G)
~
fflfflfflffl
T W E ^ T W F______//cW*G
and by [D1 , Lemma B.3] the lower left corner is a cofibrant tower; so there is
a lifting. The tower Q(cWG) consists of cofibrations, and so augmenting by the
cofibers gives a rigid tower. The new pairing automatically passes to cofibers*
* to
give (T W E, CE) ^ (T W F, CF ) ! (Q(cWG), CQ(cWG)).
MULTIPLICATIVE STRUCTURES ON HOMOTOPY SPECTRAL SEQUENCES II 11
Finally, consider
Q(cW*G)
~
fflfflfflffl
fW*G __~__//_cW*G.
The tower fW*G was cofibrant, so there is a lifting. This will be a levelwise e*
*quiv
alence, and therefore induces an equivalence on the cofibers. So we get an equi*
*v
alence of augmented towers (fW*G, eC*G) ! (Q(cWG), CQ(cWG)). Thus, we have
constructed the required realization of our homotopypairing.
For each cofibrant space X consider the tower of function spectra F(X+ , fWE?*
* )
(recall that the fW*E are all fibrant). This is a tower of rigid homotopy cofib*
*er se
quences, and we will call the associated spectral sequence E*(X, E) the Whitehe*
*ad
spectral sequence for X based on E. The homotopypairing fWE? ^_fWF? !
fWG? induces a homotopypairing on towers of function spectra, and by [D1 , Pro*
*p.
6.10] this is locallyrealizable and so induces a pairing of spectral sequences*
*: for any
cofibrant spaces X and Y we have E*(X, E) E*(Y, F ) ! E*(X x Y, G). What is
left is to identify the pairing on E1terms with the pairing on singular cohomo*
*logy
(up to the correct sign).
If X is a spectrum with a single nonvanishing homotopy group in dimension m,
there is a unique isomorphism in the homotopy category Sm ^_H(ßm X) ! X with
the property that the composite
ßm X ! ß0H(ßm X) oel!ßm (Sm ^ H(ßm X)) ! ßm X
is the identity map (the first map in the composite is the one provided by [D1 ,
Section C.7]). If X ^ Y ! Z is a pairing of spectra where X, Y , and Z each have
a single nonvanishing homotopy group in dimensions m, n and m + n, then the
diagram in Ho(Spectra)
Sm ^_H(ßm X) ^_Sn ^ H(ßnY )___//_Sm+n ^_H(ßm+n Z)
 
 
fflffl fflffl
X ^ Y ______________________//Z
is commutative. Here the top map interchanges the Sn and the H(ßm X) and
then uses the map H(ßm X) H(ßnY ) ! H(ßm+n Z) induced by the pairing
ßm X ßnY ! ßm+n Z (cf. [D1 , Section C.7]). The above observations are simple
calculations in the homotopy category of spectra.
In our situation we have specific isomorphisms ßm eCmE ~=ßm E, and the same
for F and G. This is because fWmE ! CemE induces an isomorphism on ßm ,
fWmE is connected by a chosen zigzag of weak equivalences to Wm E, and the map
Wm E ! E induces an isomorphism on ßm as well. The pairing eCmE ^ eCnF !
eCm+nG induces a pairing on homotopy groups which corresponds to the expected
pairing ßm E ßnF ! ßm+n G under these isomorphisms. Putting all the above
statements together, we have proven:
12 DANIEL DUGGER
Lemma 4.4. In Ho (Spectra) there exist isomorphisms Sn ^ H(ßnE) ! CenE,
Sn ^ H(ßnF ) ! CenF , and Sn ^ H(ßnG) ! CenG for all n 2 Z, such that the
diagrams
Sm ^ H(ßm E) ^ Sn ^ H(ßnF )___//_Sm+n ^ H(ßm+n G)
 
 
fflffl fflffl
CemE ^ eCnF_________________//eCm+nG
all commute (in the homotopy category).
