ALGEBRAIC STRUCTURE OF THE LOOP SPACE
BOCKSTEIN SPECTRAL SEQUENCE
JONATHAN A. SCOTT
Abstract. Let X be a finite, n-dimensional, r-connected CW
complex. We prove the following theorem:
If p n=r is an odd prime, then the loop space homology Bock-
stein spectral sequence modulo p is a spectral sequence of universal
enveloping algebras over differential graded Lie algebras.
Introduction
Let X be the Moore loop space on a pointed topological space X. If
R Q is a principal ideal domain, then H*(X; R) has a natural Hopf
algebra structure via composition of loops, as long as there is no torsion.
The submodule P H*(X; R) of primitive elements is a graded Lie
subalgebra; in [5], Milnor and Moore showed that if R = Q and X is
simply connected then H*(X; Q) is the universal enveloping algebra of
P . In [4], Halperin established the same conclusion for R Q when X
is a finite, simply-connected CW complex, provided that H*(X; R) is
torsion-free and the least non-invertible prime in R is sufficiently large.
In the presence of torsion, the loop space homology algebra does not
have a natural Hopf algebra structure. However, in [2] Browder showed
that the Bockstein spectral sequence
H*(X; Fp) ) (H*(X; Z)=torsion) Fp
is a spectral sequence of Hopf algebras. Halperin also proved in [4]
that for large enough primes, H*(X; Fp) is the universal enveloping
algebra of a graded Lie algebra. The present article establishes this for
every term in the Bockstein spectral sequence.
Theorem 1. Let X be a finite, n-dimensional, q-connected CW com-
plex (q 1). If p is an odd prime and p n=q, then each term in the
mod p homology Bockstein spectral sequence for X is the universal
enveloping algebra of a differential graded Lie algebra (Lr; fir).
____________
Date: December 13, 1999.
1991 Mathematics Subject Classification. Primary 55P35, Secondary 16S30.
Key words and phrases. loop space homology, Bockstein spectral sequence, uni-
versal enveloping algebra.
1
2 JONATHAN A. SCOTT
In [1], under the hypotheses of Theorem 1, Anick associates to X
a differential graded Lie algebra LX over Z(p) and a natural quasi-
isomorphism ULX '! C*(X; Z(p)) of graded Hopf algebras. The in-
clusion X : LX ,! ULX then induces a transformation of Bockstein
spectral sequences Er(X ) : Er(LX ) ! Er(X):
Theorem 2. The image of each Er(X ) is contained in Lr.
Theorems 1 and 2 follow from the work of Anick in [1] and the
following:
Theorem 3. Let (L; @) be a differential graded Lie algebra over Z(p)
which is connected, free as a graded module, and of finite type. The
mod p homology Bockstein spectral sequence of U(L; @) is a sequence
of universal enveloping algebras, Er(UL) = U(Lr; fir): Furthermore, if
: L ,! UL is the inclusion, then the image of Er() is contained in
Lr.
The proof of Theorem 3 utilizes in a fundamental way the divided
powers structure of the dual of a universal enveloping algebra.
The structure of the article is as follows.
Section 1. Notation and review of graded Lie algebras, divided pow-
ers algebras, Bockstein spectral sequences, acyclic closures and minimal
models.
Section 2. In [4], Halperin showed that for a differential graded Lie
algebra (L; @) over Fp, H(UL) = UE for a graded Lie algebra E. We
show that the inclusion : (L; @) ,! U(L; @) satisfies im H() E.
Section 3. Proof of Theorem 3.
Section 4. We show that a Hopf algebra morphism UL1 ! UL2 is
of the form U(') if and only if its dual respects divided powers.
Section 5. Two examples. The first gives a differential graded Lie
algebra whose Bockstein spectral sequence collapses after the first term,
while the spectral sequence of its universal enveloping algebra never
does. The second shows that the sequence of Lie algebras given by
Theorem 3 is not natural.
Acknowledgments. This paper is the result of work for my Ph.D.
thesis. I would like to take this opportunity to thank my Ph.D. ad-
visor, Steve Halperin, who suggested the problem of studying torsion
in loop space homology, and who patiently and consistently provided
encouragement, deep insight, and support.
