SEPARATED LIE MODELS AND THE HOMOTOPY
LIE ALGEBRA
PETER BUBENIK
Abstract. A simply connected topological space X has homo-
topy Lie algebra ss*( X) Q. Following Quillen, there is a con-
nected differential graded free Lie algebra (dgL) called a Lie model,
which determines the rational homotopy type of X, and whose ho-
mology is isomorphic to the homotopy Lie algebra. We show that
such a Lie model can be replaced with one that has a special prop-
erty we call separated. The homology of a separated dgL has a
particular form which lends itself to calculations.
1. Introduction
All of our topological spaces are assumed to be simply connected, and
have finite rational homology in each dimension. For such a space X,
there exists a differential graded Lie algebra (LV, d), where LV denotes
the free graded Lie algebra on the rational vector space V , called a Lie
model, which determines the rational homotopy type of X, and whose
homology is isomorphic to ss*( X) Q, the (rational) homotopy Lie
algebra of X. While this description is pleasing in principle, it is less
satisfying when explicit calculations are desired. Here, we want to
put a certain structure on a Lie model that will prove conducive to
calculation.
Assume that our Lie algebras are finite in each dimension and con-
centrated in positive dimension. Our assumption on our spaces implies
that their Lie models satisfy this assumption. We are particularly in-
terested in dgLs where V can be bigraded as follows: V* = Ni=1Vi,*,
such that
(1.1) dVi,* L( i-1j=1Vj,*).
We will call the first gradation degree and the second, usual gradation
dimension.
____________
Date: May 7, 2007.
2000 Mathematics Subject Classification. Primary 55P62; Secondary 17B55.
Key words and phrases. rational homotopy Lie algebra, Lie models, homology
of differential graded Lie algebras, cell attachment, Schreier property.
1
2 PETER BUBENIK
Our separated condition will say that the attaching map from degree
i + 1 has no effect on the homology coming from degree i - 1. We
need to introduce some notation to make this precise.
Given a Lie algebra L = (LV, d) as above, let
(1.2) Li = (LV i,*, d).
Let ZLi and BLi denote the cycles and boundaries in Li. There is an
induced map
(1.3) "di: Vi,*d-!Z(Li-1) i HLi-1.
The inclusion Li ,! Li+1 induces inclusions BLi ,! BLi+1 and ZLi ,!
ZLi+1. Thus there is an induced map HLi ! HLi+1.
We introduce some notation for two Lie subalgebras of HLi to which
we will often refer. Let HL-i be the Lie subalgebra of HLi given by
im(HLi-1 ! HLi), and let HL+ibe the Lie ideal generated by im(d"i+1:
Vi+1,*! HLi).
Definition 1.1. Let L be a dgL satisfying (1.1). Say that L is separated
if for all i,
HL+i\ HL-i= 0.
Let (LW, d) and (LW 0, d0) be two dgLs satisfying (1.1). Say that
(LW 0, d0) is a bigraded extension of (LW, d) if W 0= W W~ as bigraded
R-modules and d0|W = d.
Theorem 1.2. Let L be a dgL satisfying (1.1). Then L has a bigraded
extension L0such that L0is separated and the inclusion L ,! L0induces
an isomorphism on homology.
Corollary 1.3. Let X be a simply connected space with finite rational
homology in each dimension and finite (rational) LS category. Then X
has a separated Lie model.
Define
(1.4) L_i= (HLi-1 q LVi,*, "di),
where L q L0 denotes the free product of Lie algebras (i.e., their co-
product), "di|HLi-1 = 0 and "di|Vi,*is given in (1.3).
Theorem 1.4. Let L = (LV, d) be a separated dgL. Let
L^i= L((HL_i)1,*)=[d"i+1Vi+1,*].
L
Then HL ~= i^Lias Q-modules. In particular, if L is a Lie model for
a space X, then its homotopy Lie algebra has the form ss*( X) Q ~=
i^Lias rational vector spaces.
SEPARATED LIE MODELS AND THE HOMOTOPY LIE ALGEBRA 3
To state a more precise version of this theorem, which elucidates
some of the Lie algebra structure, we need to define some more nota-
tion.
Given an increasing filtration . . . Fi-1M FiM Fi+1M . . .
ofLan R-module M, there is an associated graded R-module gr(M) =
igriM where griM = FiM=Fi-1M. If M has the structure of an
algebra or Lie algebra, then there is an induced algebra or Lie alge-
bra structure on gr(M). If M has a separate grading, then gr(M) is
bigraded. If M has a differential d and dFiM Fi-1M then the filtra-
tion is called a differential filtration and there is an induced filtration
on H(M, d).