The lemma tells us that if A and B are spectra the pairing
~ ~ ~ ~
p,qHo(Sp^_A, eCqE) s,tHo(Ss^_B, eCtF ) ! u,vHo(Su ^_A ^_B, eCvG)
is globally isomorphic to the pairing obtained from the maps
Ho(Sp^_A, Sq^_H(ßqE)) Ho(Ss^_B, St^_H(ßtF ))


fflffl
Ho (Sp+s^_A ^_B, Sq+t^_H(ßq+tG)).
Now, the leftsuspension map gives an isomorphism
Ho(Sp^_B, Sq^_H(ßqE)) ~=Ho(Spq^_B, H(ßqE)) = Hqp(B; ßqE),
and similarly for the F and G terms. This allows us to rewrite the above pairin*
*g as
a pairing of singular cohomology groups, but the suspension maps introduce sign*
*s.
For any spectra M and N, the diagram
oeql^oetl
Ho (Sa^_A, M) Ho(Sb^_B, N)___//Ho(Sq+a^_A, Sq^_M) Ho(St+b^_B, St^_N)
 
 
fflffl oeq+tl fflffl
Ho (Sa+b^_A ^ B, M ^_N)________//Ho(Sq+t+a+b^ A ^ B, Sq+t^ M ^ N)
commutes up to the sign (1)ta(one compares the string `qatb' to the string `qt*
*ab'
and sees that the t and a must be commuted). Taking A = 1 X+ and B = 1 Y+ ,
we now conclude:
Theorem 4.5 (Multiplicativity of the Postnikov/Whitehead spectral sequence).
For cofibrant spaces X and Y there is a pairing of Whitehead spectral sequences*
* in
which the E1term Ep,q1 Es,t1! Ep+s,q+t1is globally isomorphic to the pairing
Hqp(X; ßqE) Hts(Y ; ßtF ) ! Hq+tps(X x Y ; ßq+tG)
up to a sign of (1)t(pq).
Remark 4.6. At first glance the sign given here doesn't agree with the sign we
obtained in Theorem 3.4: if we were to reindex the AtiyahHirzebruch spectral
sequence in the above form, the sign would be (1)q(ts). While this is not the
same as the above sign, the two are consistent. Using the family of isomorphisms
(1)pq:Hqp(X; ßqE) ~=Hqp(X; ßqE) and similarly for the F and G terms, the
sign (1)t(pq)transforms into (1)q(ts). See Exercise 2.4.
MULTIPLICATIVE STRUCTURES ON HOMOTOPY SPECTRAL SEQUENCES II 13
5. Bockstein spectral sequences
In this section we consider two spectral sequences: one is the classical Bock*
*stein
spectral sequence for the homotopy cofiber sequence HZ xn!HZ ! HZ=n. The
other is the Bockstein spectral sequence for inverting the Bott element in conn*
*ective
Ktheory.
5.1. The Bockstein spectral sequence for HZ. Consider the following tower
(W*, B*)* 0:
HZ=nOO HZ=nOO HZ=nOO
  
  
  n 
. ._._n__//HZ___n__//_HZ______//_HZ.
We extend this to negative degrees by taking Wq = HZ, Bq = *, and Wq+1 !
Wq to be the identity map. This is a tower of rigid homotopy cofiber sequences,
and there is an obvious pairing (W, B) ^ (W, B) ! (W, B) which comes from the
multiplications on HZ and HZ=n (cf. [D1 , Appendix C.7]).
For any cofibrant space X, let W? X denote the augmented tower whose levels
are F (X+ , Wn+1) ! F (X+ , Wn) ! F (X+ , Bn); these are rigid homotopy cofiber
sequences, since HZ and HZ=n are fibrant. The homotopy spectral sequence for
W? X is called the mod n Bockstein spectral sequence, and has the form
Ep,q1= Hp (X; Z=n) ) Hp (X; Z).
