FREE AND SEMI-INERT CELL ATTACHMENTS
PETER BUBENIK
Abstract.Let Y be the space obtained by attaching a finite-type wedge of
cells to a simply-connected, finite-type CW-complex.
We introduce the free and semi-inert conditions on the attaching map
which broadly generalize the previously studied inert condition. Under t*
*hese
conditions we determine H*( Y ; R) as an R-module and as an R-algebra re-
spectively. Under a further condition we show that H*( Y ; R) is generat*
*ed
by Hurewicz images.
As an example we study an infinite family of spaces constructed using *
*only
semi-inert cell attachments.
1.Introduction
In this article we will work in the usual category of pointed, simply-connect*
*ed
topological spaces with the homotopy-type of finite-type CW-complexes. We will
assume that the ground ring R is either Fp with p > 3 or is a subring of Q which
contains 1_6.
We are interested in the following problem, perhaps first studied by J.H.C.
Whitehead around 1940 [Whi41 , Whi39].
The cell attachment problem: Given a topological space X, what is the effect
on the loop space homology and the homotopy-type if one attaches one or more ce*
*lls
to X?
We approach this problem from the point of view that one is interested in un-
derstanding finite cell complexes localized away from finitely many primes [Ani*
*92].
Given a space X and a map f : W ! X where W = _j2JSnj, the adjunction
space `
Y = X [f enj+1 ,
j2J
is a homotopy cofibre of f. Let i denote the inclusion X ,! Y .
The cell attachment problem has been studied in two special cases. One approa*
*ch
is to place a strong condition on the space X. This was done by Anick [Ani89] w*
*ho
considered the case where X is a wedge of spheres. Another approach is to place*
* a
strong condition on the attaching map f. This was done by Lemaire and Halperin
[Lem78 ],[HL87 ] and F'elix and Thomas [FT89 ] who assumed that f is inert.
The attaching map f : W ! X is said to be inert over a ring R if the induced
map H*( i; R) : H*( X; R) ! H*( Y ; R) is a surjection.
In this article we generalize these two approaches, with our development foll*
*ow-
ing [Ani89]. We generalize Anick's assumption to the more general condition that
____________
Date: December 19, 2003.
2000 Mathematics Subject Classification. Primary 55P35 ; Secondary 16E45.
Key words and phrases. cell attachments, loop space, loop space homology, Ad*
*ams-Hilton
models.
1
2 PETER BUBENIK
H*( X; R) is R-free and is generated by Hurewicz images. This is trivial in the
case where R = Q, and we will give conditions under which this holds for more g*
*en-
eral coefficient rings (see Corollary 1.7). Furthermore we give two generalizat*
*ions
of the inert condition, one of which is strictly stronger than the other.
We now define these conditions in the case where R is a field. We will use the
following notation.
Notation 1.1. Given a space X, let LX denote the image of the Hurewicz map
hX : ß*( X) R ! H*( X; R). LX is a graded Lie algebra under the commutator
bracket of the Pontrjagin product. Given a map W ! X, let LWX denote the image
of the induced Lie algebra map LW ! LX . Note that the map is omitted from the
notation. Let [LWX] LX denote the Lie ideal generated by LWX.
Definition 1.2. Define a cell attachment f : W ! X to be free if [LWX] is a free
Lie algebra.
There are examples of spaces given by non-free cell attachments that can be
constructed by free cell attachments if one changes the order in which the cell*
*s are
attached (eg. CP 2[f e3 where f is the inclusion S2 ,! CP2, see [HL96 , Example
4.5]). When R is a field, it is conceivable that any cell complex can be constr*
*ucted
using only free cell attachments if one chooses an appropriate cellular structu*
*re.
Free cell attachments are convenient to work with because of the following fa*
*ct
about universal enveloping algebras, which we prove as Lemma 3.8. Since it may
be of independent interest we state it here. U denotes the universal enveloping
algebra functor.
Proposition 1.3. Let L be a connected, finite-type Lie algebra over a field. Le*
*t J
be a Lie ideal of L which is a free Lie algebra, LW . Take I to be the two-side*
*d ideal
of UL generated by J. Then the multiplication maps UL W ! I and W UL ! I
are isomorphisms of left and right UL-modules respectively.
W W ffj
Assume that Y is obtained by a free cell attachment j2JSnj ---! X. Let
^ffjdenote the adjoint of ffj. We will show that H*( Y ; R) can be determined
by calculating the homology of the following simple differential graded Lie alg*
*ebra
(dgL)
L_= (LX q Lj2J, d), where dyj = hX (f^fj).
In addition to the usual grading, L_has a second grading given by letting LX be*
* in
degree 0 and letting each yj be in degree 1. Remarkably, we will show that for *
*free
cell attachments one only needs to calculate HL_in degrees 0 and 1.
Let (HL_)i denote the component of HL_in degree i. For degree reasons, (HL_)0
acts on (HL_)1 by the adjoint action.
Definition 1.4. Define f to be a semi-inert cell attachment if it is a free cell
attachment and (HL_)1 is a free (HL_)0-module.
We will see in Section 2 that there is an obvious filtration on H*( Y ; R). L*
*et
gr*(H*( Y ; R)) be the associated graded object. We will show (see Definition 4*
*.2)
that a free cell attachment is semi-inert if and only if gr1(H*( Y ; R)) is a f*
*ree
gr0(H*( Y ; R))-bimodule.
FREE AND SEMI-INERT CELL ATTACHMENTS 3
iW j
Theorem A. Let Y = X [f j2J enj+1. Assume that f is free.
(i) Then as algebras
gr(H*( Y ; R)) ~=U((HL_)0 n L((HL_)1))
with (HL_)0 ~=LX =[LWX] as Lie algebras.
(ii) Furthermore if f is semi-inert then for some K0,
H*( Y ; R) ~=U(LXYq LK0)
as algebras.
The above result is given more precisely in Theorem 4.3. This theorem is a
nearly direct translation of a purely algebraic result given in Theorem 3.12. T*
*his
algebraic result may have other applications such as the calculation of the mod
p Bockstein spectral sequence (BSS) of finite CW-complexes. Scott [Sco02] has
shown that for sufficiently large p each term in the mod p BSS of such spaces i*
*s a
universal enveloping algebra of a dgL.
Corollary 1.5. Free cell attachments are nice in the sense of Hess and Lemaire
[HL96 ].
When R = Q, Milnor and Moore [MM65 ] proved the fabulous result that
the canonical algebra map U(ß*( Y ) Q) ! H*( Y ; Q) is an isomorphism.
Scott [Sco03] generalized this result to R Q for finite CW-complexes when cer*
*tain
primes are invertible in R. For an R-module M, let F M = M= Torsion(M). Let
P denote the primitive elements of F H*( Y ; R) which are a Lie subalgebra. Sco*
*tt
~=
showed that UP -! F H*( Y ; R) and that F (ß*( Y ) R) injects into P . However
he showed that in general this injection is not a surjection.
We will give sufficient conditions under which one obtains the desired isomor-
phism UF (ß*( Y ) R) ! F H*( Y ; R)
In Section 5 we will assume that the Hurewicz map hX has a right inverse. Usi*
*ng
this map we will define the set of implicit primes of Y . Intuitively, they are*
* the
primes p for which p-torsion is used in the attaching map f. Let
S = {S2m-1 , S2m+1 | m 1}.