Recall the definition of Li and "di+1from (1.2)and (1.3). There is an
increasing differential filtration {FkLi} on Li = (L(V i), d) ~=(LV*., d) where vk has di-
mension 2k - 1 and dvk = 1_2 i+j=k[vi, vj]. The Lie subalgebra Li =
(L, d) is a Lie model for CP n. Then L1 = (L, 0) = L_1
and L2 = (L, d) = L_2 where dv2 = 1_2[v1, v1]. We remark that
for any dgL, L2 is always separated, since HL-1= 0 and with respect
to L2 (not L), HL+2 = 0. Now (HL_2)0,*= L=[v1, v1] ~= ^L1 and
(HL_2)1,*~= Q{u2} where u2 = [v1, v2]. By Theorem 1.5, the Lie algebra
structure of HL2 is determined by the action of (HL_2)0 on L((HL_2)1,*)
in HL_2. Since d[v2, v2] = [v1, [v1, v2]] in L2, [v1, u2] = 0 in HL_2. So as
Lie algebras, HL2 ~=Lab, where Lab denotes the free abelian Lie
algebra.
Next we will show by induction that Ln is separated, as Lie algebras,
HLn ~= Lab, where un has dimension 2n, and HL-n= Lab.
By definition L_n~= (Lab q L, "dn) where "dnv1 = "dnun-1 =
0 and d"nvn = un-1. So HL-n-1\ HL+n-1 = Lab \ L = 0.
Since (HL_n)0 ~= Lab, (HL_)1 ~= Q{[v1, vn]} and [v1, [v1, vn] = 0 by
the Jacobi identity, the claim follows.
Finally, this implies that HL ~=Lab.
Example 1.9. In the next example we show that the Lie model obtained
from the minimal spherical cone decomposition of a product of spheres
SEPARATED LIE MODELS AND THE HOMOTOPY LIE ALGEBRA 5
is separated, and use this to calculate the homotopy Lie algebra for the
wedges of spheresQof various "thickness". P
Let X = ri=1Sni, where ni 2. Let N = dim X = ri=1ni. Let
Xk denote the subcomplex of X consisting of those points in X such
that at least r - k of the coordinates are the basepoint. In particular,
X1 is the wedge, and Xr-1 is the fat wedge. Also, Xk+1 can be obtained
from Xk by attaching a wedge of spheres. Then,
(1.5) * = X0 X1 . . .Xr = X
is a spherical cone decomposition for X. (It is minimal, since the cone
length of X is bounded below by the rational LS categoryfoffX, which
is r.) In addition, Xk has a Lie model Lk = L ki=1Vi , d , where
Vi = Q{ffi,j} with the ffi,jin one-to-one correspondence with the i-
fold products of the spheres Sn` and the dimension of ffi,j, |ffi,j|, is
one less thanfthefdimension of the corresponding product, and dVi
L i-1j=1Vj. Let LXk denote the homotopy Lie algebra of Xk. Then
HLk ~= LXk and HULk ~= UHLk ~= ULXk where U denotes the uni-
versal enveloping algebra functor.
For a non-negativelyPgraded Q-vector space M, let M(z) be the
formal power series i 0(dim Mi)zi. M(z) is called the Hilbert series
for M, and H*( X; Q)(z) = (ULX )(z) is called the Poincar'e series for
X. Let M(z)-1 denote the power series __1_M(z). Define Ai(z) by
Yr Xr
1 - zni-1x = Ai(z)xi.
i=1 i=0
Q r P r
Let A(z)P= i=1(1 - zni-1) = i=0 Ai(z), and finally define Bk(z) =
(-z)k-1 ri=k+1Ai(z).
Theorem 1.10. As Lie algebras, LX1 ~= L, where |xi| =
ni - 1, LX ~= ri=1L and for r 3, LXr-1 ~= LX q L*__ where
|u| = N - 2. Furthermore, for k 2, ULXk (z)-1 = A(z) - Bk(z).
Proof. The first two Lie algebra isomorphisms in the statement of the
theorem follow directly from the well-known formulas for the homotopy
Lie algebra of a wedge and a product.
The remainder of the statement of the theorem is obtained induc-
tively. We assume that k 2. Notice that the inclusion X1 ,! X
induces a surjection LX1 i LX . So, LX ~= HL-2 ,! HL-k. Assume
that Lk is separated, which is trivial for k = 2. Then using Theo-
rem 1.5 one can show that in fact, HL-k ~=LX . Thus HL-k ,! LX
and hence HL-k\ HL+k= 0. Therefore Lk+1 is separated. Thus, the
minimal cone decomposition (1.5)yields a separated Lie model for X.
