A spectral sequence for string cohomology
Marcel Bökstedt & Iver Ottosen*
5 December 2002
Abstract
Let X be a 1-connected spaces with free loop space X. We in-
troduce two spectral sequences converging towards H*( X; Z=p) and
H*(( X)hT; Z=p). The E2-terms are certain non Abelian derived func-
tors applied to H*(X; Z=p). When H*(X; Z=p) is a polynomial alge-
bra, the spectral sequences collapse for more or less trivial reasons. If
X is a sphere it is a surprising fact that the spectral sequences collapse
for p = 2.
AMS subject classification (2000): 55N91, 55P35, 18G50
1 Introduction
Let X be a space and let X denote its free loop space. The circle group T
acts on X by rotation of loops. The associated homotopy orbit space XhT
is sometimes called the string space.
Consider the cohomology H*X as given. The purpose of this paper is to
study the cohomology of the free loop space and of its homotopy orbit space.
In some cases, it is relatively easy to compute this cohomology. For
instance, suppose that X is an Eilenberg-Maclane space. Then there is a
homotopy splitting X ' X x X. The space X is also a Eilenberg-
MacLane space, so that the cohomology of X is known.
The cohomology of the homotopy orbits XhT is more difficult to com-
pute. However, this is achieved in [BO ] and [O2 ].
The main idea of the present paper is to use these computations to study
the case of a general X. In essence, this application is done using a Postnikov
______________________________
*The second author was supported by the University of Copenhagen and by the *
*Euro-
pean Union TMR network ERB FMRX CT-97-0107: Algebraic K-theory, Linear Algebraic
Groups and Related Structures.
1
decomposition of X. From our point of view, the simplest case is when X
is a product of Eilenberg-MacLane spaces, and correspondingly, the more
k-invariants a space X has, the more complicated it appears. In particular,
the spheres are very complicated spaces for this approach.
Formally, we will study two spectral sequences converging towards the
cohomology groups H*( X; Fp) and H*( XhT; Fp). Both spectral sequences
have origin in the Bousfield homology spectral sequence [B1 ].
This is a remarkable spectral sequence that under fortunate circumstances
converges to the homology of the total space of a cosimplicial space.
Let X be a simply connected space. We re-write its Postnikov tower
as a cosimplicial space, whose total space is the p-completion of X. This
cosimplicial space is the cosimplicial resolution R X of X with R = Fp. Given
this, we can form two cosimplicial spaces RX and ( RX)hT by applying
the functors (-) and (-)hT in each codegree. The total space of these new
cosimplicial spaces are the completions of X respectively ( X)hT. These
cosimplicial spaces have associated Bousfield homology spectral sequences
{E^r} and {Er} respectively.
For 1-connected X it is well known that {E^r} converges strongly towards
H*( X; Fp). We show that {Er} converges strongly towards H*( XhT; Fp)
under the additional assumption that H*(X; Fp) is of finite type.
For the dual cohomology spectral sequences, {E^r} and {Er}, we give
an interpretation of the E2-terms. The idea is that the E1-terms are given
by the cohomology of the respective functors (from spaces to spaces) ap-
plied to the Eilenberg-MacLane spaces. This cohomology_can,_according to
[BO ],[O1 ] and [O2 ] be written as certain functors respectively ` (from al-
gebras with a certain extra structure to algebras), applied to the cohomology
of the Eilenberg-MacLane spaces.
This means that the E2-term is the homology of a chain complex, where
the chains are given by these functors applied to the cohomology of Eilenberg-
MacLane spaces. Since the cohomology of an Eilenberg-MacLane space turns
out to be a free object, we can compute the E2-terms as derived_functors.
To be precise, they are the non Abelian derived functor of applied
to H*(X; Fp) respectively the non Abelian derived functor of ` applied to
H*(X; Fp). When H*(X; Fp) is a polynomial algebra the higher derived
functors vanish so the spectral sequences collapse at the E2-terms.
So far, the results are of a theoretical nature. As a concrete example, we
finally study the case X = Sn and p = 2. We develop homological algebra
sufficient for computing the relevant E2-terms.
For these spaces, there are other methods for computing H*( X; Z=p)
and H*(( X)hS1; Z=p).
2
Comparing our E2 terms with these results, we show that for X = Sn
with n 2 and p = 2 the spectral sequences collapse at the E2-terms.
We emphasize that this collapsing is not something to be expected a
priory. Since spheres have complicated Postnikov systems, from the point of
view of our spectral sequence, one would naively expect that these spectral
sequence could have many nontrivial differentials.
A natural question is: For how large a class C of 1-connected spaces do
the two spectral sequences collapse at the E2-terms ? As a first approach
we conjecture that C contains any suspension X = Y where Y is path
connected and H*(Y ; F2) is of finite type.
Finally, we want to thank the referee of [BO ] for suggesting that we look
at the Bousfield spectral sequence in this connection.
2 Bousfield homology spectral sequences
Let X be a fibrant cosimplicial space and let A be an Abelian group. In [B1 ]
Bousfield constructs a spectral sequence {Er(X; A)} with the homology of
the total space H*(Tot X; A) as expected target.
The precise convergence statement is as follows. Recall that there is a
tower of fibrations
. . .! Totm X ! Totm-1 X ! . .!.Tot0X
with inverse limit TotX. Hence for each n 0 there is a tower map
Pn(X) : {Hn(Tot X; A)}m 0 ! {Hn(Tot mX; A)}m 0
where the domain tower is constant. Let A X denote the cosimplicial
simplicial Abelian group with (A X)mt= A Xmtwhere A S = x2SA for
a set S. Bousfield forms the double normalized complex and let T (A X)
denote its total complex. It is filtered by subcomplexes F mT (A X) and
the quotient complex T (A X)=F m+1T (A X) is denoted Tm (A X). A
comparison map is defined
n(X) : {Hn(Tot mX; A)}m 0 ! {HnTm (A X)}m 0
and the following result is proved:
Lemma 2.1. {Er(X; A)} converges strongly to H*(Tot X; A) if and only if
the tower map n(X) O Pn(X) is a pro-isomorphism for each n.
3
If n(X) is a pro-isomorphism for each n then X is called an A-pro-
convergent cosimplicial space and {Er(X; A)} is called pro-convergent.
We are interested in two special cases of this spectral sequence. Let
R = Fp be the field on p elements where p is a fixed prime. For a space X we
let RX denote the cosimplicial resolution of X in the sense of [BK ]. Note that
(RX)n = Rn+1X. The free loop space on X is by definition the simplicial
mapping space X = map (T, X) where we take T = BZ. By applying
codegree wise we get a cosimplicial space RX. We can also form the T
homotopy orbit space codegree wise and get the cosimplicial space ( RX)hT.
We are interested in the Bousfield homology spectral sequences for these two
spaces. As a corollary of [B2 ] Proposition 9.7 we have
Proposition 2.2. If X is a 1-connected and fibrant space then Pn( RX)
and n( RX) are pro-isomorphisms for each n and the spectral sequence
{Er( RX; Fp)} converges strongly to H*( (X^p); Fp) ~=H*( X; Fp).
In order to handle the other spectral sequence we need some results on
cosimplicial spaces which are presented in the next section.
3 Cosimplicial spaces with group actions
In this section the category of simplicial sets is denoted S and the category
of cosimplicial spaces cS . For A, B 2 S we let map (A, B) = BA denote the
simplicial mapping space. We write cA for the constant cosimplicial space
with (cA)n = A for each n.
The category cS is a model category with weak equivalences, cofibrations
and fibrations as described in [BK ] X x4. The fibrations are here defined in
terms of matching spaces. By this definition it is clear that if f : A ! B is
a fibration in S then c(f) : cA ! cB is a fibration in cS .
The category cS is in fact a simplicial model category in the sense of [Q ]
with X K 2 cS, XK 2 cS and Map (X, Y) 2 S defined as follows for K 2 S
and X, Y 2 cS:
(X K)(ff) = X(ff) x K
(XK )(ff) = X(ff)K
Map (X, Y)n = Hom cS(X n, Y)
where ff is a morphism in the simplicial category and n = [n] 2 S de-
notes the standard n-simplex. In case K is a simplicial group, this notation
potentially clashes with the usual notation for fixed points. In this paper,
we are not going to consider fixed points.
4
Let be the cosimplicial space which in codegree n equals n. We write
[m]for the simplicial m-skeleton and put [1]= . By [BK ] X.4.3 we have
that [m]is a cofibrant cosimplicial space for each 0 m 1.
The total space of a cosimplicial space X is defined as TotX = Map ( , X)
If X is not fibrant, the total space might not give you the "right" homotopy
type. In this case, we have to choose a fibrant replacement_Z_of X, that is a
weekly equivalent, fibrant cosimplicial space, and define TotX = TotZ.
When the cosimplicial space has a group action one can choose an equiv-
ariant fibrant replacement in the following sense:
Lemma 3.1. Let G be a simplicial group and X a cosimplicial G-space.
Assume that Xn is a fibrant simplicial set for each n 0. Then there is
a cosimplicial G-space E(X) such that both E(X) and E(X) =G are fibrant
cosimplicial spaces and such that the following diagram commutes:
EG x X --~-! E(X)
? ?