1. Preliminaries
Let R be a commutative ring in which 2 is invertible. All objects are
graded by the integers unless otherwise stated. Fix an odd prime p.
LOOP SPACE BOCKSTEIN SPECTRAL SEQUENCE 3
The ring of integers localized at p is denoted Z(p)while the prime field
is denoted Fp. Differential graded modules, algebras, coalgebras, and
Hopf algebras are shortened to dgm, dga, dgc, and dgh, respectively;
a comprehensive treatment of these objects is given in [3].
1.1. Graded modules. Let M be a graded module over R. If x 2 Mk
then we say that x has degree k, and write |x| = k. A free graded
module M is of finite type if each Mk is of finite rank. We raise and
lower degrees by the convention Mk = M-k . We denote by sM the
suspension of M: (sM)i = Mi-1. The dual of M is the graded module
M] = Hom (M; R). If M is finite type and N = (sM)], then M = (sN)]
via x(sf) = -f(sx), for x 2 M, f 2 N.
If V is a graded module over R, then we denote by T V and V the
tensor algebra and free commutative algebra on V , respectively. The
tensor coalgebra on V is denoted by TC V . The shuffle product ([4],
Appendix) makes TC V into a graded commutative (not cocommuta-
tive) Hopf algebra. Note that T V = k0 T kV , V = k0 kV and
TC V = k0 TCkV , with T kV , kV , and TCkV consisting of words in V
of length k. Elements of TCkV are denoted [v1| . .|.vk].
The symmetric group Sk acts on T kV via oe . (x1 . . .xk) =
xoe(1) . . .xoe(k), where the sign is determined by the rule x y 7!
(-1)|x||y|y x.
1.2. Graded Lie algebras. A graded Lie algebra is a graded R-module
L = k0 Lk along with a degree-zero linear map [ ; ] : LL ! L, called
the Lie bracket, satisfying graded anti-commutativity, the graded Ja-
cobi identity, and the further condition [x; [x; x]] = 0 if x 2 Lodd.
For example, any non-negatively graded associative algebra A is a
graded Lie algebra via the graded commutator bracket [a; b] = ab -
(-1)|a||b|ba, for a; b 2 A.
A graded Lie algebra is connected if it is concentrated in strictly
positive degrees.
The graded abelian Lie algebra on {xj}, denoted Lab(xj), is the free
graded module on the basis {xj}, with the trivial Lie bracket.
Let L be a graded Lie algebra, and denote by L[ the underlying
graded module. The universal enveloping algebra of L is the associative
algebra UL = (T L[)=I, where I is the ideal generated by elements of
the form x y - (-1)|x||y|y x - [x; y], for x; y 2 L. UL has the natural
structure of a graded Hopf algebra; the comultiplication is defined by
declaring the elements of L to be primitive and then using the universal
property.
A Lie derivation on a graded Lie algebra L is a linear operator on L
of degree k such that for x; y 2 L, ([x; y]) = [(x); y]+(-1)k|x|[x; (y)]:
4 JONATHAN A. SCOTT
A differential graded Lie algebra (dgl for short) is a pair (L; @), where
L is a graded Lie algebra, and @ is a Lie derivation on L of degree -1
satisfying @@ = 0. If (L; @) is a dgl, then @ extends to a derivation on
UL, making U(L; @) into a dga.
1.3. Divided powers algebras. Divided powers algebras arise here
as the duals of universal enveloping algebras.
Definition 1. A divided powers algebra, or -algebra, is a commuta-
tive graded algebra A, satisfying either A = A0 or A = A0 , equipped
with set maps flk : A2n ! A2nk for k 0 and n 6= 0 satisfying the
following list of conditions.
1. fl0(a) = 1; fl1(a)P= a for a 2 A;
2. flk(a + b) = kj=0flj(a)flk-j(b) for a; b 2 A2n;
j + k j+k 2n
3. flj(a)flk(a) = fl (a) for a 2 A ;
j
(jk)! j+k 2n
4. flj(flk(a)) = _____fl (a) for a 2 A ;
ae j!kk!
akflk(b) if |a| and |b| even, |b| 6= 0,
5. flk(ab) = 0 if |a| and |b| odd.