6 PETER BUBENIK
For k 2, the Poincar'e series for Xk is obtained as follows. By
Theorem 4.2 and [Bub05 , Lemma 3.8 and Theorem 3.5]) we can apply
Anick's formula [Ani82 , Theorem 3.7],
(ULXk )(z)-1 = (U(HL_k)0)(z)-1 - [Vk+1(z) + z[(ULXk-1)(z)-1-
(U(HL_k)0)(z)-1]]
= A(z) + (-z)k-1Ak(z) - z[(ULXk-1)(z)-1 - A(z)],
where the second equality is by Theorem 1.5 and since (ULX )(z)-1 =
A(z). By induction, this is equal to A(z) - Bk(z).
The Lie algebra isomorphism for LXr-1 follows since the cell attach-
ment from Xr-2 to Xr-1 is semi-inert [Bub05 ].
We remark that the fact that the top-cell attachment of X is in-
ert [HL87 ] is witnessed by L_r= (LX q L____, d), where d|LX = 0 and
dv = u.
From Theorem 1.10 it is easy to check that a fat wedge of odd-
dimensional spheres has the maximum possible gap in the rational ho-
motopy groups [FHT01 , Theorem 33.3] if and only if all the spheres
have the same dimension.
Example 1.11. For the final example, we calculate the homotopy Lie
algebra of a connected sum of products of spheres. For example, a Lie
group M is rationally equivalent to a product of odd spheres (and so
LM is a free abelian Lie algebra).
For s 2 and 1 i s, let Mi be simply-connected, of dimension
N and rationally equivalent to a product of at least three spheres (e.g.
SU(n), n 4). Let X = #si=1Mi. Applying the previous example
gives:
Theorem 1.12. As Lie algebras,
as
LX ~= LMi q L=(u1 + . .+.us),
i=1
where |ui| = N - 2.
Proof. By the previous example Mi has a separated Lie model of the
form (L(i)q L, d) such that L_ri= (LMi`, 0) q (L, "d) with "dvi =
ui. Thus X has a separated Lie model ( si=1L(i)q L, d) and L_r=
` s
( i=1 LMi , 0) q (L, "dv = u1 + . .+.us), where r = max iri.
In particular, LX contains a free Lie algebra on 2s - 1 generators.
So X satisfies the Avramov-F'elix conjecture.
SEPARATED LIE MODELS AND THE HOMOTOPY LIE ALGEBRA 7
In the appendix we state and prove a generalization of the Schreier
property of free Lie algebras: that any Lie subalgebra is also free, which
may be of independent interest. This generalization is used in the proof
of Theorem 1.5.
1.1. Acknowledgments. I would like to thank Kathryn Hess for many
helpful discussions and Greg Lupton and John Oprea for help in rewrit-
ing the introduction and suggesting Examples 1.9 and 1.11.
2. Background
In their landmark papers [Qui69 , Sul77 ], Quillen and Sullivan con-
struct algebraic models for rational homotopy theory. Sullivan con-
structs a contravariant functor APL which serves as a fundamental
bridge between topology and algebra. For a space X, APL (X) is a com-
mutative cochain algebra which has the property that H(APL (X)) ~=
H*(X; Q) as algebras. Quillen gives a construction for a differential
graded Lie algebra (dgL) (LV, d), where LV denotes the free Lie al-
gebra on a rational vector space V , one of whose properties is that
H(LV, d) ~=ss*( X) Q. We will follow convention and call this a free
dgL even though it is almost always not a free object in the category
of differential graded Lie algebras. For an excellent reference on these
models and their applications, the reader is referred to [FHT01 , Parts
II and IV].
A quasi-isomorphism is a morphism which induces an isomorphism
in homology. A Lie model for a space X is a differential graded Lie
algebra (L, d) equipped with a quasi-isomorphism m : C*(L, d) '-!
APL (X). Here, C*(L, d) = Hom (C*(L, d), Q), is the contravariant
functor induced by Quillen's functor C*. C*(L, d) is called the Cartan-
Eilenberg-Chevalley construction on (L, d). Quillen's free dgL above
is a Lie model. Given a Lie model (L, d) for a space X and a quasi-
isomorphism (L, d) -'! (L0, d0), their is an induced quasi-isomorphism
C*(L0, d0) -'! C*(L, d) -'! APL (X). In particular, a quasi-isomorphic
bigraded extension of a Lie model for X is also a Lie model for X.