The d1differential is the usual Bockstein homomorphism. The multiplication on
(W, B) gives rise to pairings of towers W? X ^W? Y ! W? (X xY ), and therefore *
*to
pairings of spectral sequences by [D1 , Thm 6.1]. The following result is immed*
*iate,
and unlike the examples in sections 3 and 4 there are no extra signs floating a*
*round.
Theorem 5.2 (Multiplicativity of the Bockstein spectral sequence). For cofibrant
spaces X and Y there is a pairing of Bockstein spectral sequences whose E1term*
* is
isomorphic to the usual pairing Hp (X; Z=n) Hs(Y ; Z=n) ! Hps(XxY ; Z=n)
of singular cohomology groups. The spectral sequence converges to the usual pai*
*ring
on H*(; Z).
5.3. The Bockstein spectral sequence for bu. Let bu denote a commutative
ring spectrum representing connective Ktheory, and assume we have a map of ring
spectra bu ! HZ. Assume there is a map S2 ! bu which represents a generator in
Ho(S2, bu) ~=Z (this is automatic if bu is a fibrant spectrum). Consider the in*
*duced
map
fi :S2 ^ bu fi^id!bu ^ bu ~!bu.
It can be shown that S2^ bu ! bu ! HZ is a homotopy cofiber sequence. If we let
(W, B) be the tower
S4 ^OHZO S2 ^OHZO HZOO S2 ^OHZO
   
   
4^fi  S2^fi  fi  S2^fi  S4^fi
. ._.S___//_S4 ^_bu____//S2 ^ bu_____//_bu_____//_S2 ^ bu____//_//_. . .
14 DANIEL DUGGER
then one sees that there is a pairing (W, B) ^ (W, B) ! (W, B) (this uses that *
*bu is
commutative). Unfortunately we are not yet in a position to apply [D1 , Thm 6.1*
*]:
(W, B) is not a rigid tower, because we don't know that S2^ bu ! bu ! HZ is null
rather than just nullhomotopic. We don't get a long exact sequence on homotopy
groups until we choose nullhomotopies for each layer, and these must be accoun*
*ted
for. In this particular case any two nullhomotopies are themselves homotopic, *
*and
so there should be no problems with compatibility, but it's awkward to formulate
results along these lines. The best way I know to proceed is actually to discar*
*d the
HZ's and consider just the tower W* consisting of suspensions of bu. We are then
in a position to apply [D1 , Thm 6.2], but we must work a little harder to iden*
*tify
what's happening on the E1terms.
First, one can replace bu by a cofibrant commutative ring spectrum, and this
will be cofibrant as a symmetric spectrum [SS, Thm. 4.1(3)]. Because of this, we
will just assume that our bu was cofibrant in the first place. By [D1 , B.4] t*
*here
is an augmented tower T W* consisting of cofibrations between cofibrant spectra,
a weak equivalence T W* ! W*, and a pairing T W ^ T W ! T W which lifts the
pairing W ^ W ! W . We let Cn denote the cofiber of T Wn+1 ! T Wn and note
that our pairing T W ^ T W ! T W extends to (T W, C) ^ (T W, C) ! (T W, C).
For any cofibrant space X, we consider the derived tower of function spec
tra F der(X+ , T W? ) from [D1 , B.7]. By [D1 , Theorems 6.1,6.10] there is an*
* in
duced pairing of spectral sequences. Note that we have Ep,q1= ßpF_(X+ , Cq) ~=
ßpF (X, 2qHZ) ~=H2qp(X; Z). As usual, what we want is to identify the pairing
on E1terms with a pairing on singular cohomology. The following does this:
Lemma 5.4. In Ho(Spectra) it is possible to choose a collection of isomorphisms
Cn ! S2n^_HZ such that the following diagrams commute:
Cm ^_Cn _____________________________________//_Cm+n
 
 
fflffl 1 ^_t ^_1 1 ^_~ fflffl
S2m ^_HZ ^_S2n^_HZ_____//S2m ^_S2n^_HZ ^_HZ____//S2(m+n)^_HZ.