? ?
y y (1)
EG xG X --~-! E(X) =G
Here the vertical maps are the obvious quotient maps, and the horizontal
maps are weak equivalences. The map E(X) ! E(X) =G is the pullback of
the principal G-fibration cEG ! cBG over a fibration E(X) =G ! cBG.
Proof. By the model category properties we can factor the projection map
EGxG X ! cBG as a composite pOi where i : EGxG X ! Y is a cofibration
which is simultaneously a weak equivalence, and p : Y ! cBG is a fibration.
BG is a fibrant space by [GJ ] Lemma I.3.5 so cBG is a fibrant cosimplicial
space. Thus Y is fibrant.
We form the codegree wise pullback of ß : cEG ! cBG over p.
~p
E(X) _______E(X) --- ! cEG
? ? ?
? p? ?
y i y iy
~= p
E(X) =G -- - ! Y --- ! cBG
The principal G-action (in the sense of [M3 ]) of G on EG gives a principal
G-action on E(X) n for each n and an isomorphism of cosimplicial spaces
E(X) =G ~=Y as written in the diagram. By [B1 ] Lemma 7.1 it follows that
ßp is a fibration so E(X) is fibrant.
By the pullback property we can lift the map i to a map EGxX ! E(X) .
This constructs the missing map in the statement of the lemma. In each
codegree (1) is a map of fibrations over BG and we conclude that the lifting
__
is also a weak equivalence. |__|
5
Theorem 3.2. Let X be a fibrant cosimplicial space and G a simplicial group.
Then XG is a cosimplicial G-space and we can form its equivariant fibrant
replacement E(XG ). There is a natural map of fibrations of simplicial sets
for each m with 0 m 1:
(Tot mX)G --- ! EG xG (Tot mX)G -- - ! BG
? ? ?
~?y ~?y ~=?y
Totm (E(XG )) --- ! Totm (E(XG )=G) -- - ! Totm (cBG)
The first and middle vertical maps are weak equivalences and the right vertical
map is an isomorphisms of simplicial sets.
Proof. Since X is fibrant each Xn is fibrant such that (XG )n = (Xn)G is
fibrant by [M3 ] Theorem 6.9. Hence we can form E(XG ).
By [M3 ] Definition 20.3 and Theorem 20.5 we have that the top vertical
line in the diagram is a fiber bundle. By [BK ] X.5. SM7 and the fact that
[m] 2 cS is cofibrant we see that if p : A ! B is a fibration in cS then
Tot m(p) : Tot mA ! Tot mB is a fibration in S. In particular Tot mX is
fibrant since X is fibrant and by [M3 ] Theorem 6.9 we have that (Tot mX)G
is fibrant. Thus the top vertical line is a Kan fiber bundle and hence a
fibration by [M3 ] Lemma 11.9. The lower vertical line is Totm of a fibration
and hence a fibration.
There is a commutative diagram as follows:
(Tot mX)G --- ! EG xG (Tot mX)G -- - ! BG
? ? ?
~=?y fm?y ~=?y
Totm (XG ) --- ! Tot m(EG xG XG ) -- - ! Totm (cBG)
? ? fl
~?y ?y flfl
Totm (E(XG )) --- ! Totm (E(XG )=G) -- - ! Totm (cBG)
The isomorphism (Tot mX)G ~=Tot m(XG ) is one of the axiomatic isomor-
phisms in a simplicial model category. We examine it closer in order to
define fm . A cosimplicial space is a diagram in S and the axiomatic isomor-
phism comes from the corresponding isomorphism in the simplicial model
category S. For A, B, C 2 S this isomorphism is the composite
F : (AB )C ~= ABxC ~=ACxB ~=(AC )B
The following commutative diagram shows that F is equivariant with
6
respect to actions of the monoid CC .
CC x (AB )C _______________O______________//_(AB )C
| |
| |
fflffl|i2x1 O fflffl|
CC x ABxC _____//(B x C)BxC x ABxC _____//ABxC
| |
| |
fflffl|i1x1 O fflffl|
CC x ACxB _____//(C x B)CxB x ACxB _____//ACxB
| |
| |
fflffl| ix1 (O)B fflffl|
CC x (AC )B _______//_(CC )B x (AC )B_____//_(AC )B
For Z 2 S the action of G on the mapping space ZG is defined by
ad(~)x1 G G O G
G x ZG -- - - ! G x Z --- ! Z
where ad(~) denotes the adjoint of the product ~ : G x G ! G. So taking
C = G in the above we see that F is G-equivariant such that we have a map
1 xG F : EG xG (AB )G ! EG xG (AG )B
The composite
EG x (AG )B --ix1-!(EG x AG )B --- ! (EG xG AG )B
factors through EG xG (AG )B and we compose with 1 xG F to get a map
EG xG (AB )G ! (EG xG AG )B
The morphism fm in the theorem is codegree wise given by this map.
The lover part of the diagram is induced by (1). The functor (-)K :
cS ! cS where K 2 S preserves fibrations as one sees from the right lifting
property by taking adjoints. Hence XG is fibrant since X is fibrant. By [BK ]
X.5.2 we get a weak equivalence when applying Totm to a weak equivalence
between fibrant cosimplicial spaces. Thus the left vertical map is a weak
__
equivalence. The result follows. |__|
4 Strong convergence
In this section we discuss convergence of the Bousfield homology spectral
sequence associated with ( RX)hT where R = Fp, the field on p elements.
We use Fp coefficients everywhere unless stated otherwise.
7
Proposition 4.1. If X is a 1-connected space then X and XhT are nilpo-
tent spaces. In fact we have ß1( X)- respectively ß1( XhT)-central series as
follows for each i 1:
ßi( X) ßi( X) 0, (2)
ßi( XhT) ßi( X) ßi( X) 0. (3)
Proof. (2) The fibration X ! X ! X splits by the constant loop inclu-
sion X ! X. So we have ßi( X) ~= ßi( X) ßi(X) for i 1. Since the
action of the fundamental group is natural there is a commutative diagram
ß1( X) x ßi( X) --- ! ßi( X)
? ?
? ?
y y
ß1( X) x ßi( X) --- ! ßi( X)
? ?
? ?
y y
ß1(X) x ßi(X) --- ! ßi(X)
We have ß1( X) ~=ß1( X) since X is simply connected. Further ß1( X)
acts trivially on ßi( X) since X is an H-space. From the upper square we
see that the filtration (2) is ß1( X)-stable and that the action on ßi( X)
is trivial. Since ß1(X) = 0 the lower square shows that the action on the
quotient ßi( X)=ßi( X) is trivial.
(3) The fibration X ! XhT ! BT splits by a map constructed from a
constant loop. So for i 1 we have ßi( XhT) ~=ßi( X) ßi(BT). Especially
ß1( XhT) ~=ß1( X). By naturality there is commutative diagram
ß1( X) x ßi( X) -- - ! ßi( X)
? ?
? ?
y y
ß1( XhT) x ßi( XhT) -- - ! ßi( XhT)
? ?
? ?
y y
ß1(BT) x ßi(BT) -- - ! ßi(BT)
From the upper square we see that the inclusion ßi( XhT) ßi( X) is
ß1( XhT)-stable. The lower square shows that the action on the quotient
ßi( XhT)=ßi( X) is trivial. The rest of the sequence (3) has the desired
__
properties since (2) is a ß1( X)-central series. |__|
Proposition 4.2. If X is a 1-connected space then the cosimplicial space
E( RX)=T is R-pro-convergent.
8
Proof. This is a consequence of [B1 ] 3.3. Via the weak equivalences from
Lemma 3.1 we can use the filtrations from Proposition 4.1 in each codegree.
Then the quotients are ßi(cBT), ßi(RX) and ßi+1(RX). Hence it suffices to
show that when n 0 the following holds for all m 0:
ßm ßm+n (cBT) = 0, ßm ßm+n (RX) = 0, ßm ßm+n+1 (RX) = 0. (4)
Clearly ßm ßm+n (cBT) = 0 unless m + n = 2 and ß2-nß2(cBT) = 0 since
the differentials in the complex ß2(cBT) are alternating zeros and ones.
By the proof of 6.1 in [BK ], Ch. I and Proposition 6.3 in Ch. X The
following holds for any space Y : If eHi(Y ; R) = 0 for i k then ßjßi(RY ) = 0
__
for i k + j. So the last two groups in (4) are also zero. |__|
Lemma 4.3. Let X be a 1-connected space with H*X of finite type. Then
RsX is 1-connected and H*RsX is of finite type for each 0 s < 1.
Proof. By [BK ] I.6.1 we have thatQRsX is 1-connected for each s. Recall
that R(Y ) is weakly equivalent to 1n=0K(H~n(Y ), n) for any space Y . So if
H*Y is of finite type then H*R(Y ) is also of finite type and ßiR(Y ) = H~iY
is finite for each i. Hence ßi((RX)m ) is finite for each i, m. From [S] Lemma
2.6 we see that ßi(RsX) is finite for each i, s. By the Postnikov tower for
__
RsX we conclude that H*RsX is of finite type for each s. |__|
Lemma 4.4. Let . . .! C*(2) ! C*(1) ! C*(0) be a sequence of maps of
chain complexes. If for all n and m the group Cn(m) is finite, then there is
an isomorphism Hn(lim C*(m)) ~=lim Hn(C*(m)) for all n.