A -morphism is an algebra morphism which respects the divided
powers operations. A -derivation on a -algebra A is a derivation on
A satisfying (flk(a)) = (a)flk-1(a) for a 2 A2n, k 1. A differential
graded -algebra, or -dga, is a pair (A; @), where A is a -algebra,
and @ is a -derivation of degree -1 satisfying @@ = 0.
Let V be a free graded R-module. Let k(V ) be the graded submod-
ule of TCkV of elements fixed by the action of the symmetric group Sk.
Then (V ) = kk(V ) is a Hopf subalgebra of TC (V ), called the free
-algebra on V . Divided powers are defined on (V ) by
1. fl0(v) = 1, fl1(v) = v for v 2 V ,
2. flk(v) = [v| . .|.v]for v 2 V 2n
____-z___"
k times
and then extending via conditions (4) and (5) of Definition 1. If f :
V ! A is any linear map of degree zero from V into a -algebra A,
then f extends to a unique -morphism f : (V ) ! A. If V is R-free
on a countable, well-ordered basis {vi}, then (V ) is R-free, with basis
consisting of elements flk1(v1) . .f.lks(vs) where kj 0 and kj = 0 or 1
if |vj| is odd.
<;>
If V W ! R is a pairing, then there is an induced pairing
(1) T V TC W ! R
LOOP SPACE BOCKSTEIN SPECTRAL SEQUENCE 5
given by = 0 if j 6= k, and
(2) = . .<.vk; wk>
where is the sign of the permutation
v1; : :;:vk; w1; : :;:wk 7! v1; w1; : :;:vk; wk:
The pairing (1) in turn induces a pairing
(3) V W ! R:
Suppose that V is R-free of finite type, V = V<0 or V = V>0, and W =
V ]. Then (1) and (3) induce Hopf algebra isomorphisms TC (V ]) ~=
(T V )] and (V ]) ~=(V )]:
1.4. The Cartan-Chevalley-Eilenberg-Cartan complex. Denote
by B(A) the bar construction on the augmented dga (A; @) ([4], Sec-
tion 1); recall that the underlying coalgebra of B(A) is TC (sA ), where
A is the augmentation ideal. Let (L; @) be a dgl. Then (sL)
____
(sUL ) B(UL) and ((sL); @0 + @1) is a sub-dgc of B(UL), de-
noted by C*(L; @), called the chains on (L; @).
The Cartan-Chevalley-Eilenberg-Cartan complex on (L; @) is the
commutative cochain algebra C*(L; @) = (V; d), dual to C*(L; @),
where V = (sL)], and the differential d is the sum of derivations d0
and d1. The linear part d0 preserves word length and is dual to @ in
that = (-1)|v| for v 2 V , x 2 L. The quadratic part
d1 increases word length by one and is dual to the Lie bracket in L:
(4) = (-1)|sy|
where the pairing is (3) above with W = sL = V ]. We will usually refer
to the Cartan-Chevalley-Eilenberg-Cartan complex as the cochains on
(L; @).
1.5. Bockstein spectral sequences. Fix a prime p. Let C be a free
chain complex over Z(p). Applying C - to the short exact sequence
of coefficient modules
xp
0 ! Z(p)! Z(p)! Fp ! 0
leads to a long exact sequence in homology which may be wrapped into
the exact couple
H*(C) __________________H*(C)w
'''* [[
' [[^
H*(C; Fp)
6 JONATHAN A. SCOTT
from which we get the homology Bockstein spectral sequence modulo p of
C, (Er(C); fir), mod p bss for short [2]. If C = C*(X) is the normalized
singular chain complex of a space X, then we refer to the homology bss
mod p of C*(X) as the mod p homology bss of X, denoted (Er(X); fir).
There is the corresponding notion of cohomology Bockstein spectral
sequence defined in the obvious manner, using the functor Hom (C; -)
rather than C -.
The mod p bss of C measures p-torsion in H*(C): if x; y 2 Er,
x 6= 0, satisfy fir(y) = x, then x represents a torsion element of order
pr in H*(C).