Proof.This is very similar to the proof of Lemma 4.4, making use of the fact th*
*at
the graded ring nHo (S2n, Cn) is isomorphic to the ring of Laurent polynomials
Z[t, t1]. Details are left to the reader.
The spectral sequence converges to
fi. fi.
colimß*F_(X+ , bu) ! ß*F_(X+ , bu) ! . . .
which is fi1[ß*F_(X+ , bu)] = fi1bu*(X). If we write fi1bup(X) for the pth g*
*raded
piece of fi1(bu*X) then we can state the final result as follows:
Theorem 5.5. For any space X there is a conditionally convergent spectral se
quence
Ep,q1= H2qp(X; Z) ) fi1bup(X).
If X and Y are two cofibrant spaces there is a pairing of spectral sequences wh*
*ose
E1term is globally isomorphic to the usual pairing on singular cohomology, and
which converges to the pairing fi1bu*(X) fi1bu*(Y ) ! fi1bu*(X x Y ).
The above spectral sequence of course has the same form (up to reindexing) as
the AtiyahHirzebruch spectral sequence for complex Ktheory. One finds in the
end that fi1bu*(X) ~=K*(X).
MULTIPLICATIVE STRUCTURES ON HOMOTOPY SPECTRAL SEQUENCES II 15
6.The homotopyfixedpoint spectral sequence
Suppose E is a fibrant spectrum with an action of a discrete group G. The
homotopy fixed set EhG is defined to be F(EG+ , E)G .
Theorem 6.1. If E is a fibrant ring spectrum and G is a discrete group acting
via ring automorphisms, then there is a multiplicative spectral sequence of the*
* form
Ep,q2= Hq(G; ßp+qE) ) ßp(EhG). Here the pairing on E2terms is the pairing on
group cohomology with graded coefficients, defined analagously to what was done*
* in
section 2.
Remark 6.2. The above means that the pairing on E2terms Hq(G; ßp+qE)
Ht(G; ßs+tE) ! Hq+t(G; ßp+q+s+tE) is (1)t(p+q)times the `standard' pairing on
group cohomology induced by ß*E ß*E ! ß*E.
Proof.Take any model for EG which is an equivariant GCWcomplex (with G
acting freely on the set of cells in each dimension). Write EGk for skkEG, and
let Wn = F(EG=EGn1, E), Bn = F(EGn=EGn1, E). The spectral sequence for
this tower is just the AtiyahHirzebruch spectral sequence based on EG and E,
from section 3.3. The sequences Wn+1 ! Wn ! Bn are actually Gequivariant
homotopy fiber sequences, which implies that they give homotopy fiber sequences
on Hfixed sets for any H. We'll consider the associated tower (WG*, BG*).
The E1term of the spectral sequence is Ep,q1= ßp[F (EGq=EGq1, E)G ]. The
map
ßp[F (EGq=EGq1, E)G ] ! ßpF (EGq=EGq1, E) = Ho (Sp ^ [EGq=EGq1], E)
has its image in the Gfixed set of the righthandside, and one can check that*
* this
gives an isomorphism Ep,q1~=Ho (Sp ^ [EGq=EGq1], E)G . Since EGq=EGq1
is a wedge of qspheres indexed by the set of qcells, with G action induced
by that on the indexing set, one obtains a natural isomorphism with the group
Hom G(C*(EG), ßp+qE). The identification works the same as in section 3.3, and
the description of the differential carries over as well. In fact the map of t*
*owers
(WG*, BG*) ! (W*, B*) lets us compare the differential with the one on the Atiy*
*ah
Hirzebruch spectral sequence, which we have already analyzed. So the identifica*
*tion
of the E2term follows.
The pairing of augmented towers W(EG, E) ^ W(EG, E) ! W(EG x EG, E)
from section 3 obviously restricts to fixed sets, giving W(EG, E)G ^ W(EG, E)G !