Proof. This is a consequence of the lim1-sequence which can be found in e.g.
__
[M2 ] Appendix A5. |__|
Proposition 4.5. Let G be a simplicial group such that Hn(BG) is finite
for all n. Let {Zm } be a tower of G-spaces and put Z1 = limZm . Assume
that {H*(Z1 )}m 0 ! {H*(Zm )}m 0 is a pro-isomorphism and that Hn(Zm )
is finite for all integers n, m. Then {H*((Z1 )hG )}m 0 ! {H*((Zm )hG )}m 0
is also a pro-isomorphism.
Proof. We have Leray-Serre spectral sequences for 0 m 1 as follows:
E2(m) = H*(BG; H*(Zm )) ) H*((Zm )hG ).
The tower map {E2i,j(1)}m 0 ! {E2i,j(m)}m 0 is a pro-isomorphism for all i
and j by the pro-isomorphism in the assumption, so E2i,j(1) ~=lim E2i,j(m).
By the assumptions on the homology of BG and Zm , the groups E2i,j(m)
with m < 1 are all finite so by Lemma 4.4 we have E3i,j(1) ~=lim E3i,j(m).
9
By induction Eri,j(1) = limEri,j(m) for each r and since we have only finite
filtrations E1i,j(1) ~= lim E1i,j(m). Since E1i,j(m) is finite for all i, j, m*
* it
follows that {E1 (1)}m 0 ! {E1 (m)}m 0 is a pro-isomorphism. The result
__
follows by the five lemma [BK ] III 2.7. |__|
Theorem 4.6. If X is a 1-connected fibrant space with H*(X; Fp) of fi-
nite type, then the Bousfield spectral sequence {Er( RXhT; Fp)} converges
strongly to H*( (X^p)hT; Fp) ~=H*( XhT; Fp).
Proof. Let Y = RX. The spectral sequence abuts to the homology of the
total space of a fibrant replacement of YhT. We choose the fibrant replace-
ment E(Y)=T from Lemma 3.1. The total space of this fibrant replacement
is weakly equivalent to (X^p)hT by Theorem 3.2. Thus the spectral se-
quence converges to the stated result if it converges. (A Leray-Serre spectral
sequence argument shows that we can remove the p-completion inside the
homology group.)
We have shown in Proposition 4.2 that the spectral sequence is pro-
convergent. Hence it suffices to show that Pn(E(Y)=T) or equivalently
Pn(YhT) is a pro-isomorphism. By the Eilenberg-Moore spectral sequence
and Lemma 4.3 we see that H*(Tot sY) ~= H*( RsX) is of finite type for
each 0 s < 1. By Proposition 4.5 and Proposition 2.2 the result fol-
__
lows. |__|
We now change to cohomology. The dual of Proposition 2.2 and of The-
orem 4.6 is as follows:
Theorem 4.7. If X is a 1-connected and fibrant space with H*X of finite
type then we have strongly convergent Bousfield cohomology spectral sequences
E^r) H*( X), ^E-m,t2= (ßm H*( RX))t
Er ) H*(( X)hT), E-m,t2= (ßm H*(( RX)hT))t.
We are going to give a description of the E2-terms as certain non Abelian
derived functors evaluated at H*X. In the next section we set up categories
relevant for this purpose.
5 The category F and the simplicial model
category sF
For a fixed prime p we let A denote the mod p Steenrod algebra and K
the category of unstable A-algebras. The category of non-negatively graded
10
unital Fp-algebras with the property that A0 is a p-Boolean algebra (ie. x =
xp for all x 2 A0) is denoted Alg . In [O1 ], [O2 ] we defined a category F with
forgetful functors K ! F ! Alg as follows:
Definition 5.1. An object in F consists of an object A in Alg which is
equipped with an Fp-linear map ~ : A ! A with the following properties:
o |~x| = p(|x| - 1) + 1 for all x 2 A.
o ~x = x when |x| = 1 and if p is odd and |x| is even then ~x = 0.
o ~(xy) = ~(x)yp + xp~(y) for all x, y 2 A.
Furthermore A is equipped with an Fp-linear map fi : A ! A with the
following properties:
o |fix| = |x| + 1 for all x 2 A.
o fi O fi = 0 and if |x| = 0 then fix = 0.
o fi(xy) = fi(x)y + (-1)|x|xfi(y) for all x, y 2 A.
If p = 2 we require that fi = 0. A morphism f : A ! A0 in F is an algebra
homomorphism such that f(~x) = ~0f(x) and f(fix) = fi0f(x).
Remark 5.2. For an object K 2 K the map ~ : K ! K is defined by
~x = Sq|x|-1x when p = 2 and ~x = P (|x|-1)=2x when p is odd and |x| is odd.
The map fi is the Bockstein operation when p is odd.
There is an obvious product on F. There is also a coproduct. For two
objects A and A0 in F the coproduct A A0 is the tensor product of the
underlying objects in Alg equipped with maps ~ * ~0 and fi * fi0 as follows;
~ * ~0(x y) = ~(x) yp + xp ~0(y)
fi * fi0(x y) = fi(x) y + (-1)|x|x fi0(y)
In appendices in [O1 ] and [O2 ] we showed that F is complete and cocomplete.
It is well known that K and Alg also possess these properties.
In the following R denotes any one of the categories K, F or Alg . Let
nFp denote the category of non-negatively graded Fp-vector spaces. The free
functor SR : nFp ! R is by definition the left adjoint of the forgetful functor
R ! nFp. If X is a non-negatively graded set we put SR (X) = SR (Fp X)
where Fp X is the free graded Fp-vector space with basis X. In particular
we have free objects SR (xn) on one generator xn of degree n.
11
Remark 5.3. Note that SF (V ) = SAlg(V~) where
M
~V= V ~iV * 2 , p = 2
i 1
M
~V= V fiV * 1 fi ~i(fiV even,* 2 V odd,* 2) , p > 2.
i 1, 2{0,1}
In the following we use [Q ] II.4 Theorem 4 to see that the category sR
of simplicial objects in R is a simplicial model category. The arguments are
standard but we have included them anyhow.
We start by verifying that R has enough projectives. Recall that a mor-
phism f : X ! Y in a category D is called an effective epimorphism if for
any object T and morphism ff : X ! T there is a unique fi : Y ! T with
fi O f = ff provided ff satisfies the necessary condition that ff O u = ff O v
whenever u, v : S ' X are maps such that f O u = f O v ([Q ] II.4 proof of
Proposition 2).
Proposition 5.4. Let f be an effective epimorphism in a category D. Then
f is an epimorphism. Furthermore if f can be factored as f = i O p where i
is a monomorphism then i is an isomorphism.
Proof. Assume that f is an effective epimorphism. Let r, s be two parallel
arrows such that r O f = s O f. Then for ff = r O f we have fi O f = ff both
for fi = r and fi = s. So by uniqueness r = s. Thus f is an epimorphism.
Assume that f = i O p where i is a monomorphism. If f O u = f O v for
two parallel arrows u, v then i O p O u = i O p O v and p O u = p O v since i is
a monomorphism. Hence there exists an arrow j such that p = j O f. Now
i O j O f = i O p = f which implies that i O j = id since f is an epimorphism.
Furthermore i O j O i = id O i = i which implies that j O i = id since i is a
__
monomorphism. |__|
Proposition 5.5. A morphism in R is an effective epimorphism if and only
if it is a surjection on underlying graded sets.
Proof. Any morphism f : X ! Y may be factored as X ! f(X) ! Y where
the last map is clearly a monomorphism. So by the previous proposition we
see that an effective epimorphism is surjective.
Assume that f : X ! Y is a surjection and let fi : X ! T be a map
which satisfies fi O u = fi O v whenever f O u = f O v. For a given x 2 kerf
let n = |x| and define u, v : SR (xn) ' X by u(xn) = x and v(xn) = 0. Then
x 2 kerfi so we have kerf kerfi. Now ff : Y ! T with ff(f(a)) = fi(a) is
__
well defined and has ff O f = fi. |__|
12
Recall that in [Q ] an object P in a category D is called projective if
Hom D(P, -) sends any effective epimorphism to an surjection of hom-sets.
Proposition 5.6. The following statements hold in the category R:
1. SR (V ) is projective for any object V in nFp.
2. R has enough projectives.
3. {SR (xn)|n 0} is a set of small projective generators.
Proof. (1) By taking adjoints and applying the previous proposition we see
that SR (V ) is projective. (2) Let U : R ! nFp denote the forgetful functor
and let X be an object in R. The adjoint j : SR (U(X)) ! X of idU(X)
is surjective and hence an epimorphism. Thus R has enough projectives.
