Exponential growth of Lie algebras of finite global dimension

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Exponential growth of Lie algebras of finite global dimension

Yves Félix, Steve Halperin, Jean-Claude Thomas

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Yves Félix, Steve Halperin, Jean-Claude Thomas. Exponential growth of Lie algebras of finite global dimension. 2005. �hal-00008857�

ccsd-00008857, version 1 - 19 Sep 2005

An asymptotic formula for the ranks of the homotopy groups of a finite complex

Yves Felix, Steve Halperin and Jean-Claude Thomas September 23, 2005

Abstract

Let X be a finite simply connected CW complex of dimension n. The loop space homology H_∗(ΩX;Q) is the universal envelop- ing algebra of a graded Lie algebra LX isomorphic with π_∗−1(X)⊗ Q. Let QX ⊂ LX be a minimal generating subspace, and set α = lim supi

logrkπi(X)

i . Theorem:

If dimLX =∞and lim sup(dim (QX)k)^1/k <lim sup(dim (LX)k)^1/k, then

n−1

X

i=1

rkπk+i(X) =e^(α+εk)k, whereεk→0 ask→ ∞.

In particular

n−1

X

i=1

rkπk+i(X) grows exponentially ink.

AMS Classification: 55P35, 55P62, 17B70

Key words: Homotopy Lie algebra, graded Lie algebra, exponential growth

1 Introduction

SupposeX is a finite simply connected CW complex of dimension n. The homotopy groups of X then have the form

π_i(X) =Z^ρi⊕T_i,

whereT_i is a finite abelian group andρ_i = rkπ_i(X) is finite. It is known [6]

that eitherπ_i(X) =T_i,i≥2n(X isrationally elliptic) or else for all k≥1, Pn−1

i=1 rkπ_k+i(X)>0. In this case X is called rationally hyperbolic.

In [7] it is shown that in the rationally hyperbolic case^Pn−1_i=1 rkπ_k+i(X) grows faster than any polynomial in k. Here we show that with an addi- tional hypothesis this sum grows exponentially in k and, in fact, setting α= lim sup^{log rk}_i ^πi we have

n−1

X

i1

rkπ_k+i =e^(α+εk)k, whereε_k→0 as k→ ∞.

In subsequent papers we will identify a large class of spaces for which the additional hypothesis holds: in fact it may well hold for all finite simply connected CW complexes.

Note that rkπ_i(X) = dimπ_i(X)⊗Q. Thus we work more generally with simply connected spaces X such that each H_i(X;Q) is finite dimensional.

In this case dimπ_i(X)⊗Q is also finite for each i. On the other hand, a theorem of Milnor-Moore-Cartan-Serre asserts that the loop space homology H∗(ΩX;Q) is the universal enveloping algebra of a graded Lie algebra L_X and that the Hurewicz homomorphism is an isomorphismπ∗(ΩX)⊗Q−→^∼= L_X. Since there are natural isomorphismsπ_i(X)∼=π_i−1(ΩX) it follows that the results above can be phrased in terms of the integers dim (L_X)_i.

For any graded vector spaceV concentrated in positive degrees we define thelogarithmic index of V by

log indexV = lim suplog dimV_k

k .

In ([5], [6]) it is shown that if X (simply connected) has finite Lusternik- Schnirelmann category (in particular, if X is a finite CW complex) and if X is rationally hyperbolic then

log indexL_X >0.

Now let Q_X denote a minimal generating subspace for the Lie algebra L_X, and letα denote log indexL_X.

Theorem 1. Let X be a simply connected topological space with fi- nite dimensional rational homology concentrated in degrees ≤ n. Suppose log indexQ_X <log indexL_X, then

n−1

X

i=1

dim(L_X)_k+i =e^(α+εk)k, whereε_k→0 ask→ ∞. In particular this sum grows exponentially ink.

Theorem 2. Let X be a simply connected topological space with fi- nite dimensional rational homology in each degree, and finite Lusternik- Schnirelmann category. Suppose log indexQX <log indexLX <∞. Then for somed >0,

d−1

X

i=1

dim(L_X)_k+i=e^(α+εk)k, whereε_k →0ask→ ∞. In particular,^Pd−1_i=1 dim(LX)_k+i grows exponentially ink.