W(EG x EG, E)G . This uses that E ^ E ! E is Gequivariant. If one accepts that
the diagonal map EG ! EGxEG is homotopic to a map 0which is both cellular
and Gequivariant, then we get an equivariant map of towers W(EG x EG, E) !
W(EG, E) and can restrict to fixed sets. The existence of 0 follows from the G
equivariant cellular approximation theorem. Therefore we have an induced pairing
of augmented towers WG?^ WG? ! WG?. The identification of the product on
E2terms again can be done by comparing with the AtiyahHirzebruch spectral
sequence for the tower (Wn, Bn). Everything follows exactly as in section 3.3.
Remark 6.3. Instead of filtering EG by skeleta, the spectral sequence can also
be constructed via a Postnikov tower on E, just as in section 4 (by functoriali*
*ty,
the Postnikov sections of E inherit Gactions). Instead of needing a Gequivari*
*ant
cellular approximation for the diagonal map, one instead needs to carry out G
equivariant obstruction theory.
16 DANIEL DUGGER
7.Spectral sequences from open coverings
In this section we give a second treatment of the AtiyahHirzebruch spectral *
*se
quence, together with a similar approach to the LeraySerre spectral sequence. *
*Our
towers are obtained by using open coverings and their generalization to hyperco*
*vers.
We will assume a basic knowledge of hypercovers, for which the reader can consu*
*lt
[DI]. Basically, one starts with an open cover {Ua} of a space X and then choos*
*es
another open cover for every double intersection Ua \ Ub; for every resulting `*
*triple
intersection' another covering is chosen, and so on. All of this data is compil*
*ed into
a simplicial space U*, called a hypercover.
The discussion in this section is much sloppier than the previous ones, and s*
*hould
probably be improved at some point...
7.1. The descent spectral sequence. Given a hypercover U* of a space X and
a sheaf of abelian groups F on X, we let F (U*) denote the cochain complex one
gets by applying F to the open sets in U*. If all the pieces of the hypercover *
*Un are
such that Hkshf(Un, F ) = 0 for k > 0, then Verdier's Hypercovering Theorem giv*
*es
an isomorphism Hp(F (U*)) ! Hpshf(X, F ) which is functorial for X, U, and F . *
*It
is easy to explain how to get this. First, choose a functorial, injective resol*
*ution
I* for F , and look at the double complex of sections Dmn = In(Um ). This double
complex has two edge homomorphisms I*(X) ! TotD** and F (U*) ! TotD**;
the two spectral sequences for the homology of a double complex immediately show
that these maps give isomorphisms on homology. Composing them appropriately
gives our natural isomorphism Hp(F (U*)) ! Hp(I*(X)) = Hpshf(X, F ).
Given a fibrant spectrum E and a hypercover U* of a X, we use the simplicial
space U* to set up a tower as in section 3.9. We let Wn = F(U= skn1U, E), *
*Bn =
F(sknU=skn1U, E), and denote the resulting spectral sequence by E*(U, E). *
*It
is a theorem of [DI] that the natural map U* ! X is a weak equivalence, and so
the spectral sequence converges to E*(X). We'll call this the descent spectral
sequence based on U and E.
Note that the spectral sequence is functorial in several ways. It is clearly *
*func
torial in E, and if V* ! U* is a map of hypercovers of X then there is a natural
map E*(U, E) ! E*(V, E). Also, if f :Y ! X is a map then there is a pullback
hypercover f1 U* ! Y , and a map of spectral sequences E*(U, E) ! E*(f1 U, E).
So far we have said nothing about the E2term. A space X is locally con
tractible if given any point x and any open neighborhood x 2 V , there is an
open neighborhood x 2 W V such that W is contractible. Given such a space
X, one can build a hypercover U* of X in which every level is a disjoint union *
*of
contractible opens. We'll call U* a contractible hypercover.