Q
Let S be the collection of spaces homotopy equivalent to a weak product of sp*
*aces
in S.
W
Theorem B. Let Y = X [f enj+1. Assume that f is free, that the Hurewicz
map hX has a right inverse, and that the implicit primes are invertible. Then t*
*he
canonical algebra map
(1.1) ULY ! H*( Y ; R)
is a surjection.QFurthermore if R Q then (1.1)is an isomorphism and localized
at R, Y 2 S. If in addition f is semi-inert, then
LY ~=HL_~=(HL_)0 q L((HL_)1)
as Lie algebras, and hY has a right inverse.
Note that the surjection of (1.1)implies that H*( Y ; R) is generated as an
algebra by Hurewicz images. Again, more details are given in Theorem 5.5.
4 PETER BUBENIK
Corollary 1.6. If R Q then the canonical algebra map
UF (ß*( Y ) R) ! H*( Y ; R)
is an isomorphism.
Corollary 1.7. If Z is a finite cell complex constructed using onlyQsemi-inert *
*cell
attachments then localized away from a finite set of primes, Z 2 S.
It is a long-standing conjecture of Avramov [Avr82] and F'elix [FHT84 ] that *
*if
Z has finite LS category then LZ is either finite dimensional or contains a fre*
*e Lie
subalgebra on two generators. Our final corollary provides further support for *
*this
conjecture.
Corollary 1.8. If R Q, f is a free cell attachment and dim(HL_)1 > 1 then LY
contains a free Lie subalgebra on two generators.
We conclude by giving examples of spaces constructed out of semi-inert cell
attachments, together with their Hurewicz images. In particular, we give an inf*
*i-
nite family of finite CW-complexes and an uncountable family of finite-type CW-
complexes.
Outline of the paper: In Section 2 we will translate the cell attachment prob*
*lem
to a purely algebraic problem using Adams-Hilton models. We then prove our
main algebraic results in Section 3. In Section 4 we translate our algebraic re*
*sults
to obtain Theorem A. In Section 5 we prove results about Hurewicz images and
homotopy-type to obtain Theorem B. Finally in Section 6 we apply our results to
study some examples.
Acknowledgments: The results of this paper were part of my thesis at the Uni-
versity of Toronto [Bub03 ]. I would like to thank my advisor, Paul Selick, for*
* his
encouragement and support, and Jean-Michel Lemaire for his helpful suggestions.
2.Adams-Hilton models
Let R = Fp with p > 3 or let R be a subring of Q containing 1_6.
A simply-connected space X has an Adams-Hilton model [AH56 ] which we de-
note A(X). A(X) is a connected differential graded algebra (dga) which comes
with a chain map A(X) ! C*( X; R) which induces an isomorphism of algebras
~=
H(A(X)) -! H*( X; R). W
Given a cell attachment f : W !iX where W = j2J Snj and f = _j2Jffj, let Y
W j
be the adjunction space Y = X [f j2Jenj+1 . It is a property of Adams-Hilton
models that one can choose the following Adams-Hilton model for Y :
A(Y ) = A(X) q Tj2J,
where T denotes the tensor algebra. The differential on yj is determined using *
*the
attaching map ffj.
Filter A(Y ) by the `length in yk's' filtration. That is,Plet F-1A(Y ) = 0, *
*let
F0A(Y ) = A(X), and for i 0, let Fi+1A(Y ) = FiA(Y ) + ik=0FkA(Y ) .
R{yj}j2J . Fi-kA(Y ). This filtration makes A(Y ) a filtered dga.
This filtration induces a first quadrant multiplicative spectral sequence with
E0p,q= [FpA(Y )=Fp-1A(Y )]p+q which converges from gr(A(Y )) to gr(HA(Y )).
Assume that H*( X; R) ~= ULX as algebras and that it is R-free. Then
(E1, d1) ~= (ULX q ULj2J, d0), where L denotes the free Lie algebra and d0
FREE AND SEMI-INERT CELL ATTACHMENTS 5
~=
is determined by the induced map d0: R{yj} d-!ZA(X) i H*( X; R) -! ULX .
It follows from the definition of the differential that d0yj = hX (^ffj) where *
*^ffjis the
adjoint of ffj. Therefore d0yj 2 LX . Thus
(E1, d1) ~=UL_, whereL_= (LX q L, d0).
3.Differential Graded Algebra Extensions
Let R = Q or Fp where p > 3 or let R be a subring of Q containing 1_6. All of
our R-modules are graded and will be assumed to have finite type.
If R Q, then let P be the set of invertible primes in R and let
~P= {p 2 Z | p is prime andp =2P } [ {0}.
Notation 3.1. Let F0 denote Q. If M is an R-module then for each p 2 ~Pwe will
denote M Fp by M~ omitting p from the notation. Similarly if d is a different*
*ial
we will denote d Fp by ~d.
Let (A, d) be a connected finite-type differential graded algebra (dga) over R
which is R-free. Let ZA denote the subalgebra of cycles of A. Let V1 be a conne*
*cted
finite-type free R-module and let d : V1 ! ZA. Then there is a canonical dga
extension B = (A q TV1, d).
Assume that for some Lie algebra L0 which is a free R-module, H(A, d) ~=UL0
~=
as algebras. There is an induced map d0: V1 d-!ZA ! HA -! UL0. Assume that
L0 can be chosen such that d0V1 L0.
Taking d0L0 = 0, there is a canonical differential graded Lie algebra (dgL)
L_= (L0 q LV1, d0).
Then L_is a bigraded dgL where the usual grading is called dimension and a seco*
*nd
grading, called degree, is given by taking L0 and V1 to be in degrees 0 and 1
respectively. Then the differential d has bidegree (-1, -1).
Notation 3.2. Subscripts of bigraded objects will denote degree, eg. M0 is the
component of M in degree 0.
The following lemma is a well-known fact, and the subsequent lemma is part of
lemmas from [Ani89]. We remind the reader that all of our R-modules have finite
type.
Lemma 3.3. Let R Q. A homomorphism f : M ! N is an isomorphism if and
only if for each p 2 ~P, f Fp is an isomorphism.
Let L be a connected bigraded dgL. The inclusion L ,! UL induces a natural
map
(3.1) _ : UHL ! HUL.
Lemma 3.4 ([Ani89, Lemmas 4.1 and 4.3]). Let R = Fp with p > 3 or let R Q
containing 1_6. Suppose that HUL is R-free in degrees 0 and 1. Then HL is R-free
in degrees 0 and 1 and the map _ in (3.1)is an isomorphism in degrees 0 and 1.
B is a filtered dga underPthe increasing filtration given by F-1B = 0, F0B = *
*A,
and for i 0, Fi+1B = ij=0FjB.V1.Fi-jB. Letting E0p,q(B) = [Fp(B)=Fp-1(B)]p+q
gives a first quadrant spectral sequence of algebras:
(E0(B), d0) = gr(B) =) E1 = gr(HB).
6 PETER BUBENIK
It is easy to check that (E1, d1) ~=UL_ and hence E2 ~=HUL_. The following
theorem follows from the main result of Anick's thesis [Ani82, Theorem 3.7]. An*
*ick's
theorem holds under either of two hypotheses. We will use only one of these.