(3) The object SR (xn) is projective by (1). Since Hom R(SR (xn), X) = Xn
we have that Hom R(SR (xn), -) commutes with filtered colimits so SR (xn)
is small. Finally, for two different morphisms f, g : X ' Y there exist an
x 2 X such that f(x) 6= g(x). Hence the map SR (xn) ! X with xn 7! x
where n = |x| separates f and g such that we have a set of generators as
__
stated. |__|
We now turn to the category sR of simplicial objects in R. The homotopy
groups of an object R in R is defined as the homology ß*R = H*(R, @) where
@ is the differential given by the alternating sums
Xn
@ = (-1)idi : Rn ! Rn-1.
i=0
Especially ß0(R) = R=(d0 - d1)R and we have a morphism ffl : R ! ß0(R) in
sR given by projection where we view ß0(R) as a constant simplicial object.
If f : X ! Y is a morphism in R we can form the diagram
X -- ffl-!ß0X
? ?
f?y i0f?y
Y -- ffl-!ß0Y
One says that f is surjective on components if the map from X into the
pullback (f, ffl) : X ! Y xi0Y ß0X is a surjection. Note that if ß0(f) is an
isomorphism then f is surjective on components if and only if f is surjective.
Proposition 5.7. There is a simplicial model category structure on sR as
follows:
13
o f : X ! Y is a weak equivalence if ß*f : ß*X ! ß*Y is an isomor-
phism.
o f : X ! Y is a fibration if it is surjective on components and an acyclic
fibration if it is both a fibration and a weak equivalence.
o f : X ! Y is a cofibration if for any commutative diagram
X -- - ! A
? ?
f?y p?y (5)
Y -- - ! B
where p is an acyclic fibration, there exist a map Y ! A making both
triangles commute.
The solution to the arrow diagram (5) is unique up to simplicial homotopy
under X and over B.
Proof. This is a special case of [Q ] II x4 Theorem 4. The uniqueness part
__
follows from [Q ] II x2 Proposition 4. |__|
Note that the cofibrations are described in an indirect way. The concept
of an almost free map make up for this weakness. See [Q ] II page 4.11 Remark
4 and the main source [M4 ] x3, [M5 ] x2 or [G ].
Definition 5.8. Let e denote the subcategory of the simplicial category
with objects [n] = {0, 1, . .,.n} for n 0 and morphisms the order preserving
maps which sends 0 to 0. An almost simplicial object in a category C is a
functor from e opto C.
Definition 5.9. A morphism f : X ! Y in sR is called almost free if there
is an almost simplicial sub vector space V of Y such that for each n 0, the
natural map Xn SR (Vn) ! Yn is an isomorphism.
Proposition 5.10. (1) Almost free morphisms are cofibrations in sR
(2) Any morphism A ! B may be factored canonically and functorially
as A ! X ! B where the first map is almost free and the second is an
acyclic fibration.
(3) Any cofibration is a retract for an almost free map.
__
Proof. Similar to the one given in [M4 ]. See also [G ]. |_*
*_|
Definition 5.11. A simplicial resolution of an object A 2 sR is an acyclic
fibration P ! A in sR with P cofibrant. An almost free resolution of A is
an acyclic fibration Q ! A such that Fp ! Q is almost free.
14
Note that an almost free resolution is a resolution and that almost free
resolutions always exist by the above proposition.
Example 5.12. Let RX be the cosimplicial resolution of a space X. Then
H*(RX) is an almost free resolution of H*X in each of the categories K, F
and Alg .
6 Derived functors
In this section R denotes any of the categories K, F or Alg . We use the
following notation for non Abelian derived functors:
Definition 6.1. The homology of an object R in sR with coefficients in a
functor E : R ! Alg is defined by
H*(R; E) = ß*E(P )
where P ! R is a simplicial resolution of R. By the uniqueness statement
in Proposition 5.7 this homology theory is well defined and functorial in R.
For an object R 2 R we also write R for the corresponding constant
simplicial object in sR. We are mainly interested in H*(R; E) when R 2 R.
These homology groups have certain properties which we now describe.
Let E, F and G be functors from R to Alg with natural transformations
E ! F ! G. Let V : Alg ! nFp denote the forgetful functor to graded
Fp-vector spaces. If the sequence 0 ! V E ! V F ! V G ! 0 is short exact
when evaluated on any free object in R then we get a long exact sequence
. . .Hi(R; E) Hi(R; F ) Hi(R; G) Hi+1(R; E) . ...
The 0th homology group is sometimes given by the following result:
Lemma 6.2. Define the category R0 as we defined the category R except
that we do no longer require that objects are unital. Let F : R0 ! Alg 0be
a functor. Assume that for every surjective morphism f : A ! B in R0 the
following two conditions hold:
1. F (f) : F (A) ! F (B) is surjective
2. F (ker f) ! F (A) ! F (B) is exact
then H0(C; F ) ~=F (C) for all objects C in R.
15
Proof. Let P ! C be a simplicial resolution of C. From the normalized
chain complex N*F (P ) we see that H0(C; F ) = F (P0)=F (d1)(ker F (d0)).
The maps d0, d1 : P1 ! P0 are surjective by the simplicial identities.
Let i : ker d0 ! P1 denote the inclusion. By condition 2. we have that
kerF (d0) = F (i)(F (ker d0)). Thus
F (d1)(ker F (d0)) = F (d1) O F (i)(F (ker d0)).
There is a commutative diagram
d01
kerd0 -- - ! d1(ker d0)
? ?
i?y j?y
P1 --d1-! P0
where d01denotes the restriction of d1 and j is the inclusion. By this diagram
F (d1) O F (i) = F (j) O F (d01). Furthermore F (d01)(F (ker d0)) = F (d1(ker d*
*0))
by condition 1. So we have
F (d1)(ker F (d0)) = F (j) O F (d01)(F (ker d0)) = F (j)(F (d1(ker d0)))
and H0(C; F ) = F (P0)=F (j)(F (d1(ker d0))).
Using condition 1. and 2. on the projection map P0 ! P0=d1(ker d0) we
__
see that H0(C; F ) ~=F (P0=d1(ker d0)) ~=F (C). |__|
The following result can sometimes be used to compute derived functors
of pushouts. We denote the pushout of a diagram A0 A ! A00in sR or R
by A0 A A00.
Proposition 6.3. Let E : R ! Alg be a functor.
(1) If there is a natural isomorphism E(A0 A00) ~= E(A0) E(A00) for
objects A0, A00in R then there is an isomorphism
H*(B0 B00; E) ~=H*(B0; E) H*(B00; E) for B0, B002 R.
(2) Assume that there is a natural isomorphism
E(A0 A A00) ~=E(A0) E(A) E(A00)
for diagrams A0 A ! A00in R. Assume further that B0 B ! B00is
a diagram in R such that Tor Bi(B0, B00) = 0 for i > 0. Then there is a first
quadrant spectral sequence as follows:
E2i,j= TorH*(B;E)i(H*(B0; E), H*(B00; E))j ) Hi+j(B0 B B00; E).
16
Proof. Let P ! B be a simplicial resolution of B. By the factorization axiom
we get a diagram
P 0oo__oPo__//_//_P 00
| | |
|~ |~ |~
fflffl|fflffl|fflffl|fflffl|fflffl|fflffl|
B0 oo___B _____//B00
where the vertical maps are acyclic fibrations and the upper horizontal maps
are cofibrations as indicated. Since Fp ! P is a cofibration and cofibrations
are stable under composition we see that P 0! B0and P 00! B00are simplicial
resolutions.
Now form the map of pushouts f : P 0 P P 00! B0 B B00and consider
the corresponding map of derived tensor products in the sense of [Q ] II x6:
Lf : P 0 P P 00= P 0 LPP 00! B0 LBB00.
By [Q ] II x6 Theorem 6 there are second quadrant spectral sequences
Tor i*Pi(ß*P 0, ß*P 00)j ) ßi+j(P 0 P P 00)
Tor i*Bi(ß*B0, ß*B00)j ) ßi+j(B0 LBB00)
The above diagram gives a map of spectral sequences which is an isomorphism
at the E2-terms. Hence Lf is a weak equivalence. By the Corollary following
Quillen's Theorem 6 we have that B0 LBB00! B0 B B00is a weak equivalence.
Thus f is itself a weak equivalence.
Since f is surjective it is a fibration. Since the pushout of a cofibration
is a cofibration P 0! P 0 P P 00is a cofibration and thus the domain of f is
cofibrant. So f is a simplicial resolution.
For the proof of (1) take B = Fp and apply E codegree wise. The re-
sult follows by the Eilenberg-Zilber theorem. For the proof of (2) apply E
__
codegree wise. The result follows by Quillen's Theorem 6. |__|
If one knows that the higher derived functors vanish on a certain class
of objects, they can be used to compute derived functors by the following
result.
Theorem 6.4. Let E : R ! Alg be a functor and let A 2 R. Assume that
Qi~ A is an acyclic fibration in sR and that
(
F (Qj) , i = 0
Hi(Qj; E) =
0 , i > 0.
Then H*(A; E) ~=ß*E(Q).
17
Proof. We have shown that sR is a simplicial model category. So ssR is a
simplicial model category by the Reedy structure [GJ ] VII 2.13. A fibration
in ssR is especially a level fibration and a cofibration is especially a level
cofibration by [GJ ] VII 2.6. A weak equivalence is a level weak equivalence
by definition.