If U* ! X is a hypercover there is a natural isomorphism Ep,q2(U, E) ~=
Hq(Epq(U*)). If we assume X is locally contractible and U* is a contractible
hypercover, we can simplify this further. Let ~Eqdenote the sheafification of *
*the
presheaf V 7! Eq(V ), and note that this is a locally constant sheaf on X. It
follows that Eq(V ) ! ~Eq(V ) is an isomorphism for every contractible V . In p*
*ar
ticular, Hq(Epq(U*)) ! Hq(~Epq(U*)) is an isomorphism. But since the sheaves
~E*are locally constant, it follows that if V is a contractible open set in X t*
*hen
Hpshf(V, ~Eq) = 0 for p > 0 (see [Br, II.11.12]). The fact that U* is a contra*
*ctible
MULTIPLICATIVE STRUCTURES ON HOMOTOPY SPECTRAL SEQUENCES II 17
hypercover now shows that
Ep,q2(U, E) ~=Hq(Epq(U*)) ~=Hq(~Epq(U*)) ~=Hqshf(X, ~Epq)
(and all these isomorphisms are natural).
Remark 7.2. Let us for a moment forget about contractible hypercovers, and look
at all of them. For each map of hypercovers V* ! U* there is an induced map of
spectral sequences E*(U, E) ! E*(V, E). Two homotopic maps V* ' U* induce
the same map of spectral sequences from E2 on. So if we forget about E1terms,
we have a diagram of spectral sequences indexed by the homotopy category of
hypercovers HCX . We can take the colimit of these spectral sequences, and the
Verdier Hypercovering Theorem identifies the E2term with sheaf cohomology. So
the hypothesis that the base X be locally contractible is not really necessary *
*for the
development of our spectral sequence. It does make things easier to think about,
though, so we will continue to specialize to that case.
Now suppose that E is a fibrant ring spectrum, Y is another locally contracti*
*ble
space, and V* ! Y is a contractible hypercovering. The product simplicial space
U* x V* is a contractible hypercover of X x Y . By the discussion in section 3*
*.9
we get a pairing of spectral sequences E*(U, E) E*(V, E) ! E*(U x V , E)
and a map of spectral sequences E*(U x V, E) ! E*(U x V , E). It follows fr*
*om
Proposition 3.10 that the latter map is an isomorphism (from the hypercover U o*
*ne
defines a simplicial set ß0U by applying ß0() in every dimension, and in our c*
*ase
U ! ß0(U) is a levelwise weak equivalence; the same holds for V and U x V , and
so we are really just dealing with simplicial sets). Therefore we get the follo*
*wing:
Theorem 7.3 (Multiplicativity of the descent spectral sequence). Let X and Y be
locally contractible spaces, with contractible hypercovers U* ! X and V* ! Y . *
*Then
there is a pairing of descent spectral sequences E*(U, E) E*(V, E) ! E*(U x V*
*, E)
in which the E2term is globally isomorphic to
Hqshf(X, Epq) Htshf(Y, Est) ! Hq+tshf(X x Y, Epqst)
up to a sign difference of (1)t(p+q).
Proof.To identify the product on E2terms we use the fact that if F and G are
two sheaves on X and Y , and U and V contractible hypercovers of X and Y , then
the pairing of cosimplicial abelian groups F (U*) G(V*) ! (F G)(U* x V*)
induces the cup product on sheaf cohomology via an EilenbergZilber map and the
isomorphisms Hp(F (U*)) ~=Hp(X, F ), etc. Someone should write down a careful
proof of this someday, but I have proven enough `trivial' things for one paper.
The sign comes from an implicit use of the suspension isomorphism. The E2term
of E*(U, E) is most naturally identified with Hqshf(X, Gp,q) where Gp,qdenotes *
*the
sheafification of V 7! Ep(Sq ^ V+ ). Clearly Gp,qis isomorphic to the sheaf ~E*
*pq
via the leftsuspension isomorphism, but this introduces signs into the pairings
much like in Theorem 4.5. We leave the details to the reader.