Recall that the Hilbert series of an F-module is given by the power series A(*
*z) =
1n=0(Rank FAn)zn. Assuming that A0 6= 0, the notation (A(z))-1 denotes the
power series 1=(A(z)).
Theorem 3.5. Let R = F. If the two-sided ideal (d0V1) UL0 is a free UL0-
module then the above spectral sequence collapses at the E2 term. That is, gr(H*
*B) ~=
HUL_as algebras. Furthermore the multiplication map
: T(_(HL_)1) (HUL_)0 ! HUL_
is an isomorphism and (HUL_)0 ~=UL0=(d0V1). In addition,
(3.2) HB(z)-1 = HUL_(z)-1 = (1 + z)(HUL_)0(z)-1 - z(UL0)(z)-1 - V1(z).
Proof.[Ani82, Theorem 3.7] shows that the spectral sequence collapses as claimed
and that the multiplication map TW (HUL_)0 ! HUL_is an isomorphism where
W is a basis for (HUL_)1 as a right (HUL_)0-module. By Lemma 3.4 and the
Poincar'e-Birkhoff-Witt Theorem the homomorphism _(HL_)1 (HUL_)0 ! (HUL_)1
induced by multiplication in HUL_is an isomorphism. So we can let W = _(HL_)1.
The remainder of the theorem follows directly from [Ani82, Theorem 3.7].
Corollary 3.6. If R Q and for each p 2 ~P, the two-sided ideal (d~~V1) UL~0*
*is
a free UL~0-module, then HB is R-free if and only if HUL_ is R-free if and only*
* if
L0=[d0V1] is R-free.
Proof.First (HUL_)0 ~=UL0=(d0V1) ~=U(L0=[d0V1]). So (HUL_)0 is R-free if and
only if L0=[d0V1] is R-free. Since UL0 and V1 are R-free, the corollary follows
from (3.2).
We now prove a version of Theorem 3.5 for subrings of Q.
Theorem 3.7. Let R Q. If L0=[d0V1] is R-free and for each p 2 ~P, the two-sid*
*ed
ideal (d~~V1) UL~0is a free UL~0-module, then HB is R-free and the multiplica*
*tion
map
: T(_(HL_)1) (HUL_)0 ! HUL_
is an isomorphism. Also gr(HB) ~=HUL_as algebras and (HUL_)0 ~=UL0=(d0V1).
Proof.Since L0=[d0V1] is R-free, by Corollary 3.6 so are HUL_and HB. It follows
from the Universal Coefficient Theorem that 8p 2 ~P, HB Fp ~=H(B Fp) and
HUL_ Fp ~=HU(L_ Fp). In particular 8p 2 ~P, (HUL_)0 Fp ~=(HU(L_ Fp))0.
Using Lemma 3.4,
_(HL_)1 Fp ~=(HL_)1 Fp ~=H(L_ Fp)1 ~=_H(L_ Fp)1.
Thus 8p 2 ~P,
Fp : T(_(HL_)1 Fp) (HUL_)0 Fp ! HB Fp
corresponds under these isomorphisms to the multiplication map
T(_(H(L_ Fp))1) (HU(L_ Fp))0 ! H(B Fp).
But this is an isomorphism by Theorem 3.5. Therefore is an isomorphism by
Lemma 3.3.
The last two isomorphisms also follow from Theorem 3.5.
FREE AND SEMI-INERT CELL ATTACHMENTS 7
The next lemma will prove that if the Lie ideal [d0V1] L0 is a free Lie alg*
*ebra
then the hypothesis in Anick's Theorem (Theorem 3.5) holds. That is, (d0V1) is a
free UL0-module.
Lemma 3.8. Given a dgL L over a field F, denote UL by A. Let J be a Lie ideal of
L which is a free Lie algebra, LW . Take I to be the two-sided ideal of A gener*
*ated
by J. Then the multiplication maps A W ! I and W A ! I are isomorphisms
of left and right A-modules respectively.
Proof.From the short exact sequence of Lie algebras
0 ! J ! L ! L=J ! 0
we get the short exact of sequence of Hopf algebras
F ! U(J) ! U(L) ! U(L=J) ! F
and so UL ~= UJ U(L=J) as F-modules. Since J is a free Lie algebra LW ,
UJ ~=TW . It is also a basic fact that U(L=J) ~=UL=I. Hence we have that
(3.3) A ~=T W A=I
as F-modules. Furthermore
(3.4) A ~=I A=I
as F-modules.
Let M(z) denote the Hilbert series for the F-module M, and to simplify the
notation let B = A=I. Then from equations (3.3)and (3.4)we have the following
(using (T W )(z) = 1=(1 - W (z))).
B(z) = A(z)(1 - W (z)), I(z) = A(z) - B(z).
Combining these we have I(z) = A(z)W (z). That is, I ~=A W as F-modules.
Let ~ : A W ! I be the multiplication map. To show that it is an isomorphism
it remains to show that it is either injective or surjective.
We claim that ~ is surjective. Since I is the ideal in A generated by W , any
x 2 I can be written as
(3.5) x = iaiwibi1. .b.imi, whereai2 A, wi2 W and bik2 L.
Each such expression gives a sequence of numbers {mi}. Let M(x) = min[maxi(mi)],
where the minimum is taken over all possible ways of writing x as in (3.5). We
claim that M(x) = 0.
Assume that M(x) = t > 0. Then x = x0+ iaiwibi1. .b.it, where M(x0) < t.
Now wibi1= [wi, bi1] bi1wi. Furthermore since J is a Lie ideal [wi, bi1] 2 J ~=*
*LW ,
so
[wi, bi1] = jcj[[wj1, . .,.wjnj] = kdkwk1. .w.kNk= lalwl,
where al2 A and wl2 W . So x = x0+ i liaialiwlibi2. .b.it. But this is of the
form in (3.5)and shows that M(x) < t which is a contradiction.
Therefore for each x 2 I, M(x) = 0 and we can write x = iaiwi where ai2 A
and wi2 W . Then x 2 im(~) and hence ~ is an isomorphism.
Since A is associative, ~ is a map of left A-modules.
The second isomorphism follows similarly.
8 PETER BUBENIK
We are now almost ready to prove our main algebraic results. Recall that B =
(A q TV1, d) where dA A and dV1 ZA. Also H(A, d) ~=UL0 as algebras and if
d0: V1 ! UL0 is the induced map then d0(V1) L0. Let L_= (L0 q LV1, d0) with
d0L0 = 0.
We introduce the following terminology.
Definition 3.9. If R is a field say B is free dga extension if the Lie ideal [d*
*0V1] L0
is a free Lie algebra. If R Q say B is free dga extension if L0=[d0V1] is R-f*
*ree and,
using Notation 3.1, for every p 2 ~P, the Lie ideal [d~0~V1] ~L0is a free Lie*
* algebra.
Definition 3.10. In either of the cases of the previous definition we say B is
a semi-inert dga extension if in addition there is a free R-module K such that
(HL_)0 n L(HL_)1 ~= (HL_)0 q LK. At the end of this section we will give two
simpler equivalent conditions (see Lemma 3.13).
Note that it follows from [HL87 , Theorem 3.3] and [FT89 , Theorem 1] that the
semi-inert condition is a generalization of the inert condition.