We use a dot to denote a simplicial direction in the following. Let cQoo
denote the object in ssR defined by (cQ)ij = Qj for all i. Let Poo be a
resolution of cQoo ie. (Fp)oo æ Pooi~cQoo.
We have that (Fp)o æ Pio~iQo for each i by the above. By composition
with the acyclic fibration Qoi~A we see that Piois a resolution of A.
So the horizontal homotopy of F (Poo) is given by ßhjF (Pio) = Hj(A; F ).
We apply vertical homotopy on this and obtain
(
Hj(A; F ) , i = 0
ßvißhjF (Poo) ~=
0 , i > 0.
We also have that Poj is a resolution of Qj for each j. So ßviF (Poj) ~=
Hi(Qj; F ) which equals F (Qj) for i = 0 and equals 0 for i > 0. We apply
horizontal homotopy on this and obtain
(
ßjF (Qo) , i = 0
ßhjßviF (Poo) ~=
0 , i > 0.
Thus both spectral sequences associated with F (Poo) collapse and the result
__
follows. |__|
7 The E2 terms seen as derived functors
__
In [BO ], [O1 ] and [O2 ] we introduced a functor : F ! Alg as follows:
__
Definition 7.1. (R) is the quotient of the free graded commutative and
unital R-algebra on generators {dx|x 2 R} of degree |dx| = |x| - 1, modulo
the ideal generated by the elements
d(x + y) - dx - dy, d(xy) - d(x)y - (-1)|x|xd(y),
d(~x) - (dx)p, d(fi~x).
__ __
There is a differential d : (R) ! (R) given by d(x) = dx for x 2 R.
__
Note that for p = 2 the Bockstein is trivial so here the functor is the
same as the functor which we originally denoted ~.
18
__
It was shown that there is a lift to a functor : K ! K and that this lift
is nothing but_Lannes' division functor (- : H*(T))K . In particular there is
a morphism (H*X) ! H*( X) for any space X which is an isomorphism
when H*X is a free object in K.
An other functor ` : F ! Alg was also defined by generators and rela-
tions. The explicit definition was rather complicated so we do not repeat it
here. The functor ` also lifts to an endofunctor on K and it comes with a
natural morphism `(H*X) ! H*( XhT) which is an isomorphism if H*X is
a free object in K.
Via Example 5.12 we can now restate Theorem 4.7 in an appropriate
form.
Theorem 7.2. If X is a 1-connected and fibrant space with H*X of finite
type then we have strongly convergent Bousfield cohomology spectral sequences
E^r) H*( X) and Er ) H*( XhT) with the following E2 terms:
^E-m,t2~=Hm (H*(X); __)t and E-m,t2~=Hm (H*(X); `)t.
We now introduce other functors in order to study the derived functors
of `. Recall that the functors L and e from F to Alg are defined by L(R) =
`(R)=(u) and e (R) = L(R)=(ffi(x)|x 2 R).
Proposition_7.3._For each object R 2 F there are isomorphisms as follows:
H0(R; ) ~= (R), H0(R; e) ~=e (R) and H0(R; L) ~=L(R).
Proof._We use Lemma 6.2 to prove this. By their definitions we may consider
, e and L as functors from F0 to Alg 0.
Let A be an object in F0 and let I A be an ideal. We must verify
condition 1. and 2. in Lemma 6.2 for these functors where f : A ! A=I is
the natural projection. We do this for the functor L. The verification for the
other functors is similar but easier.
The map L(f) is surjective with kernel
J = (OE(x) - OE(y), q(x) - q(y), ffi(x) - ffi(y)|x - y 2 I) L(A)
so L(A)=J ~= L(A=I). We must check that L(I) = J.
The inclusion L(I) J holds since OE(0) = q(0) = ffi(0) = 0.
For the inclusion L(I) J assume first that p = 2. Since OE and q
are additive we have that ffi(x) - ffi(y) and OE(x) - OE(y) lie in L(I). Furth*
*er
q(x) - q(y) = q(x - y) + ffi(xy) but ffi(xy) = ffi(x(y - x)) so also q(x) - q(y*
*) 2
L(I). Thus the inclusion holds.
19
For p odd ffi is additive, OE is additive on elements of even degree and q is
additive on elements of odd degree. For |x| = |y| odd we have
p-2X
OE(x) - OE(y) = OE(x - y) + ffi(x)iffi(y)p-2-iffi(xy)
i=0
and again ffi(x(y - x)) = ffi(xy) such that this lies in L(I). For |x| = |y| ev*
*en
we have
p-1X
1 i p-i
q(x) - q(y) = q(x - y) - ffi( __x y )
i=1i
so it suffices to see that y - x divides the sum inside the ffi(-). The followi*
*ng
equation in Fp[x, y] shows that this is the case:
p-3X p-1X k
1 i p-i X 1
xy(y - x) akxkyp-3-k = __x y where ak = _____.
k=0 i=1 i j=0 j + 1
P p-1 __
The equation holds since by Euler's sum formula n=1 n = 0 modulo p. |__|
Definition 7.4. Let Z, B, H : F ! Alg denote the functors given by
Z(R) = ker(d), B(R) = im(d), H(R) = Z(R)=B(R)
__
where d is the differential on (R).
Recall from [BO ], [O1 ] and [O2 ] that there are natural transformations
of functors : e ! H and Q : L ! Z. It was shown that if A 2 F is
a free object, or its underlying algebra is polynomial, then A and QA are
isomorphisms. We can now give a nice interpretation of the functor L, which
was originally defined by generators and complicated relations.
Theorem 7.5. For any R 2 F one has L(R) ~=H0(R; Z).
Proof. The induced map Q* : H*(R; L) ! H*(R; Z) is an isomorphism since
Q is an isomorphism on free objects. The 0th derived functor of L was
__
computed in Proposition 7.3. |__|
For any functor E : F ! Alg we have that Hi(A; F ) = 0 for i > 0 when A
is a free object since we can use the trivial almost free resolution to compute
the derived functors. For polynomial algebras we also have nice results.
Theorem 7.6. Assume that the underlying_algebra of A 2 F is a polynomial
algebra. Then one has Hi(A; ) = 0, Hi(A; e) = 0, Hi(A; L) = 0 and
Hi(A; `) = 0 for each i > 0.
20
__
Proof. We first prove the statements for and e. Let : Alg ! Alg denote
the usual de Rham complex functor. Pick an almost free resolution P 2 sF
of A. The forgetful functor U : F ! Alg takes free objects to free objects.
So we can apply U to P and get an almost free resolution of U(A) in sAlg .
Thus there is an isomorphism
HFi(A; U) ~=HAlgi(U(A); )
and the last group is trivial for_i_> 0 since U(A) is a free object in Alg .
There is a linear map U(A) ! (A); x0dx1 . .d.xn 7! x0dx1 . .d.xn.
The map is not multiplicative and it does not commute with the de Rham __
differential, but it is an isomorphism of graded vector spaces. Thus Hi(A; )
is additively isomorphic to Hi(A; U) which is trivial for i > 0. A similar
isomorphism gives the result for the functor e .
__ Next we consider the functor L. The short exact sequence 0 ! Z !
! B ! 0 gives a long exact sequence of derived functors. By the above
this sequence breaks up into the exact sequence
__
0 ! H1(A; B) ! H0(A; Z) ! H0(A; ) ! H0(A; B) ! 0
together with the isomorphisms Hi(A; Z) ~=Hi+1(A; B) for i 1.
There is also a short exact sequence 0 ! B ! Z ! H ! 0 with corre-
sponding long exact sequence of derived functors. Since is an isomorphism
on free objects we have a natural isomorphism * : H*(-; e) ~= H*(-; H).
By the above vanishing result for H*(A; e) the long exact sequence breaks
up into the short exact sequence
0 ! H0(A; B) ! H0(A; Z) ! H0(A; H) ! 0
and the isomorphisms Hi(A; B) ~=Hi(A; Z) for i 1.
Using Proposition 7.3 and Theorem 7.5 we can rewrite the exact sequences
involving 0th derived functors as
__
0 ! H1(A; B) ! L(A) ! (A) ! H0(A; B) ! 0
0 ! H0(A; B) ! L(A) ! H(A) ! 0.
Since Q : L(A) ! Z(A) is an isomorphism we see that H1(A; B) = 0. By
the isomorphisms H1(A; B) ~=H1(A; Z) ~=H2(A; B) ~=. . .we conclude that
Hi(A; Z) is trivial for i > 0. But H*(-; L) is isomorphic to H*(-; Z) so we
are done.