Notation. If c 2 C is such that [c] 2 E1 lives until the Er term then
we will denote the corresponding element of Er by [c]r.
1.6. Acyclic closures and minimal models. (Reference: [4], Sec-
tions 2 and 7.) Consider the graded algebra V (sV ) over R. Extend
the divided powers operations on (sV ) to R V + (sV ) via rule
5 of Definition 1.
Definition 2. ([4], Section 2) An acyclic closure of the dga (V; d) is
a dga of the form C = (V (sV ); D) in which D is a -derivation
restricting to d in V and H(C) = H0(C) = R.
Let (L; @) be a connected dgl over R which is R-free of finite type.
Then C*(L) = (V; d) where V = (sL)]. Let C be an acyclic closure
for C*(L), and set ((sV ); D ) = R C*(L)C. By the work of Halperin
in [4], we identify H(UL) = H([(sV ); D ]]) and UL = ((sV ); D )].
Let R = Z(p)or R = Fp, and consider a commutative algebra of the
form (W; d) over R, where W = W 2 is R-freePand of finite type.
We may write the differential as a sum d = j0 dj where dj raises
wordlength by j.
Definition 3. If R = Z(p), the dga (W; d) above is Z(p)-minimal if
d0 : W ! pW . If R = Fp, (W; d) is Fp-minimal if d0 = 0.
Suppose (A; @) is a cochain algebra satisfying H0(A) = R, H1(A) =
0, H2(A) is R-free, and H*(A) is of finite type. Then by [4], Theorem
7.1, there exists a quasi-isomorphism m : (W; d) !' (A; @) from an
R-minimal algebra. This quasi-isomorphism is called a minimal model.
Associated to an Fp-minimal model m : (W; d) !' (A; @) is its
homotopy Lie algebra, E. As a graded vector space, E = (sW )]; the
bracket is defined by the relation
= (-1)|sy|
for w 2 W , x; y 2 E.
LOOP SPACE BOCKSTEIN SPECTRAL SEQUENCE 7
2. The image of H(L) ! H(UL)
Let (L; @) be a connected dgl over Fpof finite type. By [4], the choice
of minimal model m : (W; d) !' C*(L) determines an isomorphism
of graded Hopf algebras, H(UL) ~=UE, where E is the homotopy Lie
algebra of m.
Proposition 4. With the notation above, the image of H() : H(L) !
H(UL) lies in E.
Proof. It suffices to prove that the following diagram commutes.
H(L)
H(L; @) ______wH(UL)
| |
(5) | |~=
| |
|u E |u
E ___________wUE
Recall that C*(L; @) = (V; d), where V = (sL)] and d = d0 + d1.
Recall further that the minimality condition on (W; d) implies that
the linear part of its differential vanishes. The linear part of m is the
linear map m0 : (W; 0) ! (V; d0) defined by the condition m - m0 :
W ! 2 V . Recall that E = (sW )] and UE = (sW )] ([4], Theorem
6.2).
The model m extends to a morphism of constructible acyclic closures
([4], Section 2) ^m : (W (sW ); D) ! (V (sV ); D) by Propo-
sition 2.7 of [4]. Since (W; d) is Fp-minimal, d0 = 0. By Corollary 2.6
of [4], d0 = 0 is equivalent to D = 0 in ((sW ); D ). Apply Fpm - to
^m to get a -morphism m : ((sW ); 0) ! ((sV ); D ).
Let ssL : ((sV ); D ) i s(V; d0) and ssE : ((sW ); 0) i s(W; 0) be
the projections. The maps ssL and ssE fit into the diagram
ssE
((sW ); 0) ______ws(W; 0)
| |
| |
(6) m | |sm0
| |
|u ss |u
L
((sV ); D ) ______ws(V; d0)
For w 2 W , Proposition 2.7 of [4] states that m^(1 sw) - 1 sm0w
has total wordlength at least two. It follows that m (sw) - sm0w has
(sV )-wordlength at least two, so ssL(m (sw)) = sm0w = sm0(ssE (sw)),__
so Diagram (6) commutes. Dualize and pass to homology to get (5). |__|
8 JONATHAN A. SCOTT
3. Bockstein spectral sequence of a universal enveloping
algebra
In this section, we prove the main algebraic result, Theorem 3. Un-
less otherwise stated, our ground ring will be Z(p), the integers localized
at p.