Recall that there is a map __ : UHL_ ! HUL_. B is a filtered dga under
thePincreasing filtration given by F-1B = 0, F0B = A, and for i 0, Fi+1 =
i
k=0FkB . V1 . Fi-kB. There is an induced filtration on HB. We prove one last
lemma.
Lemma 3.11. There exists a quotient map
(3.6) f : F1HB ! (HUL_)1.
Given ~w2 (HL_)1 there exists a cycle w 2 F1B such that f([w]) = ~w.
Proof.By Theorem 3.5 or Theorem 3.7, (gr(HB))1 ~= (HUL_)1. So there is a
quotient map
~=
f : F1HB i (gr(HB))1 -! (HUL_)1.
By Lemma 3.4 (HL_)1 ~=(__HL_)1 (HUL_)1. So for ~wone can choose a represen-
tative cycle w 2 ZF1B such that f([w]) = ~w.
Recall that R = Fp with p > 3 or R Q containing 1_6. Also all of our R-modu*
*les
are connected, R-free, and have finite type. Let (A, d) be a dga and let V1 be*
* a
R-module with a map d : V1 ! A. Assume that there exists a Lie algebra L0 such
that H(A, d) ~=UL0 as algebras and d0V1 L0 where d0is the induced map.
Theorem 3.12. Let B = (A q TV1, d). Assume that B is a free dga extension in
the sense of Definition 3.9. Let L_= (L0 q LV1, d0).
(a) Then as algebras
gr(HB) ~=U((HL_)0 n L(HL_)1)
with (HL_)0 ~= L0=[d0V1] as Lie algebras. If R Q then additionally (HL_)0 n
L(HL_)1 ~=__HL_as Lie algebras.
(b) Furthermore if B is semi-inert (that is, there is an R-module K such that
(HL_)0 n L(HL_)1 ~=(HL_)0 q LK) then as algebras
HB ~=U((HL_)0 q LK0)
~=
for some K0 F1HB such that f : K0 -! K, where f is the quotient map in
Lemma 3.11.
FREE AND SEMI-INERT CELL ATTACHMENTS 9
Proof.(a) If R = Fp then by Lemma 3.8, (d0V1) UL0 is a free UL0-module. If
R Q then by Lemma 3.8, for each p 2 ~P, (d~0~V1) UL~0is a free UL~0-module.
So we can apply either Theorem 3.5 or Theorem 3.7 to show that gr(HB) ~=HUL_
as algebras and that the multiplication map
: T(__(HL_)1) (HUL_)0 ! HUL_
is an isomorphism.
By Lemma 3.4 (HUL_)0 ~=U(HL_)0 and __(HL_)1 ~=(HL_)1. By the definition of
homology (HL_)0 ~=L0=[d0V1].
Let N = __(HL_). Then N0 acts on N1 so we can define
L0= N0 n LN1.
Note that L00= N0 and L01= N1. There is a Lie algebra map u : L0! N and an
induced algebra map ~u: UL0-Uu-!UN ! HUL_.
Recall that as R-modules, L0~=N0xLN1. The Poincar'e-Birkhoff-Witt Theorem
shows that the multiplication map
~= 0
OE : TN1 (HUL_)0 -! ULN1 UN0 ! UL
is an isomorphism. Since ~uis an algebra map, = ~uOOE. Thus ~uis an isomorphi*
*sm.
Therefore HUL_~=UL0as algebras and hence gr(HB) ~=UL0. If R = Fp then this
finishes (a).
If R Q then we will show that u : L0! N is an isomorphism.~Let ' : N ,!
HUL_ be the inclusion. Since the composition L0 u-!N -'!HUL_ =-!UL0 is the
canonical inclusion L0 ,! UL0, u is injective. The inclusion L0 ! UL0 splits as
R-modules; so as R-modules N ~= L0 N=L0. Since L0 and N are R-free, so is
N=L0.
Recall that the composition ' O u induces the isomorphism ~u: UL0-Uu-!UN !
HUL_. Tensor these maps with Q to get the commutative diagram
(3.7) UL0 Q _Uu_Q_//UN Q .
MM
MMM |
~=MMM&&MMfflffl||
HUL_ Q
It is a classical result that the natural map
~=
(3.8) __Q: UH(L_ Q) -! HU(L_ Q)
is an isomorphism. Notice that
N Q = (__HL_) Q ~=__QH(L_ Q) ~=H(L_ Q)
and HUL_ Q ~=HU(L_ Q). Under these isomorphisms the vertical map in (3.7)
corresponds to the isomorphism in (3.8).
Therefore Uu Q is an isomorphism and hence u Q is surjective. As a result
cokeru = N=L0 is a torsion R-module. But we have already shown that N=L0 is
R-free. Thus N=L0= 0 and N ~=L0. Hence HUL_~=UN.
(b) Recall that N0 acts on N1 = (__HL_)1 ~=(HL_)1 via the adjoint action. Assume
that B is semi-inert. That is, there exists {w~i} N1 such that L0 ~=N0 q LK,
10 PETER BUBENIK
where K = R{w~i}. Recall from (a) that HUL_~= gr(HB) and that the inclusions
N0 HUL_and ~wi2 HUL_induce a Lie algebra map
u : L0! gr(HB).
By Lemma 3.11, there exists wi2 F1B such that f([wi]) = ~wiwhere f is the map
~=
in (3.6). Let K0 = R{[wi]} F1HB, and let L00= N0 q LK0. Then f : K0 -! K
~=
and f induces an isomorphism L00-! L0.
By part (a), N0 (grHB)0. Since F-1HB = 0, (grHB)0 ,! HB, so N0 ,!
HB. Since N0, K0,! HB, there are induced maps
j
L00_____//HB;;x,
| xxx
| xx`
fflffl|xx
UL00
where j is a Lie algebra map and ` is an algebra map.
Grade L00by letting N0 be in degree 0 and K0 be in degree 1. This also filters
L00. Then j is a map of filtered objects.
From this we get the following commutative diagram
gr(j)
gr(L00)_____//gr(HB)99BB.
| gr(`)rrrrr
| rrr
fflffl|rr
gr(UL00) æ
~=||
fflffl|
U gr(L00)
Now gr(L00) ~=L00~=L0and gr(j) corresponds to u under this isomorphism. So
æ corresponds to ~uwhich is an isomorphism. Thus gr(`) is an isomorphism, and
hence ` is an isomorphism. Therefore HB ~=UL00which finishes the proof.
As promised we now give two simpler equivalent conditions for semi-inertness.
Lemma 3.13. Let B be a free dga extension (in the sense of Definition 3.9). Then
the following conditions are equivalent:
(a) (HL_)0 n L(HL_)1 ~=(HL_)0 q LK for some free R-module K (HL_)1,
(b) (HL_)1 is a free (HL_)0-module, and
(c) gr1(HB) is a free gr0(HB)-bimodule.
Proof.(b) =) (a) Let K be a basis for (HL_)1 as a free (HL_)0-module. Then
(HL_)0 n L(HL_)1 ~=(HL_)0 q LK.
(a) =) (c) Since B is a free dga extension, by Theorem 3.12(a), gr*(HB) ~=
U ((HL_)0 n L(HL_)1). So by (a),
gr*(HB) ~=U ((HL_)0 q LK)~=gr0(HB) q TK,
for some free R-module K (HL_)1. Therefore
gr1(HB) ~=[gr0(HB) q TK]1~=gr0(HB) K gr0(HB).