Finally we consider the functor `. By definition of L there is a short
exact sequence 0 ! u` ! ` ! L ! 0. From the corresponding long exact
21
sequence of derived functors we find that Hi(A; u`) ~=Hi(A; `) for i > 0. By
Theorem 10.3 from the appendix and Proposition 4.4 from [O2 ] there is a
short exact sequence
0 ! uj+1`(B) ! uj`(B) ! uj e(B) ! 0 , j > 0 (6)
when B is a free object in F or when the underlying algebra of B is a poly-
nomial algebra. The corresponding long exact sequence of derived functors
shows that Hi(A; uj`) ~= Hi(A; uj+1`) so we have Hi(A; `) ~= Hi(A; uj`) for
all j 0. But (uj`)k = 0 for k < 2j so Hi(A; uj`)k = 0 for k < 2j and the
__
result follows. |__|
Proposition 7.7. If the underlying algebra of an object A 2 F is a polyno-
mial algebra, then H0(A; `) ~=`(A).
Proof. The short exact sequence 0 ! u` ! ` ! L ! 0 gives a short exact
sequence of 0th derived functors since H1(A; L) = 0. Furthermore, there is
a natural map H0(-; F ) ! F for any functor F : F ! Alg . So we have a
commutative diagram with exact rows as follows:
0 --- ! H0(A; u`) -- - ! H0(A; `) -- - ! H0(A; L) --- ! 0
? ? ?
? ? ?
y y y
0 --- ! u`(A) -- - ! `(A) -- - ! L(A) --- ! 0.
The right vertical map is an isomorphism so it suffices to show that the left
vertical map is also an isomorphism.
Since H1(A; e) = 0 the short exact sequence (6) gives a commutative
diagram as follows for j > 0:
0 --- ! H0(A; uj+1`) -- - ! H0(A; uj`) -- - ! H0(A; uj e) --- ! 0
? ? ?
? ? ?
y y y
0 --- ! uj+1`(A) -- - ! uj`(A) -- - ! uj e(A) --- ! 0.
where the right vertical map is an isomorphism. Fix a degree n. For j + 1 >
n=2 the map H0(A; uj+1`)n ! (uj+1`(A))n is an isomorphism since both
__
domain and target space are zero. The result follows by induction. |__|
Corollary 7.8. Let X be a 1-connected space such that H*X is of finite
type and H*X is a polynomial algebra. Then then the spectral sequences of
Theorem 7.2 collapses at the E2 terms. So there are isomorphisms
__* * * * *
H*(H*(X); ) ~=H ( X) and H*(H (X); `) ~=H (( X)hT).
22
8 The derived functors of an exterior algebra
In the rest of this paper we take p = 2. Let = (oe) 2 F be an exterior
algebra on one generator of degree |oe| 2. Note that ~oe = 0 for dimensional
reasons. We intend to compute the higher derived functors of the various
functors we have been considering for this algebra.
Proposition 8.1. There are isomorphisms
__ __
H*( ; ) ~= ( ) [!] , H*( ; e) ~=e ( ) [e!].
The inner degrees are |fli(!)| = i(2|oe| - 1), |fli(e!)| = i(4|oe| - 1) and the
grading of the homology groups are given by
__ __
Hi( ; ) ~= ( ) fli(!) , Hi( ; e) ~=e ( ) fli(e!).
Proof. The algebra is the pushout of F2 F2[y] ! F2[x] where y 7! x2.
Put ~x = 0 and ~y = 0. By Proposition 6.3 we find
__
Hi( ; E) ~=Tor E(F2[y])i(F2, E(F2[x])) for E = , e.
__
The result follows by standard computations. |__|
In order to compute derived functors of the other functors we need an
explicit simplicial resolution of . By Theorem 6.4, Theorem 7.6, Proposition
7.3 and Proposition 7.7 we may use an almost free resolution of in sAlg
and equip it with ~ = 0.
Proposition 8.2. There is an almost free resolution Ro 2 sAlg of the algebra
with Rn = F2[x, y1, y2, . .,.yn] for n 0. The structure maps di : Rn !
Rn-1 and si : Rn ! Rn+1 are given by
si(x) = x
(
yj , j i
si(yj) =
yj+1 , j > i
di(x) = x
8
>>x2 , i = 0, j = 1
><
yj-1 , i < j, j > 1
di(yj) =
>>yj , i j, j < n
>:
0 , i = n, j = n
The degrees of the generators are |x| = |oe| and |yi| = 2|oe| for all i.
23
Proof. We first give a description of the simplicial set 1o= Hom (-, [1])
suited for our purpose.
Define the elements yj 2 1nfor n 0 and 0 j n + 1 by yj(i) = 0
if i < j and yj(i) = 1 if i j. We have 1n= {y0, . .,.yn+1}. The structure
maps are as follows:
( (
yj-1 , i < j yj+1 , i < j
diyj = and siyj =
yj , i j yj , i j
Let F2[-] denote the functor which takes a graded set to the polynomial
algebra generated by that set. Let F2[ 1o, *] denote the pushout of F2
F2[a] ! F2[ 1o] where F2 and F2[a] are constant simplicial algebras. In degree
n the maps are as follows: a 7! 0 2 F2 and a 7! yn+1 2 F2[ 1o]. Note that
F2[ 1o] ' F2[*] by the simplicial contraction of 1o. The spectral sequence
[Q ] II x6 Theorem 6 gives that ßi(F2[ 1o, *]) ~=F2 for i = 0 and 0 otherwise.
Define Ro as the pushout of F2[x] F2[z] ! F2[ 1o, *] where in degree
n the maps are z 7! x2 and z 7! y0. For this pushout Quillen's spectral
sequence gives that
(
, i = 0
ßi(Ro) ~=Tor F2[z]i(F2, F2[x]) =
0 , i > 0
Thus Ro has the right homotopy groups. Further Rn is as stated and the
structure maps are as stated. Note that Ro is almost free. The degrees are
__
correct since the structure maps must be degree preserving. |__|
Lemma 8.3. Hn( ; ) has the following F2-basis:
dy1 . .d.yn, xdy1 . .d.yn, dxdy1 . .d.yn, xdxdy1 . .d.yn.
Proof. Using the formulas in Proposition 8.2 it is easy to check that the four
given classes are in the kernel of di for all i. To check linear independency,
we introduce two gradings of .
Firstly, the wedge grading on (Rn) is defined as the number of wedge
factors, ie. the number of d's in a homogeneous element. Secondly the
polynomial grading is defined as follows: Give x grading 1, yj grading 2 and
each dx or dyj grading 0 and extend multiplicatively. Note that the maps di
preserve both gradings. We write q,t(Rn) for the elements in (Rn) of wedge
degree q and polynomial degree t. Thus there is a direct sum decomposition
M
Hn( ; ) = Hn( ; q,t).
q,t 0
24
The classes we consider sit in different bigradings, so we only have to
check that they individually do not represent the trivial class.
We have the following bases for n,0(Rn+1), n,0(Rn) and n,0(Rn-1)
respectively:
{dxdy1 . .c.dyj.d.y.n-1} [ {dy1 . .c.dyj.c.d.yk.d.y.n+1},
{dxdy1 . .c.dyj.d.y.n-1} [ {dy1 . .d.yn},
{dxdy1 . .d.yn-1}.
We use the normalized complex consisting of \i>0ker(di) with differential
d0 to compute the homology. For this normalized complex we have the
respective bases ;, {dy1^. .^.dyn}, {dx^dy1^. .^.dyn-1}. Taking homology
and using that n,0(Rn-2) = 0 we see that the classes dxdy1 . .d.yn-1 and
dy1 . .d.yn do not represent zero.
Similarly, xdxdy1 . .d.yn-1 and xdy1 . .d.yn do not represent zero. The
__
result follows by shifting dimensions. |__|
Lemma 8.4. Hn( ; H) has an F2-basis as follows:
y1 . .y.ndy1 . .d.yn, x2y1 . .y.ndy1 . .d.yn,
xdxy1 . .y.ndy1 . .d.yn, x3dxy1 . .y.ndy1 . .d.yn.
__
Proof. This follows from Lemma 8.3 and the Cartier isomorphism. |__|
Corollary 8.5. The long exact sequence associated to B ! Z ! H splits
into short exact sequences when evaluated at :
0 ! H*( ; B) ! H*( ; Z) ! H*( ; H) ! 0.
Proof. The generators from Lemma 8.4 are visibly in the image of the map
__
Hn( ; Z) ! Hn( ; H). |__|
Definition 8.6. Let fli,j2 (Rj) denote the following elements:
fli,j= y1y2 . .y.idy1dy2 . .d.yj for i, j > 0
fl0,j= dy1dy2 . .d.yj for j > 0 and fl0,0= 1.
These elements are actually in Z( ) if i j, and in B( ) in case i < j.
They are even in the normalized chain complex, so they define classes in
H*( ; Z) respectively H*( ; B).
25
Theorem 8.7. For n 0 the homology group H2n( ; B) has F2-basis
{dxfl0,2n} [ {fl2i,2n, x2fl2i,2n, xdxfl2i,2n, x3dxfl2i,2n|0 i < n}
and the homology group H2n+1( ; B) has F2-basis
{fl0,2n+1, dxfl0,2n+1, xdxfl0,2n+1}[
{fl2i+1,2n+1, x2fl2i+1,2n+1, xdxfl2i+1,2n+1, x3dxfl2i+1,2n+1|0 i < n}.