Corollary: The conclusion of Theorem 1 holds for finite simply connected CW complexesX for whichL_X is infinite, but finitely generated. The con- clusion of Theorem 2 holds for simply connected spaces of finite LS category and finite rational Betti numbers provided that LX is infinite, but finitely generated, and, log indexL_X <∞.

Recall that the depth of a graded Lie algebra L is the least m (or ∞) such that Ext^m_{U L}(Q, U L)6= 0.

The key ingredients in the proofs of Theorems 1 and 2 are

• A growth condition forL_X established in ([5],[6])

• The fact that depthL_X <∞, established in ([3],[6])

We shall use Lie algebra arguments in Theorem 3 below to deduce the conclusion of Theorems 1 and 2 from these ingredients, and then deduce Theorems 1 and 2.

Theorems 1 and 2 may be compared with the results in [7] and in [8]

that assert forn-dimensional finite CW complexes (respectively for simply connected spaces with finite type rational homology and finite Lusternik- Schnirelmann category) that ^Pn−1_i=1 dim (L_X)_k+i (resp. ^Pd−1_i=1 dim (L_X)_k+i) grows faster than any polynomial ink. These results use only the fact that LX has finite depth, and require no hypothesis on QX.

The hypothesis onQ_X in Theorems 1 and 2 may be restated as requiring that the formal series ^P_qdim Tor^{U L}_1,q^Xz^q have a radius of convergence strictly greater than that of the formal series^P_qdim (U LX)qz^q. Lambrechts [9] has proved a much stronger result under the hypothesis that the formal series^P_q^P_p(−1)^pdim Tor_p,qz^qhas a radius of convergence strictly larger than that of^P_qdim (U L_X)_qz^q.

2 Lie algebras

In this section we work over any ground field lk of characteristic different from 2; graded Lie algebras L are defined as in [7] and, in particular, are assumed to satisfy [x,[x, x]] = 0, x ∈ L_odd (This follows from the Jacobi identity except when charlk= 3).

A graded Lie algebraL is connected and of finite type if L={L_i}_i≥1 and each L_i is finite dimensional.

We shall refer to these ascft Lie algebras. The minimal generating subspaces Qof a cft Lie algebraL are those subspacesQfor which Q→L/[L, L] is a linear isomorphism.

A growth sequence for a cft Lie algebra L is a sequence (r_i) such that r_i → ∞and

ilim→∞

log dimL_ri

r_i = log indexL .

Aquasi-geometric sequence (ℓ_i) is a sequence such that for some integer m, ℓ_i < ℓ_i+1 ≤ mℓ_i, all i; if additionally (ℓ_i) is a growth sequence then it is a quasi-geometric growth sequence.

Of particular interest here are the growth conditions

0<log indexL <∞; (A.1) and

log indexQ <log indexL . (A.2) Proposition. Let L be a cft Lie algebra satisfying (A.1) and (A.2), and assume L has a quasi-geometric growth sequence (r_j). Then any sequence (s_i) such that s_i → ∞ has a subsequence(s_ij) for which there are growth sequences(tj) and(pj)such that

t_j ≤s_ij < p_j and p_j/t_j →1.

Proof. First note that because of (A.2),

log indexL/Q= log indexL

Now adopt the following notation, fori≥1 : log indexL/Q=α dimL_ie^(α+εi)i dimQ_i =e^(α+σi)i dimL_i/Q_i=e^(α+τi)i dim (U L)_i =e^(α+δi)i















(1)

Then because of (A.1) and (A.2), 0 < α < ∞, and lim supε_i = 0. More- over, by a result of Babenko ([2], [6]), log indexU L = log indexL, and so lim sup(δ_i) = 0. Finally (A.2) implies that lim sup(σ_i) = σ < 0, and that τ_qi→0 as i→ ∞.