(c) =) (b) Let L0= (HL_)0nL(HL_)1. Then by Theorem 3.12(a), gr*(HB) ~=UL0
and gr1(HB) ~=(UL0)1. By (c), (UL0)1 is a free (UL0)0-bimodule. Then we claim
FREE AND SEMI-INERT CELL ATTACHMENTS 11
that it follows that N1 is a free N0-module. Indeed, if there is a nontrivial d*
*egree
one relation in L0 then there is a corresponding nontrivial degree one relation*
* in
UL0.
4. Cell-attachments
Let R = Fp with p > 3 or R Q containing 1_6. Let X be a finite-type
simply-connected CW-complex such that H*( X; R) is torsion-free and as alge-
bras H*(WX; R) ~= ULX where LX is the Lie algebra of Hurewicz images. Let
W = j2JSnjWbe a finite-type wedge of spheres and let f : W ! X. Let
Y = X [f j2Jenj+1 . Using the Adams-Hilton models of Section 2, we de-
fined a Lie algebra L_= (LX q L, d0). Mirroring Definitions 3.9 and 3.10, we
introduce the following terminology.
Definition 4.1. If R is a field call f a free cell attachment if the Lie ideal *
*[LWX]
LX is a free Lie algebra. If R Q call f a free cell attachment if LX =[LWX] i*
*s R-free
and for every p 2 ~P, the Lie ideal [~LWX] ~LXis a free Lie algebra.
Definition 4.2. In either of the cases of the previous definition say that f is*
* a
semi-inert cell attachment if in addition one of the following three equivalent*
* (by
Lemma 3.13] conditions is satisfied:
(a) gr1(HA(Y )) is a free gr0(HA(Y ))-bimodule,
(b) (HL_)1 is a free (HL_)0-module, or
(c) there is a free R-module K such that
(HL_)0 n L(HL_)1 ~=(HL_)0 q LK.
Theorem 3.12 gives most of the following topological result directly.
W
Theorem 4.3. Let Y = X [f j2Jenj+1 and let L_= (LX q L, d0). Assume
that f is free.
(a) Then H*( Y ; R) and gr(H*( Y ; R)) are R-free and as algebras
gr(H*( Y ; R)) ~=U(LXYn L(HL_)1)
with LXY~=LX =[LWX] as Lie algebras.
(b) Furthermore if f is semi-inert then as algebras
H*( Y ; R) ~=U(LXYq LK0)
for some K0 F1H*( Y ; R).
Proof.It remains to show that (HL_)0 ~=LXY. By Theorem 3.12 we have the algebra
isomorphism
gr(HA(Y )) ~=U((HL_)0 n L(HL_)1)
with (HL_)0 ~=LX =[LWX]. Therefore
(4.1) F0HA(Y ) ~=(gr(HA(Y )))0 ~=U(HL_)0 ~=U(LX =[LWX]).
~=
The inclusion i : A(X) -! F0A(Y ) induces a map H(i) : HA(X) ! F0HA(Y ).
~=
Now under the isomorphism (4.1)and ULX -! HA(X) the map H(i) corresponds
to a map ULX ! U(LX =[LWX]) where U(LX =[LWX]) ULY . It is easy to check
that this is the canonical map. In other words LXY~=LX =[LWX]. Therefore (HL_)0*
* ~=
LXY.
Corollary 1.5 follows from Theorem 4.3.
12 PETER BUBENIK
Proof of Corollary 1.5.The cell attachment f is nice in the sense of Hess and
Lemaire [HL96 ] if and only if ULX =(LWX) injects in H*( Y ; R). Recall the sta*
*ndard
fact that ULX =(LWX) ~=U(LX =[LWX]). By Theorem 4.3,
U(LX =[LWx]) ~=ULXY~=gr0(H*( Y ; R))
which injects in H*( Y ; R).
5. Hurewicz images
Let R = Fp with p > 3 or R Q be a subring containing 1_6.
W
Recall that we have a homotopy cofibrationWW -f!X ! Y where W = j2J Smj
is a finite-type wedge of spheres, f = j2Jffj, H*( X; R) is torsion-free, and*
* as
algebras H*( X; R) ~=ULX . Let ^ffdenote the adjoint of ff.
Assume that f is free. That is, the Lie ideal [LWX] is a free Lie algebra.
Recall that hX : ß*( X) R ! LX H*( X; R) is a Lie algebra map. In order
to identify Hurewicz images in H*( Y ; R) we will need to be able to construct
maps from information about the loop space homology. In particular, we will need
to assume that there exists a Lie algebra map oeX : LX ! ß*( X) R such that
hX O oeX = idLX . This map exists if R = Q or if X is a wedge of spheres. We wi*
*ll
give a sufficient condition for the existence of this map later in this section.
If R is a subring of Q with invertible primes P {2, 3}, then we may need to
exclude those primes p for which an attaching map ffj 2 ß*(X) includes a term
with p-torsion. Following Anick [Ani89] we define the set of implicit primes be*
*low.
Definition 5.1. By the Milnor-Moore theorem [MM65 ], hX , oeX are rational iso-
morphisms, so im(oeX O hX - id) is a torsion element of ß*( X) R. Let flj =
oeX hX (f^fj) - ^ffjwhere ^ffj: Snj-1 ! X is the adjoint of ffj. Then tjflj =*
* 0 2
ß*( X) R for some tj > 0. Let tj be the smallest such integer. Define PY , the
set of implicit primes of Y as follows. A prime p is in PY if and only if p 2 P*
* or
p|tj for some j 2 J.
One can verify that the implicit primes have the following properties.
Lemma 5.2. (a) Let {xi} be a set of Lie algebra generators for LX and let
fii = oeX xi. If all of the attaching maps are R-linear combinations of iterat*
*ed
Whitehead products of the maps fii, then PY = P .
(b) If P = {2, 3} and n = dim(Y ) then the implicit primes are bounded by max(3*
*, n=2).
By replacing R with R0= Z[PY -1] if necessary, we may assume that the implicit
primes are invertible. This implies that for all j 2 J, oeX hX ^ffj= ^ffj, and*
* hence
oeX (dyj) = ^ffj.
Remark 5.3. If R = Fp then we will also need that for all j 2 J, oeX hX ^ffj= ^*
*ffj. So
that we can state the cases R = Fp and R Q together, when R = Fp and we say
the implicit primes are invertible we mean that for all j 2 J, oeX hX ^ffj= ^ff*
*j.
We now consider both cases R = Fp or R Q. Recall that in Section 2 we
defined the differential graded Lie algebra
L_= (LX q Lj2J, d0).
By Theorem 4.3, H*( Y ; R) is torsion-free and gr(H*( Y ; R)) ~=U(LXYnL(HL_)1)
as algebras. From this we want to show that H*( Y ; R) ~=ULY as algebras.
FREE AND SEMI-INERT CELL ATTACHMENTS 13
This situation closely resembles that of torsion-free spherical two-cones, an*
*d we
will generalize Anick's proof for that situation [Ani89].
The proof of the following is the same as the proof of [Ani89, Claim 4.7], so*
* we
will only sketch it here. See [Bub03 , Chapter 7] for more details.