Similarly, H2n( ; Z) has F2-basis
{dxfl0,2n} [ {fl2i,2n, x2fl2i,2n, xdxfl2i,2n, x3dxfl2i,2n|0 i n}
and H2n+1( ; Z) has F2-basis
{fl0,2n+1, dxfl0,2n+1, xdxfl0,2n+1}[
{fl2i+1,2n+1, x2fl2i+1,2n+1, xdxfl2i+1,2n+1, x3dxfl2i+1,2n+1|0 i n}.
Proof. Recall that Hm ( ; ) ~= ( ) fl0,m and Hm ( ; H) ~= e( ) flm,m .
From the splitting Hm ( ; Z) ~= Hm ( ; B) Hm ( ; H) together with the
computation of Hm ( ; H) in Lemma 8.4 , it follows that the statements
about Hm ( ; Z) and Hm ( ; B) are equivalent.
Recall the long exact sequence
Hm+1 ( ; B) --b-! Hm ( ; Z) --i*-! Hm ( ; ) --d*-! Hm ( ; B) (7)
Claim: The image of d* is a one dimensional vector space, generated by a
class, which in the normalized chain complex is represented by dxfl0,m.
Note that three out of four of the generators of Hm ( ; ) are annihilated
by d*. So the image of d* is spanned by the single class, represented in the
normalized complex by dxfl0,m 2 B( ). This element actually represents a
non zero homology class, since the map Hm ( ; B) ! Hm ( ; ) induced by
inclusion, send it to the element represented by dxdy1 . .d.ym . But this is
non-trivial according to Lemma 8.3, and the claim follows.
We will now prove the theorem by induction on m. We first treat the
case m = 0. Here we see that the map d0 : H0( ; ) ! H0( ; B) is surjective
by the long exact sequence (7), and the statement follows directly from the
claim we just proved.
Assume that the theorem holds for m. The long exact sequence (7) gives
us a short exact sequence
0 ! im(dm+1 ) ! Hm+1 ( ; B) !b ker(im ) ! 0
26
So, in order to prove the theorem, we have to show that b maps the classes
given in the statement of the theorem with dxfl0,m+1 removed to a basis for
the kernel of im .
By induction we have a basis for Hm ( ; Z). From this we see that the
kernel of the map i2n+1 : H2n+1( ; Z) ! H2n+1( ; ) has basis
{fl2i+1,2n+1, x2fl2i+1,2n+1, xdxfl2i+1,2n+1, x3dxfl2i+1,2n+1|0 i n}
and that the kernel of the map i2n : H2n( ; Z) ! H2n( ; ) has basis
{x2fl0,2n, x3dxfl0,2n} [ {fl2i,2n, x2fl2i,2n, xdxfl2i,2n, x3dxfl2i,2n|1 i *
* n}.
It is convenient now to pass from the normalized to the unnormalized
complex. The normalized complex is a subcomplex of the unnormalized one,
and in the notation we do not distinguish between a class in the subcomplex
and its image in the unnormalized one.
Let us first consider the odd case, m = 2n + 1. ThePelement fl2i,2n+22
B(R2n+2) is a cycle with respect to the boundary @ = dk. We compute its
image under the map b.
Let ffr 2 (R2n+2) denote the following element:
ffr = y1y2 . .y.2iyrdy1dy2 . .c.dyr.d.y.2n+2 , 2i + 1 r 2n + 2
P 2n+2
where the hat means that the factor is left out. Put fi = r=2i+2ffr. We
have dffr = fl2i,2n+2for each r, so that also dfi = fl2i,2n+2. This means that
b(fl2i,2n+2) is represented by
2n+2X
@fi = (yr-1 + yr)fl2i,2n+1+ y2n+1fl2i,2n+1= fl2i+1,2n+1.
r=2i+2
Since b is linear with respect to multiplication by x2, xdx, x3dx this gives
the desired result for H2n+2( ; B).
In the even case m = 2n, a similar argument shows that b(fl2i+1,2n+1) is
represented by fl2i+2,2n.
Checking with the lists of classes above, we see that we are left to prove
that b(fl0,2n+1) = x2fl0,2n. The argument is very similar. Let
ffr = yrdy1dy2 . .c.dyr.d.y.2n+1and fi = ff1 + ff2 + . .+.ff2n+1.
Then dffr = fl0,2n+1so also dfir = fl0,2n+1. Thus b(fl0,2n+1) is represented by
__
@fi = x2fl0,2n. |__|
27
Proposition 8.8. There are short exact sequences for i 0, t 1 as follows:
0 -- - ! Hi( ; u`) --- ! Hi( ; `) --- ! Hi( ; L) -- - ! 0 ,
0 -- - ! Hi( ; ut+1`) --- ! Hi( ; ut`) --- ! Hi( ; ute ) --- ! 0.
Proof. The first short exact sequence follows if we can prove that the con-
necting homomorphism b : Hi+1( ; L) ! Hi( ; u`) is trivial. So consider the
diagram
0 --- ! u`(Ri+1) --- ! `(Ri+1) --- ! L(Ri+1) --- ! 0
? ? ?
? ? ?
y y y
0 --- ! u`(Ri) --- ! `(Ri) --- ! L(Ri) --- ! 0.
The element q(y1) . .q.(yj)ffi(yj+1) . .f.fi(yi+1) 2 `(Ri+1) maps to the ele-
ment flj,i+12 L(Ri+1). By the relations ffi(a)2 = ffi(~a), q(a)2 = OE(~a) +
ffi(a2~a) and ffi(a)q(b) = ffi(a~b) + ffi(ab)ffi(b) we see that this element ma*
*ps
down to zero in `(Ri). So the connecting homomorphism b is trivial.
__
The proof for the second short exact sequence is similar. |__|
Corollary 8.9. For each i 0 there are isomorphisms of F2-vector spaces
M t
Hi( ; `)~=Hi( ; L) u Hi( ; e)
t 1
~=Hi( ; B) F2[u] Hi( ; e) .
Let F : F ! Alg be a functor. We_define_the total degree of a class in
Hi(R; F )n to be n - i. For F = `, this corresponds through the spectral
sequences of Theorem 7.2 to the grading of cohomology groups. We write
PF (t) for the Poincar'e series corresponding to the total degree of H*( ; F )*.
Theorem 8.10. Let s denote the degree of the class oe 2 . Then we have
the following Poincar'e series:
P__(t)= (1 + ts)(1 - ts-1)-1,
Pe (t)= (1 + t2s)(1 - t2s-1)-1,
PB(t) = ts-1(1 + ts-1 - t2s-1)(1 - t2s-1)-1(1 - t2s-2)-1,
P`(t)= (1 + ts-1 - ts+1 + t2s-1)(1 - t2)-1(1 - t2s-2)-1.
Proof. The first two formulas follow from Proposition_8.1: The total degree
of fli(!) is |fli(!)| - i = (2s - 2)i and ( ) = (doe) so
P__(t) = (1 + ts)(1 + ts-1)(1 - t2s-2)-1 = (1 + ts)(1 - ts-1)-1.
28
A similar argument gives Pe (t).
To determine PB(t) we must count the classes given in Theorem 8.7,
according to the total degree.
We divide these classes into three groups. The first group are those of
the form dxfl0,m. The Poincar'e series of the subspace generated by those
classes is ts-1=(1 - t2s-2). The second group are those of the type fl0,2n+1or
xdxfl0,2n+1. These have Poincar'e series t2s-2(1 + t2s-1)=(1 - t4s-4).
The third group is the remaining classes. They span a free e ( )-module
with basis X = {fl2i,2n, fl2i+1,2n+1|0 i < n, 0 n}. We introduce the
following operation on the set X: T (fli,n) = fli+1,n+1. This operation has
total degree 4s-2. All generators are obtained by applying T a non-negative
number of times starting from one of the elements of Y = {fl0,2n|n 1}.
The set Y has Poincar'e series t4s-4=(1-t4s-4). So, the set X has Poincar'e
series t4s-4(1 - t4s-4)-1(1 - t4s-2)-1. We multiply this by (1 + t2s)(1 + t2s-1*
*),
make a small reduction, and obtain the Poincar'e series for the third group
of classes:
t4s-4(1 + t2s)(1 + t2s-1)-1(1 + t4s-4)-1.
To get the Poincar'e series of B, we add the three series obtained so far.
ts-1 t2s-2(1 + t2s-1) t4s-4(1 + t2s)
PB(t) = _________+ _______________+ _____________________
1 - t2s-2 1 - t4s-4 (1 - t2s-1)(1 - t4s-4)
The stated formula for PB(t) follows after some reductions.
Finally, Corollary 8.9 gives that P`(t) = PB(t) + (1 - t2)-1Pe (t) which
__
leads to the stated formula after some reductions. |__|
9 The spectral sequences for spheres
Let X be a pointed space, Y = X it's reduced suspension. We have es-
tablished a spectral sequence converging to H*(ET xT Y ; F2) in general.
But in this special case, we are fortunate to have a direct calculation of the
homology H*(ET+ ^T Y ; F2). If the homology of X is of finite type, the
(finite) dimensions of these homology and cohomology groups agree. So if we
for the particular space X can check that the Poincar'e series of the homology
of H*(ET xT Y ; F2) as computed in [CC ] agrees with the Poincar'e series
of the E2 term of our spectral sequence, we know that our spectral sequence
collapses. (See also [BM ] for an easier proof of the results in [CC ]).