It is the Samelson product on ss*( X), which corresponds the the
Whitehead product under the canonical isomorphism ss*( X) ~=ss*+1(X),
which gives it the structure of a graded Lie algebra, which we call the
(rational) homotopy Lie algebra.
A rational homotopy equivalence is a continuous map f : X ! Y
such that ss*(f) Q is an isomorphism. Two spaces X and Y are said
to have the same rational homotopy type, written X 'Q Y , if they are
connected by a sequence of rational homotopy equivalences (in either
8 PETER BUBENIK
direction). The Lusternik Schnirelmann (LS) category of a space X,
denoted cat(X), is the smallest integer n such that X is the union of
n + 1 open subsets, each contractible in X. The rational LS category of
a space X, denoted catQ(X), is the smallest integer n such that there
exists a space Y with cat(Y ) = n and X 'Q Y . So catQ(X) cat(X).
A space Xn is said to be a spherical n-cone if there exists a sequence
of spaces
(2.1) * = X0 X1 . . .Xn
iW j
such that for k = 0 . .n.- 1, Xk+1 = Xk [fk+1 jDnj+1j,k, where
W nj
Dn denotes the n-dimensional disk and fk+1 : j Sj,k ! Xk is an
attaching map. A space X is said to have rational cone length n,
written clQ(X) = n, if n is the smallest number such that X 'Q Xn
for some spherical n-cone Xn. (This is equivalent to the more usual
definition of rational cone length, see [FHT01 , Proposition 28.3], for
example.) It is a theorem of Cornea [Cor94 ] that if catQ(X) = n then
clQ(X) = n or n + 1. A CW complex is said to have finite type if it
has finitely many cells in each dimension.
A free dgL (LV, d) is said to have length N if we can decompose
V* = Ni=1Vi,*such that dVi,* L( i-1j=1Vj,*), where dV1,*= 0. Note
that V is now bigraded, though (LV, d) is typically not a bigraded dgL.
We will call the first gradation degree and the second, usual gradation
dimension. For any free dgL (LV, d) of length N there is a spherical
N-cone X such that (LV, d) is a Quillen model for X. We call (LV, d)
the cellular Lie model of X. Furthermore, any spherical N-cone has a
cellular Lie model of length N. We outline the construction as follows.
For each n 1 let the (n+1)-cells Dn+1ffof X correspond to a basis {vff}
of Vn. Let Xn denote the n-skeleton of X. By induction, (LV, ^d) ' (LW, d).
Thus _ is a quasi-isomorphism. It follows that OE is one as well.
Corollary 3.2. Let L = (LW, d) with {ffj, fij}j2J LW where dffj =
0 and dfij = ffj. Taking W~ = k{aj, bj}j2J with |aj| = |fij| and |bj| =
|fij| + 1, let L0 = L(W W~, d0) where d0|W = d, d0aj = ffj, and
d0bj = aj - fij. Then L0' L.
Given an k-module M, let ZM, BM denote the k-submodules of
cycles and boundaries.
Lemma 3.3. Let L = (LW, d) and let V = {vj}j2J HLn with vj 6= 0.
For each vj choose a representative cycle ^vj2 ZL. Let L0= (LW 0, d0)
where W 0= W k{aj, bj}j2J, d0|W = d, daj = ^vjand bj is in dimension
n + 2. Then H n L0~= H n L=V .
Proof. Since Z n L0= Z n L and B n L0~= B n L k{daj}j2J, H n L0~=
H n L=V .
Lemma 3.4. Given k-modules C A B and D B, let p : B !
B=C denote the quotient map. If D \ A C then pD \ pA = 0.
Proof. Let x 2 pD \ pA. Then there are y 2 D, and y0 2 A such that
py = py0 = x. Let z = y - y0. Since pz = 0, z 2 C A. Thus
y = y0+ z 2 A \ D C. Therefore x = py = 0.
The proof of theorem 1.2 will rely on a two-step inductive procedure
given in Proposition 3.8. To help with the book-keeping, we introduce
the following definitions. We will use the notation HL+i and HL-i
defined just before Definition 1.1.
10 PETER BUBENIK
Definition 3.5. Let (LW, d) be a dgL satisfying (1.1). Say that
(LW, d) is k-separated if for all i < k, HL+i\ HL-i= 0. Say (LW, d)
is (k, n)-separated if it is k-separated and H__