Proposition 5.4. Let W ! X ! Y and L_be as above with oeX hX ^ffj= ^ffj, 8j.
Then there exists an injection of LXY-modules
u1 : (HL_)1 ,! (gr(LY ))1.
Proof sketch.Let ~wi2 (HL_)1 be a homology class in dimension m + 1. It can be
represented by a cycle fli2 L_in degree 1. Using the Jacobi identity one can wr*
*ite
fli= sk=1ckuk whereck 2 R and uk = [. .[.yjk, xk1], . .,.xknk],
with [xki] 2 LX .
The sphere Sm has an Adams-Hilton model (T, 0) which can be extended to
(T, d) where db = a which is an Adams-Hilton model for the disk Dm+1 . Us*
*ing
properties of Adams-Hilton models, one can construct maps gk : (Dm+1 , Sm ) !
(Y, X) for 1 k s such that A(gk)(b) = ckuk.
Let _ !
s` `s
g0= g1 _ . ._.gs : Dm+1 , Sm ! (Y, X).
k=1 k=1
Using the fact that oeX hX ^ffj= ^ffj, one can show that g0|W kSm is contract*
*ible in
X. As a result, g0can be extended to a map g : Sm+1 ! Y whose Hurewicz image
modulo lower filtration is ~wi.
From this we will prove our final main result. Recall that R = Fp with p > 3 *
*or
R Q containing 1_6.
W W
Theorem 5.5. Let Y = X [f enj+1 with f = ffj satisfying the hypotheses
of Theorem 4.3. In addition assume that hX has a right inverse oeX and that the
implicit primes are invertible. Then
(a) the canonical algebra map
(5.1) ULY ! H*( Y ; R)
is a surjection.
(b) If R QQthen (5.1)is an isomorphism, gr(LY ) ~=LXYnL((HL_)1), and localized
at R, Y 2 S.
(c) If R Q and f is semi-inert then
(i) there exists ^K F1LY such that LY ~=LXYq LK^ as Lie algebras,
(ii) LY ~=HL_as Lie algebras, and
(iii) hY has a right inverse oeY .
Proof of (a) and (b).
~=
(a) Let g : gr(H*( Y ; R)) -! UL0 be the algebra isomorphism given by Theo-
rem 4.3(a) where L0= LXYn LK0 with K0= (HL_)1. Note that L00= LXYand that
L01= K0.
We have an injection of Lie algebras
~=
(5.2) u0 : LXY,! F0LY -! (gr(LY ))0.
14 PETER BUBENIK
Since the implicit primes are invertible, we have that for all j, hX oeX ^ffj*
*= ^ffj.
So by Proposition 5.4 we get an injection of LXY-modules u1 : K0 ,! (gr(LY ))1.
Hence for x 2 LXYand y 2 K0, u1([y, x]) = [u1(y), u0(x)]. Thus u0 and u1 can be
extended to a Lie algebra map u : L0! gr(LY ).
The inclusion LY ,! H*( Y ; R) induces a map between the corresponding
graded objects, Ø : gr(LY ) ! gr(H*( Y ; R)).
We claim that for j = 0 and 1, gOØOuj is the ordinary inclusion of L0jin UL0.*
* For
j = 0, under the isomorphisms gr0LY ~=F0LY and gr0H*( Y ; R) ~=F0H*( Y ; R),
~=
gØu0 corresponds to the map LXY,! F0LY ,! F0H*( Y ; R) -! ULXY. For j = 1,
under the isomorphism UL0 ~=HUL_, gØu1 corresponds to the inclusion K0 =
(HL_)1 ,! (HUL_)1. It follows that g O Ø O u is the standard inclusion L0,! UL0.
Since g O Ø O u is an injection,~so is u.
The canonical map U gr(LY ) -=!gr(ULY ) is an algebra isomorphism. Now u
and Ø induce the maps Uu and ~Øin the following diagram.
~=
(5.3) UL0 ___Uu__//_MU gr(LY_)___//_gr(ULY )
MMMM~=MM |~Ø o oo
g-1 MMM&&Mfflffl||~Øwwooo
gr(H*( Y ; R))
Since we showed that gØu is the ordinary inclusion L0 ,! UL0 the diagram com-
mutes. Since g-1 is surjective, the induced map ~Øis surjective. Since ~Øis t*
*he
associated graded map to the canonical map ULY ! H*( Y ; R) and the filtrations
are bicomplete, the associated ungraded map is also surjective. So the canonical
map ULY ! H*( Y ; R) is surjective which finishes the proof of (i).
(b) In the case where R Q we can tensor with Q and make use of results from
rational homotopy theory.
Recall that gr(H*( Y ; R)) ~=UL0 and that we constructed a Lie algebra map
u : L0! gr(LY ) and showed that it is an injection. We claim that for R Q, u
is an isomorphism. H*( Y ; R) and gr(H*( Y ; R)) have the same Hilbert series.
Also since H*( Y ; R) is torsion-free, it has the same Hilbert series as H*( Y *
*; Q).
Let S be the image of hY Q. Then S, LY and gr(LY ) have the same Hilbert
series. By the Milnor-Moore Theorem [MM65 ], H*( Y ; Q) ~= US. So by the
Poincar'e-Birkhoff-Witt Theorem, S has the same Hilbert series as L0, and hence
gr(LY ) ~=L0as R-modules. Since u : L0! gr(LY ) is an injection it follows that*
* it
is an isomorphism.
Using diagram (5.3)we get that (5.1)is an isomorphism.
The map u gives the desired Lie algebra isomorphism gr(LY ) ~= L0 = LXYn
L(HL_)1.
By the Hilton-Serre-Baues Theorem [Bau81 , Lemma V.3.10],[Ani89, Lemma 3.1],
that (5.1)isQan isomorphism is equivalent to the statement that localized at R,
Y 2 S.
Before we prove (c) we strengthen the result in (b) in the semi-inert case.
Lemma 5.6. Let R Q. Assume Y is a space satisfying the hypotheses of The-
orem 5.5. If furthermore f is semi-inert then there exists K^ F1LY such that
LY ~=LXYq LK^ as Lie algebras.
FREE AND SEMI-INERT CELL ATTACHMENTS 15
Proof.Assume that f is semi-inert. Recall the situation from Theorem 4.3(b). We
have that gr(H*( Y ; R)) ~= UL0 where L0 ~=LXYq LK0 where K0 ~=R{w~i}
(gr(H*( Y ; R)))1. For each ~wilet [wi] be an inverse image under the quotient *
*map
H*( Y ; R) ! gr(H*( Y ; R)). Let K00= {[wi]}. By Theorem 4.3(b) as algebras
~=
H*( Y ; R) ~=UL00where L00= LXYq LK00(see Theorem 3.12(b)). Since K00-!
~=
K0, there is an induced Lie algebra isomorphism L00-! L0. So gr(H*( Y ; R)) ~=
H*( Y ; R) as algebras.
Recall that K0= (HL_)1, so by Proposition 5.4 there exists ^K F1LY such that
~=
f : ^K-! K0, where f is the quotient map from (3.6). Let
^L= LXYq LK^.
~= ~=
So f : ^K-! K0, induces a Lie~algebra~isomorphism ^L-! L0. This in turn induces
the algebra isomorphism UL^-=!UL0-=!H*( Y ; R).