Let us now consider the special case of spheres X = Ss-1 and Y = Ss.
The purpose of this section is to show that in this case the two Poincar'e
series actually agree, forcing the spectral sequence to collapse.
29
Theorem 9.1. The Poincar'e series of H*(ET xT Ss; F2) is
(1 + ts-1 - ts+1 + t2s-1)(1 - t2)-1(1 - t2s-2)-1.
Proof. We compute a sequence of related Poincar'e series. First, let A =
H~*(Ss-1) considered as a graded vector space. This has Poincar'e series ts-1.
For each m 1, the cyclic group Cm acts on A m . This is a 1-dimensional
vector space over F2. We now consider the homology groups
H*(Cm ; A m )
We first look at homological dimension 0. H0(Cm , A m ) ~= A m , so it has
Poincar'e series tm(s-1). In higher homological degrees, there are two cases. If
m is odd, the groups all vanish, and we get a trivial Poincar'e series. If m is
even, and i 1, Hi(Cm , A m ) ~=A m Since this single group has homological
degree i, its Poincar'e series is ti+m(s-1).
Now, recall from [CC ], proposition 9.3 that
~H*(ES1+^S1 Ss) ~= m 1 H*(Cm ; A m )
(Actually, we are correcting a misprint in [CC ] here. The homology groups
on the right hand side of the formula should not be reduced).
The Poincar'e series of the right hand side contains the sum of the contri-
butionPof the homology in dimension zero. The Poincar'e series of this part
is m 1 tm(s-1) = ts-1(1 - ts-1)-1. It also contains the sums of the contri-
butions of the reduced group homologies. Since this is trivial if m is even,
we can as well put m = 2n, and the Poincar'e series of the reduced part is
X X
ti+2n(s-1)= t2(s-1)+1(1 - t)-1(1 - t2(s-1))-1
i 1 n 1
Summing, we get that the Poincar'e series for H~*(ES1+^S1 ( X)) is
(ts-1 - ts + t2s-2)(1 - t)-1(1 - t2s-2)-1
Finally, we note that there is a short exact sequence of homology groups:
0 ! ~H*(BT) ! ~H*(ET xT Ss) ! ~H*(ET+ ^T Ss) ! 0.
This shows that the Poincar'e series of H*(ET xT Ss) is
(ts-1 - ts + t2s-2)(1 - t)-1(1 - t2s-2)-1 + (1 - t2)-1
__
Bringing on common denominator and adding proves the theorem. |__|
30
Proposition 9.2. The Poincar'e series of H*( Ss; F2) is (1 + ts)(1 - ts-1)
when s 2.
Proof. The mod 2 cohomology ring of Ss is a special case of Theorem 2.2
of [KY ] except for the case s = 2. It is however shown (Remark 2.6) that
the Eilenberg-Moore spectral sequence also collapses when s = 2 so we can
compute the Poincar'e series from the E2-term. It has the following form (see
the proof of Theorem 2.2):
E*,*2~= (x) (__x) [!]
where the respective bidegrees of x, __xand fli(!) are (0, s), (-1, s) and
(-2i, 2is) such that the respective total degrees becomes s, s-1 and 2i(s-1).
Thus the Poincar'e series is
(1 + ts)(1 + ts-1)(1 - t2(s-1))-1
__
and the result follows by a small reduction. |__|
Theorem 9.3. If we let X = Ss with s 2 and use F2-coefficients, then the
spectral sequences of Theorem 7.2 collapses. Thus there are isomorphisms of
graded F2-vector spaces:
__* * s * s * * s
H*(H*(Ss); ) ~=H ( S ) and H*(H (S ); `) ~=H (( S )hT).
Proof. By Theorem 8.10, Theorem 9.1 and Proposition 9.2 the Poincar'e series
of the E2-terms agree with the Poincar'e series of the targets. So the spectral
__
sequences collapses. |__|
10 Appendix: On a filtration of the functor `
In this appendix we identify the graded object associated with the filtration
`(A) u`(A) u2`(A) . . .
in the case where p = 2 and A is a polynomial algebra.
Recall that the functors L, e : F ! Alg are defined by L(A) = `(A)=(u)
and e (A) = L(A)=Iffi(A) where Iffi(A) is the ideal (ffi(x)|x 2 A) L(A).
We want to define a map `(A) ! e (A)[t] such that the elements OE(x),
q(x) and u in the domain are send to the elements OE(x), q(x) and t2 in the
target. Unfortunately, this cannot be done by a ring map. But if we pay the
penalty of changing the multiplicative structure of the target, we can almost
get such a map.
31
Definition 10.1. tw (A) is the free graded commutative algebra on gener-
ators OE(x), q(x) for x 2 A and t, of degrees |OE(x)| = 2|x|, |q(x)| = 2|x| - 1
and |t| = 1, modulo the relations
q(x + y) = q(x) + q(y), OE(x + y) = OE(x) + OE(y),
q(xy) = OE(x)q(y) + OE(y)q(x), OE(xy) = OE(x)OE(y),
q(x)2 = OE(~x) + tq(~x).
Clearly, tw(A)=(t) ~=e (A). The ring tw(A) is just a twisted version of
the polynomial ring over e (A) in t in the following case:
Theorem 10.2. Assume that the underlying algebra of A is a polynomial
algebra. Then the graded ring Gr*( tw (A)) corresponding to the filtration of
tw (A) by powers of t equals e (A)[t].
Proof. As an intermediate step, let us consider the ring R(A) which is defined
exactly like tw (A) except that we do not include the last relation q(x)2 =
OE(~x) + tq(~x).
If A is a polynomial algebra on generators {xi|i 2 I}, then R(A) is a
polynomial ring on generators OE(xi) and q(xi). To obtain tw(A) from R(A),
we have to add the relations q(p)2 = OE(~p)+tq(~p), where p is any polynomial
in the generators xi. Actually, it is sufficient to do this for the generators
themselves, as this relation for p1p2 follows from the relations for p1 and p2.
Because, assume those are satisfied, then we calculate
q(p1p2)2= OE(p1)2q(p2)2 + OE(p2)2q(p1)2
= OE(p1)2(OE(~p2) + tq(~p2)) + OE(p2)2(OE(~p1) + tq(~p1))
= OE(~(p1p2)) + tq(~(p1p2)).
Thus we can write tw(A) as an algebra:
F2[t, OE(xi), q(xi)|i 2 I]={q(xi)2 = OE(~xi) + tq(~xi)}
From this it is clear, that tw (A) is a free F2[t, OE(xi)|i 2 I]-module, with
generators
{q(xi1) . .q.(xin)|ir 6= is forr 6= s, n 0}.
(The empty product means 1 here.) It follows that Gr*( tw (A)) is a free
module over Gr*(F2[t, OE(xi)|i 2 I]) with the same generators.
So, to finish the proof, we only have to determine the multiplicative struc-
ture of tw (A). The multiplicative relations are given by the relations. In
32
the graded ring they are q(xi)2 = OE(~xi). So, we have a presentation of the
graded ring as
F2[t, OE(xi), q(xi)|i 2 I]={q(xi)2 = OE(~xi)}.
__
But this is exactly e (A)[t]. |__|
Theorem 10.3. Let A be an object in F and i 1 an integer. Multiplication
with ui defines a natural surjective F2-linear map
ui`(A)
ui : e(A) ! _________.
ui+1`(A)
If the underlying algebra of A is a polynomial algebra, then this map is an
isomorphism and
M
Gr*(`(A)) ~=L(A) uj e(A).
j 1
Proof. Multiplication with ui gives a surjective map `(A) ! ui`(A) and
ui(Iffi(A)) = 0, ui(u`(A)) = ui+1`(A) so the map factors through e (A).
We define a natural ring map v : `(A) ! tw(A) by the formulas
v(OE(x)) = OE(x) + tq(x), v(q(x)) = q(x), v(u) = t2, v(ffi(x)) = 0.
To see that v is well defined, we have to check that the relations in the
definition of ` goes to 0. This is trivial for all relations except three which*
* is
verified as follows:
v(OE(xy))= OE(xy) + tq(xy) = (OE(x) + tq(x))(OE(y) + tq(y)) + t2q(x)q(y)
= v(OE(x)OE(y) + uq(x)q(y)),
v(q(xy)) = q(xy) = OE(x)q(y) + q(x)OE(y)
= (OE(x) + tq(x))q(y) + (OE(y) + tq(y))q(x)
= v(OE(x)q(y) + q(x)OE(y)),
v(q(x)2)= q(x)2 = v(OE(~x) + ffi(x2~x)).
By the map v we get a commutative diagram as follows:
e (A) -- - ! tw(A)=t2 tw (A)
? ?
? ?
uiy t2iy
ui`(A)=ui+1`(A) -- - ! t2i tw(A)=t2i+2 tw(A)
When the underlying algebra of A is a polynomial algebra, then Theorem
10.2 gives that the top and the right vertical maps are injective. So in this
__
case the left vertical map is also injective. |_*
*_|
33
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35