Let (W; d) be a minimal Sullivan algebra over Z(p). Let C = (W
(sW ); D) be a constructible acyclic closure for (W; d) ([4], Section 2).
Let ((sW ); D ) be the quotient Z(p)(W;d) C. C Fpis a constructible
acyclic closure for (W; d) Fp. Since (W; d) is Z(p)-minimal, p|d0, so
the linear part of the differential vanishes in (W; d) Fp. It follows
by Corollary 2.6 of [4] that the differential in ((sW ); D ) Fp is null,
so that p|D . Set Er = Er([(sW ); D ]]) and Er = Er((sW ); D ). Let
ae : (sW ) ! (sW ) Fp = E1 be the reduction homomorphism.
Proposition 5. With the hypotheses and notation above, for r 1,
the following statements hold.
1. Er is isomorphic to a free divided powers algebra,
2. there is a -morphism gr : Er ! E1 such that if g(z) = ae(a) for
some z 2 Er, a 2 (sW ), then z = [a]r.
3. there is a graded Lie algebra Lr such that (Er; fir) = U(Lr; fir).
Lemma 6. Let (UL; @) be a dgh over Fp of finite type, so (UL)] = V
as an algebra. If @] is a -derivation, then @(L) L.
Proof of Lemma 6. It suffices to prove the dual statement, namely that
@] : V ! V factors over the surjection ss : V ! V to induce a
differential in V . But ker(ss) consists of products along with elements
k ]
of the form flp (v) for v 2 V , k 1. Since @ is a -derivation,
k ] pk-1 ]
@](flp (v)) = @ (v)fl (v) is a product. It follows that @ (ker(ss))_
ker(ss), completing the proof. |__|
Proof of Proposition 5.We proceed by induction. For r = 1, let W1 =
W Fp. Since p|D , E1 = (sW1), establishing the first statement. For
the second statement we may take g1 to be the identity map. Let L1
be the homotopy Lie algebra of (W; d) Fp. Apply Theorems 6.2 and
6.3 of [4] to the minimal algebra (W; d) Fp to get a graded Hopf
algebra isomorphism E1 = ((sW1))] ~=UL1.
Since p|D in ((sW ); D ), fi1 = D=p (reduced modulo p). Thus
because D is a -derivation, so is fi1. Since (sW1) is the -algebra dual
to UL1 it follows by Lemma 6 that fi1 : L1 ! L1 and so E1 = U(L1; fi1).
Now suppose the three statements are established for r - 1. Let
C(r-1) = (Wr-1(sWr-1); D) be a constructible acyclic closure for
LOOP SPACE BOCKSTEIN SPECTRAL SEQUENCE 9
C*(Lr-1; fir-1) = (Wr-1; d). By Lemma 5.4 of [4], there is a chain iso-
~=
morphism flr-1 : (Er-1; fir-1) = U(Lr-1; fir-1) ! ((sWr-1); D )]. Fix
a well-ordered basis {xj} of Lr-1; this determines a dual basis {swj} of
sWr-1. The isomorphism flr-1 identifies the Poincare-Birkhoff-Witt
basis element xk11. .x.kjjof ULr-1 as a dual basis to the basis ele-
ment flk1(sw1) . .f.lkj(swj) of (sWr-1). It follows that flr-1 factors as
~= ~=
the composition of dgc isomorphisms ULr ! (Lr)[ ! ((sWr-1))].
Since Lr-1 and Wr-1 are finite type, we dualize to obtain the dga
~=
isomorphism ffr-1 : ((sWr-1); D ) ! (Er-1; fir-1).