Since UL^~=H*( Y ; R) as algebras there is an injection LY ,! UL^. Also, since
^K LY there is a canonical Lie algebra map v : ^L! LY . These fit into the
following commutative diagram.
L^_____________//_@UL^==
@@@ ----
v@@__@@----
LY
It follows that v is an injection.
We claim that v is an isomorphism. Since H*( Y ; R) is torsion-free, it has
the same Hilbert series as H*( Y ; Q). Let S be the image of hY Q. Then S
and LY have the same Hilbert series. By the Milnor-Moore Theorem [MM65 ],
H*( Y ; Q) ~= US. So by the Poincar'e-Birkhoff-Witt Theorem, S has the same
Hilbert series as ^L, and hence LY ~= ^Las R-modules. Since v : ^L! LY is an
injection it follows that it is an isomorphism.
Therefore LY ~=LXYq LK^ as Lie algebras, with ^K F1LY and H*( Y ; R) ~=
ULY as algebras.
Proof of Theorem 5.5(c).Lemma 5.6 proves (i) and puts us in a position to prove
(ii) and (iii).
(ii) Recall that (HL_)1 ~= ^Kand L0 = (HL_)0 q L((HL_)1). Thus there is a
canonical map : L0! HL_. Now the composition
~= 0
L0-! HL_-OE!HUL_-! UL
is just the ordinary inclusion of L' into its universal enveloping algebra. The*
*refore
is an injection. Tensoring with Q we get that L0and HL_have the same Hilbert
series. It follows that is an isomorphism.
(iii) We will construct a map oeY right inverse to hY .
Let i denote the inclusion X ,! Y . Consider the composite map
F : [LWX] ,! LX -oeX-!ß*( X) R (-i)#---!ß*( Y ) R.
We claim that F = 0. Since F is a Lie algebra map it is sufficient to show that*
* it
is zero on the Lie algebra generators of LWX. That is, show
( i)# oeX (R{hX (^ffj)}) = 0.
16 PETER BUBENIK
Since there are no implicit primes oeX hX ^ffj= ^ffj. By the construction of Y *
*, iO^ffj'
0. So F = 0 as claimed. ~
Therefore there is an induced map G : LXY-=!LX =[LWX] ! ß*( Y ) R. That
hY O G is the inclusion map can be seen from the following commutative diagram.
oeX **
i `
LX oohX_ ß*( X) R
| ||
fflfflfflffl||||
LXY (|i)#
KKK |
| KKGK |
| KKK |
fflffl|hY %%Kfflffl|
LY oo___ ß*( Y ) R
Now construct oeY : LY ! ß*( Y ) R as follows. We have shown that LY ~=
LXYq LK^ for some ^K F1LY . Since hY : ß*( Y ) R i LY , choose preimages
~K ß*( Y ) R such that hY : ~K-~=!^K. Let oeY |LX = G and let oe ^K= ~Kbe
Y Y
right inverse to hY . Now extend oeY canonically to a Lie algebra map on LY .
We finally claim that hY oeY = idLY . Since hY oeY is a Lie algebra map it su*
*ffices
to check that it is the identity for the generators.
hY oeY LXY= hY GLXY= LXY, hY oeY ^K= hY ~K= ^K
Therefore oeY is the desired Lie algebra map right inverse to hY .
Remark 5.7. Anick conjectured Theorem 5.5(c)(ii) without the semi-inert conditi*
*on
in the special case where X is a wedge of spheres [Ani89, Conj. 4.9].
Corollary 5.8 (Corollary 1.6). If R Q then the canonical algebra map
U(F (ß*( Y ) R)) ! H*( Y ; R)
is an isomorphism.
Proof.Recall that F M = M= Torsion(M). By definition, the Lie algebra map
hY : ß*( Y ) R ! LY is a surjection. By Theorem 4.3, LY H*( Y ; R) is
torsion-free. As a result, there is an induced surjection ~hY: F (ß*( Y ) R) *
*i LY .
~=
Furthermore if we tensor with Q we see that ß*( Y ) Q -! LY Q, which is a
result of Cartan and Serre (see [FHT01 ]). Thus F (ß*( Y ) R) and LY have the
same Hilbert series, and therefore ~hYis an isomorphism of Lie algebras.
So using Theorem 5.5(b), the canonical algebra map
~=
U(F (ß*( Y ) R)) ! ULY -! H*( Y ; R)
is an isomorphism.
Corollaries 1.7 and 1.8 follow immediately from Theorem 5.5.
6.Examples
(Spherical) n-cones are those spaces X such that there exists a finite sequen*
*ce
* = X0 X1 . . .Xn ' X
FREE AND SEMI-INERT CELL ATTACHMENTS 17
where for k 0, Xk+1 is the adjunction space
` n +1
Xk+1 = Xk [fk+1 e j .
j2J
In particular, any finite CW-complex is an n-cone for some n.
Example 6.1. Let X = S3a_ S3band let 'a, 'b denote the inclusions of the spheres
into X. Let Y = X [ff1_ff2(e8 _ e8) where the attaching maps are given by the
iterated Whitehead products ff1 = [['a, 'b], 'a] and ff2 = [['a, 'b], 'b].
Let W = S7 _ S7 and for i = 1, 2 let ^ffi: S6 ! X denote the adjoint of ffi.*
* Let
R = Fp with p > 3 or R Q containing 1_6. Then [LWX] = [R{hX (^ff1), hX (^ff2)*
*}].
Y has an Adams-Hilton model (see Section 2) U(L, d) where L = L, *
*|x| =
|y| = 2, dx = dy = 0, da = [[x, y], x] and db = [[x, y], y]. Furthermore hX (^f*
*f1) =
[da] and hX (^ff2) = [db].
It is well-known that over a field, a Lie subalgebra of a free Lie algebra is*
* also
free. Thus, since L is a free Lie algebra, UL is a free dga extension (in*
* the
sense of Definition 3.9).
Let u = [a, y] - [b, x]. Then du = [[[x, y], x], y] - [[[x, y], y], x] = [[x,*
* y], [x, y]] = 0.
Since u is not a boundary 0 6= [u] 2 (HL)1 and 0 6= [u] 2 (HUL)1. Thus UL is not
an inert dga extension. By the definition of homology
ffi
(HL)0 ~=L R{[[x, y], x], [[x, y], y]} .
One can check that (HL)1 is freely generated by the (HL)0-action on [u]. Thus
UL is a semi-inert extension. Therefore by Theorem 3.12(b),
HUL ~=U((HL)0 q L<[u]>)
as algebras. Thus as algebras
W
H*( Y ; R) ~= U LX =[LX ] q L<[u]>
~= H*( X; R)=(LWX) q T<[u]>.
ff1 _ ff2 is a non-inert, semi-inert attaching map.
If R = Z[1_6] then since X is a wedge of spheres there exists a map oeX right
inverse to hX . By Lemma 5.2, PY = {2, 3}. By Theorem 5.5,
H*( Y ; R) ~=ULY , LY ~=LXYq L ~=L=J,
where w = hY (^!) with ^!the adjoint of some map ! : S10 ! Y , and J is the Lie*
* Q
ideal generated by {[[x, y], x], [[x, y], y]}. Furthermore localized at R, Y *
* 2 S
and there exists a map oeY right inverse to hY .