Next, since r_j is a quasi-geometric sequence, for some fixed m we have r_j < r_j+1 ≤mr_j, all j. It follows that for each s_i in our sequence we may choose q_i in the sequence (r_j) so that

si< qi ≤msi. (2)

Thus q_i → ∞ as i → ∞. The adjoint representation of U L in L defines surjections

M

(ℓ,k,t)∈J_i

(U L)_ℓ⊗Q_k⊗L_t։ ^M

(ℓ,k,t)∈J_i

(U L)_ℓ⊗[Q_k, L_t]։L_qi/Q_qi, (3) whereJ_i consists of those triples for whichℓ+k+t=q_i and t≤s_i < t+k.

Thus

3(q_i+ 1) max

(ℓ,k,t)∈Ji

dim (U L)_ℓ dim [Q_k, L_t]≥dimL_qi/Q_qi. For each s_i we may therefore choose (ℓ_i, k_i, t_i)∈ J_i so that

3(qi+ 1) dim (U L)_ℓidim [Q_ki, Lti]≥dimLqi/Qqi. (4) Lemma 1. (k_i+t_i)≥ _m¹ q_i, and(k_i+t_i)≥ _m−1¹ ℓ_i. In particular,k_i+t_i→ ∞ asi→ ∞.

Proof. It follows from (2) thatk_i+t_i ≥s_i ≥ _m¹ q_i. Sinceq_i =k_i+t_i+ℓ_i, (k_i+t_i)≥ _m¹₋₁ℓ_i. Sinces_i → ∞so does k_i+t_i.

Next, defineε(k_i, t_i) by

dim [Q_ki, L_ti] =e^{[α+ε(ki,ti)](ki+ti)}.

Lemma 2. ε(k_i, t_i)→0 andε_ki+ti →0as i→ ∞.

Proof. Define ǫ(q_i) by 3(q_i+ 1) =e^ε(qi)qi. Then (4) reduces to ε(q_i)q_i+ (α+δ_ℓi)ℓ_i+ [α+ε(k_i, t_i)](k_i+t_i)≥(α+τ_qi)q_i. Sinceℓ_i+k_i+t_i=q_i,

ε(q_i) q_i ki+ti

+δ_ℓi ℓ_i ki+ti

+ε(k_i, t_i)≥τ_qi q_i ki+ti

.

Now asi→ ∞,qi ≥si → ∞. Thusε(qi) → 0. Moreover, since qi belongs to a growth sequence,ε_qi →0 andτ_qi →0. Use Lemma 1 to conclude that

ε(q_i) q_i

k_i+t_i →0 and τ_qi q_i

k_i+t_i →0, and hence

lim infε(k_i, t_i)≥ −lim supδ_ℓi ℓ_i k_i+t_i . Next, since lim supδ_ℓi = 0 and _k^ℓi

i+ti ≤m−1, it follows that

−lim supδ_ℓi ℓ_i

k_i+t_i = 0.

Finally, [Q_ki, L_ti] embeds inL_ki+ti/Q_ki+ti, and it follows that ε_ki+ti ≥ε(k_i, t_i).

Butk_i+t_i→ ∞and so lim sup ε_ki+ti ≤0. This, together with lim inf ε(k_i, t_i)≥

0, completes the proof of the lemma.

Next, since Q_ki⊗L_ti →[Q_ki, L_ti] is surjective we have

σ_kik_i+ε_tit_i≥ε(k_i, t_i)(k_i+t_i) (5)

Lemma 3. t_i/k_i → ∞and t_i → ∞asi→ ∞.

Proof. Ifti/ki does not converge to ∞ then we would have tiν/kiν ≤T for some subsequence (s_iν). But

σ_kiν ≥ −εt_iν

t_iν

k_iν +ε(kiν, tiν)(1 +tiν/kiν).

Since (Lemma 1)k_iν+t_iν → ∞and (Lemma 2)ε(k_iν, t_iν)→0, the lim inf of the right hand side of this equation would be≥0. Hence lim supσ_kiν ≥0, which would contradict lim supσi<0. Finally, since ti+ki → ∞it follows

thatt_i → ∞.