Let mr : (Wr; d) '! C*(Lr-1; fir-1) be a minimal model. Let C0(r)
be a constructible acyclic closure of (Wr; d) ([4], Section 2). Since
(Wr; d) is Fp-minimal, d0 = 0, so by Corollary 2.6 of [4], the differ-
ential in ((sWr); D(r)) = Fp W C0(r) is zero. By [4], Proposition
2.7, mr induces a -morphism mr : ((sWr); 0) '! ((sWr-1); D ):
Since Fp is a field, by Lemma 3.3 of [4], we may identify H(mr ) with
Tormr (Fp; Fp), where Tor is the differential torsion functor [3]. There-
fore since mr is a quasi-isomorphism, H(mr ) is an isomorphism. Let
~=
ffr : (sWr) ! Er be the composition of algebra isomorphisms
H(mr ) H(ffr-1)
(sWr) -- - ! H((sWr-1); fir-1) --- - - !Er
to establish the first statement. Note that ffr will be the isomorphism
dual to flr in the next stage of the induction.
Setting f = ffr-1mr ff-1r, we get the commutative diagram
mr
((sWr); 0) ______w((sWr-1); D )
| |
ffr|~= ffr-|1~=
| |
|u f |u
(Er; 0) __________(Er-1;wfir-1)
and it follows from the definitions that H(f) is the identity on Er.
By the inductive hypothesis, there exists a -morphism gr-1 : Er-1 !
E1 such that z = [a]r-1 whenever z 2 Er-1, a 2 (sW ) satisfy
g(z) = ae(a). We now show that gr := gr-1f satisfies statement 2.
For u 2 Er choose a 2 (sW ) so that gr-1(f(u)) = ae(a). Then
f(u) = [a]r-1, hence fir-1[a]r-1 = 0 and [a]r 2 Er is defined. Since f
induces the identity in homology, f([a]r) = [a]r-1 + fir-1(v) for some
v 2 Er-1. Thus f(u - [a]r) = fir-1(v), so u - [a]r is a boundary in
(Er; 0), whence u = [a]r. This establishes the second statement.
10 JONATHAN A. SCOTT
The model mr determines an isomorphism Er = H(U(Lr-1; fir-1)) ~=
ULr, where Lr is the homotopy Lie algebra for the model mr. As a
graded vector space, Lr = (sWr)].
Let u 2 Er, and suppose for some a 2 (sW ) that ae(a) = gr(u).
Then u = [a]r, so D a = prb for some b 2 (sW ). Thus fir(u) =
[b]r. Since gr and ae are -morphisms, ae(flj(a)) = gr(flj(u)) so flj(u) =
[flj(a)]r. Furthermore, D (flk(a)) = prb . flk-1(a) so
firflk(u) = fir[flk(a)]r = [b . flk-1(a)]r = [b]r[flk-1(a)]r = fir(u) . flk-1(u*
*):
By Lemma 6, this establishes the third statement, completing the in- __
ductive step and the proof. |__|
Proof of Theorem 3. Let m : (W; d) '! C*(L; @) be a minimal model.
Recall that the underlying algebra of C*(L; @) is V , where V = (sL)].
Let (W (sW ); D) and (V (sV ); D) be constructible acyclic
closures for (W; d) and C*(L; @), respectively. The model m deter-
mines a -morphism m : ((sW ); D ) ! ((sV ); D ) where H(m] ) is an
isomorphism. The composition
~= ] ' ]
U(L; @) ! ((sV ); D ) ! ((sW ); D )
induces an isomorphism of Bockstein spectral sequences, establishing
the first statement.
The reduced minimal model m Fp : (W; d) Fp '!C*(L; @) Fp
has homotopy Lie algebra L1, so by Proposition 4, im E1() L1.
Suppose that im Er-1() Lr-1. Let (r-1): Lr-1 ,! ULr-1 be the
inclusion. Then im Er() im H((r-1)) . The homotopy Lie algebra of
the minimal model mr : (Wr; d) !' C*(Lr-1; fir-1) is Lr, so Proposi-
tion 4 states that im H((r-1)) Lr, completing the induction and the __
proof. |__|
Proof of Theorems 1 and 2. Anick in [1] proves that there is a dgl LX
and a dgh quasi-isomorphism ULX ! C*(X; Z(p)). Thus as Hopf
algebras, for r 1, Er(ULX ) = Er(X) and Er(ULX ) = Er(X). __
The result follows by applying Theorem 3 to the dgl LX . |__|
4. Morphisms of universal enveloping algebras
Let R be a commutative ring, L1 and L2 connected, graded Lie
algebras over R which are R-free of finite type. Let ' : UL1 ! UL2 be
a morphism of Hopf algebras. The purpose of this section is to prove
Proposition 7. '(L1) L2 if and only if '] : (UL2)] ! (UL1)] is a
-morphism.