The assumption in the above theorem that X and Y be locally contractible is
probably not really necessary, as in Remark 7.2; but I have not worked out the
details. Also, when X and Y are both locally contractible and paracompact, sheaf
cohomology with locally constant coefficients is isomorphic to singular cohomol*
*ogy;
18 DANIEL DUGGER
the pairing on sheaf cohomology corresponds to the cup product under this isomo*
*r
phism [Br, Theorem III.1.1]. So the descent spectral sequence takes the same fo*
*rm
as the AtiyahHirzebruch spectral sequence in this case.
Exercise 7.4. If U* ! X is a contractible hypercover, let ß0(U) be the simplici*
*al
set defined above_obtained by replacing each Un by its set of path components.
There is a natural map U ! ß0(U), and this is a levelwise weak equivalence. Con
vince yourself that the descent spectral sequence E*(U, E) is canonically isomo*
*rphic
to the AtiyahHirzebruch spectral sequence for the space ß0(U) based on its s*
*kele
tal filtration. Conclude that everything in this section is just a restatement *
*of the
results in section 3.3.
7.5. The Leray spectral sequence. Now assume that ß :E ! B is a fibration
with fiber F , and that B is locally contractible. Choose a hypercover U* of B
and consider the pullback hypercover ß1U* of E. Once again, we know by [DI]
that ß1U* ! E is a weak equivalence. Taking E = HZ, the skeletal filtration
of ß1U* gives a spectral sequence for computing the homotopy groups of the
function space F (E, HZ)_i.e., it computes the singular cohomology groups of E.
This is the Leray (or LeraySerre) spectral sequence.
Exercise 7.6. Let Hnß1 denote the sheaf on B obtained by sheafifying V 7!
Hnsing(ß1V ). Check that Hnß1 is a locally constant sheaf on B whose stalks a*
*re
isomorphic to Hnsing(F ). So if B is simplyconnected, it is a constant sheaf. *
*We will
abbreviate Hnß1 as Hn(F ).
It follows that if U is a contractible hypercover then the E2term of the spe*
*ctral
sequence is isomorphic to Hqshf(B, Hpq(F )). When B is simply connected this *
*is
Hqshf(B, HpqF ), and when B is paracompact this can be identified with singul*
*ar
cohomology.
Once again, if E0! B0 is a second fiber bundle with fiber F 0, there is a pai*
*ring
of spectral sequences which on E2terms has the form
Hqshf(B, HpqF ) Htshf(B0, HstF 0) ! Hq+tshf(B x B0, Hpqst(F x F 0)).
One has the usual sign difference from the canonical pairing, for the same reas*
*ons
as in Theorem 7.3. If E0! B0is the same as E ! B, then we can compose with the
diagonal map to get a multiplicative structure on the LeraySerre spectral sequ*
*ence
for E ! B.
Theorem 7.7 (Multiplicativity of LeraySerre). Let E ! B and E0 ! B0 be
fibrations where B and B0are locally contractible. Then there is a pairing of L*
*eray
Serre spectral sequences for which the E2term is globally isomorphic to the pa*
*iring
Hqshf(B, HpqF ) Htshf(B0, HstF 0) ! Hq+tshf(B x B0, Hpqst(F x F 0)).
except for a sign difference of (1)t(p+q).
Exercise 7.8. Fill in the many missing details from this section.
Exercise 7.9. Of course we didn't need to take E = HZ in the above discussion.
We could have used any ring spectrum, in which case we would obtain the combina
tion AtiyahHirzebruchLeraySerre spectral sequence. Think through the details
of this.
MULTIPLICATIVE STRUCTURES ON HOMOTOPY SPECTRAL SEQUENCES II 19
References
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* Symp.
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[BK] A. Bousfield and D. Kan, A second quadrant homology spectral sequence, Tr*
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[Sp] E.H. Spanier, Algebraic Topology, SpringerVerlag, New York, 1966.
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Department of Mathematics, University of Oregon, Eugene, OR 97403
Email address: ddugger@math.uoregon.edu