Example 6.2. The 6-skeleton of S3 x S3 x S3.
This space Y is also known as the fat wedge F W (S3, S3, S3). Let X = S3a_WS3*
*b_
S3c. Let 'a, 'b and 'c be the inclusions of the respective spheres. Let W = 3j*
*=1S5.
W 3 W3
Then Y = X [f ( j=1e6j) where f : W ! X is given by j=1 ffj with ff1 = ['b, '*
*c],
ff2 = ['c, 'a] and ff3 = ['a, 'b].
Let R = Z[1_6]. Then Y has Adams-Hilton model U(L, d) where L = L,
|x| = |y| = |z| = 2, dx = dy = dz = 0, da = [y, z], db = [z, x] and dc = [x, y]*
*. Let
w = [x, a] + [y, b] + [z, c].
18 PETER BUBENIK
By the same argument as in the previous example, as algebras
H*( Y ; R) ~=ULY , LY ~=LXYq L ~=L=J,
where w = hY (^!) with ^!the adjoint of some map ! : S8 ! Y and J is the Lie Q
ideal generated by {[x, y], [y, z], [z, x]}. Furthermore localized at R, Y 2 *
* S and
there exists a map oeY right inverse to hY .
The following spherical three-cone Y , illustrates our results.
Example 6.3. Let R = Z[1_6]. All spaces here are localized at R. For i = 1, 2
let Zi= Ai[ff1_ff2(e8 _ e8) be two copies of the two-cone from Example 6.1. Let
X = Z1 _ Z2, W = S28 _ S28 and let f = fi1 _ fi2 where fi1 = [[!1, !2], !1] and
fi2 = [[!1, !2], !2]. Let Y = X [f (e29_ e29).
Now,
(6.1) LX ~=LZ1 q LZ2 ~=LA1Z1q LA2Z2q L.
If follows from this that f is a free attaching map. Thus Y satisfies the hypot*
*heses
of Theorem 4.3. Using Theorem 4.3 and Anick's formula one can calculate that if
f is semi-inert then K0(z) = z37.
Recall that L_= (LX qL, d0) where d0e = [[w1, w2], w1] and d0g = [[w1, *
*w2], w2].
Also recall (from Theorem 4.3) that (HL_)0 ~=LXY. Let ~u= [e, w2] + [g, w1] (wi*
*th
|~u| = 37). Then one can check that d~u= 0 and [~u] is a basis for (HL_)1 as a *
*free
(HL_)0-module; so f is indeed semi-inert. Let oeX = oeZ1 oeZ2. It is right in*
*verse
to hX . By Lemma 5.2, PY = {2, 3}. As a result by Theorem 5.5,
H*( Y ; R) ~=ULY where
LY ~=LXYq L__ ~=L=J,
with u = hY (^~) for some map ~ : S38 ! Y and J the Lie ideal generated by
{[[x1, y1], x1], [[x1, y1], y1], [[x2, y2], x2],
[[x2, y2], y2], [[w1, w2], w1], [[w1, w2], w2]}.
Q
Furthermore Y 2 S and there exists a map oeY right inverse to hY .
Note that
LY ~=L11q L21q L2 q L____
where L11~=L21~=R{x, y, [x, y]} and L2 ~=R{w1, w2, [w1, w1], [w1, w2], [w2, w2]*
*}.
Example 6.4. An infinite family of finite CW-complexes constructed out of semi-
inert attaching maps
The construction in the previous example can be extended inductively. By in-
duction, we will construct spaces Xn and maps !n : S~n ! Xn for n 1 such that
Xn is an n-cone constructed out of a sequence of semi-inert attaching maps. Giv*
*en
!n, let wn = hXn ([!n]) and given waiand wbi, let Li= L=Jiwhere Jiis *
*the
Lie ideal of brackets of bracket length 3.
Let R = Z[1_6]. Begin with X1 = S3 and ~1 = 3. Let !1 : S~1 ! X1 be the
identity map.
Given Xn, let Xanand Xbnbe two copies of Xn. For n 1, let
Xn+1 = Xan_ Xbn[fn+1(e~n+1_ e~n+1),
FREE AND SEMI-INERT CELL ATTACHMENTS 19
where ~n+1 = 3~n - 1 and fn+1 = [[!an, !bn], !an] _ [[!an, !bn], !bn].
By the same argument as in the previous example, fn+1 is a semi-inert cell
attachment and there exists a map
!n+1 : S~n+1 ! Xn+1
where ~n+1 = 4~n - 2, such that
i a j
H*( Xn+1; R) ~=ULXn+1 whereLXn+1 = Lji q L
1 i n
1 j 2n-i
with Ljia copy of Li and wn+1 = hXn+1(^!n+1).
Example 6.5. An uncountable family of CW-complexes constructed out of semi-
inert cell attachments
At each stage of the inductive construction in Example 6.4, we could have used
an attachment of the type in Example 6.2 instead of an attachment of the type in
Example 6.1. For ff 2 [0, 1) use the binary expansion of ff to choose the seque*
*nce
of attachments to obtain a space Xff.
Example 6.6. CW-complexes X with only odd-dimensional cells
Let X(n)denote the n-skeleton of X. From the CW-structure of X, there is a
sequence of cell attachments for k 1.
W2k f2k+1---!X(2k-1)! X(2k+1)
where W2k is a finite wedge of (2k)-dimensional spheres, X(2k+1)is the adjuncti*
*on
space of the cell attachment and X(1)= *. Assume R Q containing 1_6. Let PX(n)
be the set of implicit primes of X(n)(see Definition 5.1). We will show by indu*
*ction
that
H*( X(2k+1); Z[PX(2k+1)-1]) ~=ULX(2k+1)
where LX(2k+1)~=LV (2k+1)with V (2k+1)concentratedQin even dimensions. In
addition localized away from PX(2k+1), X(2k+1)2 S and there exists a map
oeX(2k+1)right inverse to hX(2k+1).
For k = 0 these conditions are trivial. Assume they hold for k - 1. Let L_=
(LX(2k-1)qLK, d0) where K is a free R-module in dimension 2k corresponding to t*
*he
spheres in W2k. For degree reasons LW2kX(2k-1)= d0(K) = 0. So f2k+1is automatic*
*ally
free. Furthermore L_has zero differential so HL_= LV (2k-1)q LK. Thus f2k+1 is
semi-inert. By Theorem 5.5, H*( X(2k+1); Z[PX(2k+1)-1]) ~=ULX(2k+1), where
(2k-1) (2k-1)
LX(2k+1)~=LXX(2k+1)q LK ~=LX(2k-1)q LK ~=L(V K).
Q
Also by Theorem 5.5, localized away from PX(2k+1), X(2k+1)2 S, and there
exists a map oeX(2k+1)right inverse to hX(2k+1).
Therefore by induction H*( X; Z[PX -1]) ~=ULX , where LX ~=QL(s-1H~*(X))
with s the suspension map, and localized away from PX , X 2 S.
20 PETER BUBENIK
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MR 2,323c
D'epartment de Math'ematics, Ecole Polytechnique F'ed'erale de Lausanne, CH-1*
*015,
Lausanne, Switzerland
E-mail address: peter.bubenik@epfl.ch
__