Lemma 4. Write k_iλ_it_i. Then,

λ_i →0 and ε_ti →0 as i→ ∞.

Proof. Lemma 3 asserts thatλ_i →0. Rewrite equation (5) as

ε_ti ≥(ε(k_i, t_i)−σ_ki)λ_i+ε(k_i, t_i). (6) Since lim supσ_j is finite, the σ_j are bounded above. Thus −σ_j ≥ A, some constantA. Sinceλ_i →0, lim inf(−σ_kiλ_i)≥0. Since (Lemma 2) ε(k_i, t_i)→ 0 it follows from (6) that lim infε_ti = 0. But lim supε_ti ≤lim supε_i = 0 and

soε_ti →0. .

The lemmas above establish the Proposition. Simply set p_i = t_i +k_i and note thatti → ∞(Lemma 3), pi → ∞(Lemma 1), pi/ti = 1 +λi → 1 (Lemma 4). Furthermoreε_ti →0 (Lemma 4) andε_pi →0 (Lemma 2). Thus

(t_i) and (p_i) are growth sequences.

Theorem 3. Let L be a cft Lie algebra of finite depth and satisfying the growth conditions (A.1) and (A.2). Setα=log indexL. Then for somed,

d−1

X

i=1

dimL_k+i=e^(α+εk)k, whereε_k →0ask→ ∞. In particular, this sum grows exponentially ink.

Proof. According to [4] there is a finitely generated sub Lie algebra E ⊂L such that Ext^r_{U L}(lk, U L)→Ext^r_{U E}(lk, U L) is non-zero.

Lemma 5. The centralizer, Z, ofE inLis finite dimensional.

Proof. Since E has finite depth, Z ∩E is finite dimensional [3]. Choose k so Z ∩E is concentrated in degrees < k. Suppose x ∈ Z_≥k has even degree, and put F = lkx⊕E. Then Ext_{U F}(lk, U F) → Ext_{U E}(lk, U F) is zero, contradicting the hypothesis that the composite

Ext^r_{U L}(lk, U L)→Ext^r_{U F}(lk, U L)→Ext^r_{U E}(lk, U L)

is non-zero.

It follows thatZ_≥kis concentrated in odd degrees, hence an abelian ideal inZ+E. Again

Ext^r_{U L}(lk, U L)→Ext^r_U(Z+E)(lk, U L)→Ext^r_{U E}(lk, U L)

is non-zero. Thus Ext^r_U(Z+E)(lk, U L)6= 0,Z+E has finite depth and every

abelian ideal inZ+E is finite dimensional.

Choosedso thatEis generated in degrees≤d−1. As in the Proposition, set log indexL = log indexL/Q = α. If the theorem fails we can find a sequencesi → ∞ such that

d−1

X

j=1

dimL_si+j ≤e^(α−β)si, (7) someβ >0. Apply the Proposition to find growth sequencest_i and p_i such thatt_i ≤s_i< p_i and p_i/t_i →1.

We now use (7) to prove that

dim (U E)_{(si−ti,si−ti+d)}dimL_(si,si+d)<dimL_ti, ilarge. (8) In fact since lim sup (dim (U L)_i)^1/i=e^α it follows that for someγ >0,

d−1

X

j=1

dim (U E)_j+k≤e^γ(k+1), allk . Thus it is sufficient to show that

γ(si−ti+ 1) + (α−β)si <(α+εti)ti, large i ,

where ε_ti → 0 as i→ ∞. Write s_i =µ_it_i; then µ_i → 1 and the inequality reduces to the obvious

γ/t_i+γ(µ_i−1) + (α−β)µ_i <(α+ε_ti), largei . Thus (8) is established.

Chooses=s_i,t=t_i so that (8) holds and so that Z_j = 0, j≥t. Write s−t=k. The adjoint action of U LinL restricts to a linear map

h⊕^d_j=1⁻¹(U E)_k+jⁱ⊗L_t→ ⊕^d_j=1⁻¹L_s+j

and it follows from (8) that for some non-zerox∈L_t, (ada)x= 0, a∈U E_(k,k+d).