LOOP SPACE BOCKSTEIN SPECTRAL SEQUENCE 11
Proof. Observe that, for j = 1; 2, the sequence of functors
Lj C*(Lj) Vj = R C*(Lj)(C*(Lj) Vj) (Vj)] = ULj
identifies ULj as the natural dual of the free -algebra Vj, with Lj
naturally dual to Vj, and the inclusion Lj ,! ULj naturally dual to
ssj
the projection Vj i Vj. Therefore if '|L1 : L1 ! L2, then '] is a -
morphism. Conversely, if '] is a -morphism, then '](ker ss2) kerss1,_
so '(L1) L2. |__|
5. Examples
We begin with a proposition to be used in both examples.
Proposition 8. Define a dgl over Fp by (L; @) = (Lab(e; f); @f = e),
where |f| = 2n. Then C*(L; @) = ((x; y); d) with dx = y and |x| = 2n.
A minimal model m : ((x1; y1); 0) !' C*(L; @), given by x1 7! xp
~=
and y1 7! xp-1y, induces isomorphisms (sx1; sy1) ! H([UL]]) and
~=
H(UL) ! ULab(e1; f1) with |e1| = |sx1| = 2np - 1, |f1| = |sy1| = 2np.
__
Proof. Straightforward. |__|
Example 1. Let L = Lab(e; f) over Z(p)on generators e and f of de-
grees 2n - 1 and 2n, respectively. Set @f = pe. Applying Proposition 8
recursively, we have Er(UL) = (sxr; syr) and Er(UL) = ULab(er; fr),
with |er| = |sxr| = 2npr - 1, |fr| = |syr| = 2npr, fir(sxr) = syr, and
fir(fr) = er, while the sequence Er(L) collapses after the first term.
Example 2. Define a dgl (L; @) over Z(p)by L = Lab(e; f; g), where
|e| = 2n - 1, |f| = |g| = 2, and @(f) = pe. Then L1 = Lab(e; f; g) (over
Fp), with fi1(f) = e, and C*(L1; fi1) = ((x; y); dx = y) ((z); 0).
Recall the model m from Proposition 8. Define dga morphisms i; j :
((z); 0) ! C*(L1; fi1) by i(z) = z, j(z) = z + y. Then ' = m i
and = m j are minimal models, both with homotopy Lie alge-
bra L2 = Lab(a; b; c), |a| = 2np - 1,|b| = 2np, and |c| = 2n. The two
models determine Hopf algebra isomorphisms '*; * : H(UL1) ! UL2,
given by '*[efp-1 ] = *[efp-1 ] = a, '*[g] = *[g] = c, '*[fp] = b, and
*[fp] = b + cp. The algebra isomorphism *('*)-1 : ULab(a; b; c) !
ULab(a; b; c) is not of the form U for any Lie algebra morphism :
Lab(a; b; c) ! Lab(a; b; c). Therefore the construction involved in The-
orem 3 is not natural.
References
1.D. Anick, Hopf algebras up to homotopy, J. Amer. Math. Soc. 2 (1989), no. 3,
417-453.
12 JONATHAN A. SCOTT
2.W. Browder, Torsion in H-spaces, Ann. of Math. (2) 74 (1961), 24-51.
3.Y. Felix, S. Halperin, and J.-C. Thomas, Differential graded algebras in topo*
*logy,
Handbook of Algebraic Topology, North-Holland, Amsterdam, 1995, pp. 829-
865.
4.S. Halperin, Universal enveloping algebras and loop space homology, J. Pure
Appl. Alg. 83 (1992), 237-282.
5.J. W. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. of Math.
(2) 81 (1965), 211-264.
Department of Mathematics, University of Toronto, M5S 3G3 Canada
E-mail address: scott@math.toronto.edu