On the other hand, since E is generated in degrees ≤d−1, (U E)_>k = U E·(U E)_(k,k+d). Thus (U E)_>k·x= 0 and so (U E)·x is finite dimensional.

A non-zero elementy of maximal degree in U E·xsatisfies [a, y] = (ada)(y) = 0, a∈E ,

i.e. y ∈ Z in contradiction to Z≥t = 0. This completes the proof of the

Theorem.

3 Proof of Theorems 1 and 2

Proof of Theorem 2: We show thatL_X satisfies the hypothesis of Theorem 3. Since depthL_X <∞ ([3]) and (A.2) holds by hypothesis we have only to construct a quasi-geometric growth sequence (ri).

Letα= log indexL_X. Thenα >0 by [5]. Choose a sequence u₁ < u₂<· · ·

such that (dim (L_X)_ui)^1/ui →e^α.

Next, suppose catX = m and put a 1

2(m+ 1) m+1

. By starting the sequence at someu_j we may assume dim (L_X)_ui > ¹_a, alli. Thus the formula in ([5], top of page 189) gives a sequence

u_i =v₀< v₁· · ·< v_ku_i+1 such thatv_i+1 ≤2(m+ 1)v_i and

dim (L_X)_vj

1 vj+1

≥[adim (L_X)_v0]^v0+11 , j < k . Since v₀ = u_i and u_i → ∞ it follows that a_v¹

0+1 → 1 as i → ∞. Hence interpolating the sequencesu_i with the sequencesv_j gives a quasi-geometric

growth sequence (rj).

Proof of Theorem 1: A theorem of Adams-Hilton [1] shows that U L_X =H∗(ΩX;Q) =H(T V, d)

whereT V is the tensor algebra onV andV_i∼=H_i+1(X;Q). ThusV is finite dimensional. Since T V has a strictly positive radius of convergence so do H(T V, d) and LX :

log indexL_X <∞.

Thus X satisfies the hypotheses of Theorem 2. The fact that dcan be

replaced bynis proved by Lambrechts in [10].

References

[1] J.F. Adams and P.J. Hilton, On the chain algebra of a loop space, Comment. math. Helvetici 30(1956), 305-330.

[2] I.K. Babenko, On analytic properties of Poincar´e series of loop spaces, Math. Notes 27 (1980), 759-767, Translated from Mat. Zametki 27 (1980), 751-765.

[3] Y. Felix, S. Halperin, C. Jacobsson, C. L¨ofwall and J.-C. Thomas, The radical of the homotopy Lie algebra,American Journal of Mathematics 110 (1988), 301-322

[4] Y. Felix, S. Halperin and J.-C. Thomas, Elliptic Hopf algebras,J. Lon- don Math. Soc. 43(1991), 545-555.

[5] Y. Felix, S. Halperin and J.-C. Thomas, The homotopy Lie algebra for finite complexes, Publications Math´ematiques de l’I.H.E.S. 56 (1983), 179-202.

[6] Y. Felix, S. Halperin and J.-C. Thomas, Rational Homotopy Theory, Graduate Texts in Mathematics 205, Springer-Verlag, 2000.

[7] Y. Felix, S. Halperin and J.-C. Thomas, Growth and Lie brackets in the homotopy Lie algebra, Homology, Homotopy and Applications,4 (2002), 219-225.

[8] Y. Felix, S. Halperin and J.-C. Thomas, Graded Lie algebras with finite polydepth,Ann. Scient. Ec. Norm. Sup.36(2003), 793-804.

[9] P. Lambrechts, Croissance des nombres de Betti des espaces de lacets, Thesis, Louvain-La-Neuve, 1995.

[10] P. Lambrechts, Analytic properties of Poincar´e series of spaces,Topol- ogy 37(1998), 1363-1370.

Universit´e Catholique de Louvain, 1348, Louvain-La-Neuve, Belgium University of Maryland, College Park, MD 20742-3281,USA

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