Let R s (resp. RA) be the radius of convergence of the Poincar6 series of a loop space (2(S) (resp. of the Betti-Poincar6 series of a noetherian connected graded ...
I lve tioHes mathematicae
Invent. math. 68, 257-274 (1982)
(C) Springer-Verlag 1982
The Radius of Convergence of Poincar Series of Loop Spaces Y. Felix and J.C. T h o m a s F.N.R.S.-Inst. Math. U.C.L.-B 1348 L o u v a i n la N e u v e - Belgique U.E.R. Math. E . R . A . C . N . R . S . 07590 - F 5 9 6 5 5 - Villeneuve d'Ascq - F r a n c e
Abstract. Let R s (resp. RA) be the radius of convergence of the Poincar6 series of a loop space (2(S) (resp. of the Betti-Poincar6 series of a noetherian connected graded c o m m u t a t i v e algebra A over a field IK of characteristic zero). If S is a finite 1-connected CW-complex, the rational h o m o t o p y Lie algebra of S is finite dimensional if and only if R s = 1. Otherwise R s < 1. There is an easily c o m p u t a b l e upper b o u n d (usually less than 1) for R s if S is formal or coformal. O n the other h a n d R A= + oo if and only if A is a polynomial algebra and R A = 1 if and only if A is a complete intersection (Golod and Gulliksen conjecture). Otherwise R A < 1 and the sequence dim Torf(IK, IK) grows exponentially with p.
I. Introduction I.l. A m o n g the 1-connected topological spaces S, we distinguish two classes according to the dimension H , ( S ) | M o r e precisely, S is elliptic if dim H , (S) | lI~ < + oo and hyperbolic if dim H , (S)| = + oo. 1.2. Let R s be the radius of convergence for Poincar6 series of the loop space O(S). If S elliptic, R s = 1 or + Go. As a consequence of [4] and [10] we have the Theorem (II.6). I f H*(S; Q) is a noetherian algebra, S is hyperbolic i f and only if R s < l .
It is m o r e o v e r (I1.8) easy to find a lower b o u n d for R s. 1.3. The next p r o b l e m is to find, for a hyperbolic space S, an easily c o m p u t a b l e real n u m b e r r, such that R s < r < 1. The principal result of this paper is to find AMS CLASSIFICATION
(MOS 80) 5 5 P 6 2 ; 5 5 M 3 0 ; 5 5 P 3 5
0020- 9910/82/0068/0257/$03.60
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Y. Felix and J.C. Thomas
such an r when S is formal. (Recall that a space S is formal if its rationalization S o has an automorphism 0 satisfying 0 * ( x ) = t Ixl .x for each x belonging do H*(S, Q) [24]. K~ihler compact manifolds [7], Riemannian symmetric spaces, p-connected manifolds with dimension less than 4 p + 2 [18], wedges and products of formal spaces, are examples of formal spaces). Theorem (III.4). Let S be a space which is both hyperbolic and formal, and such f
that, dim H*(S; I1~)< + oc. Then R s < r where r = i n f l l z i [ , z i running through the t
zeros of the PoincarO polynomial of S, ~ dimHi(S; Q)t i . i=0
There follows Corollary (III.6). I f S is a ( p - D-connected compact formal oriented n-manifold, 1 -n " \ l / p
then R s < ( ~ ; )
where bp is the
pth Betti
number of S.
1.4. The result of Theorem III.4 holds also under a weaker assumption namely for spaces with a homotopical weight decomposition (III.1). In a dual setting we study spaces equipped with a homological weight decomposition and obtain an upper bound for R s. Such spaces include those which are cofiber of a map between suspensions (formula VI.8). 1.5. In our second main result, we apply the same techniques in algebra and give a proof of the Golod and Gulliksen conjecture [19] for graded connected algebras and of conjecture C 2 Avramov [3]. More precisely, as a corollary of Theorem IV.5 we obtain: Theorem (IV.7). Let H be a noetherian and connected graded commutative algebra over llC I f R n denotes the radius of convergence of the Betti-Poincar( series Pu(t) = ~ dimTor~(IK, IK)tq p>0
there are only three possibilities: i) RH= + o0, in this case H is a polynomial algebra; ii) R H = 1, in this case H is a complete intersection and
dim TorH (IK, IK) ____K p m for some fixed K and m; iii) R n < 1; in this case H is not a complete intersection, and there exists a constant C > 1 such that dim Tor~(lK, IK) _>_C p for all p.
1.6. The paper is organised as follows: I) II) III) IV) V)
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radius of convergence of the Poincar6 series . . . . . . . . . . . . . . . . Formal spaces and spaces equipped with a homotopical weight decomposition Poincar6 series of a connected graded algebra . . . . . . . . . . . . . . . . Coformal spaces and spaces with a homological weight decomposition .
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The Radius of Convergence of Poincar6 Series of Loop Spaces
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1.7. Rational homotopy and minimal models are the principal tools in this paper. We use the terminology of [6, 13, 14] and [24]. It is the articles of Roos [23] and Lemaire [15, 16] which has drawn attention to the relations between topology and local algebra. The authors are deeply indebted to A. Magnus for his advice, and should also like to thank S. Halperin for many valuable conversations and suggestions. 1.8. In this text, S will be a 1-connected space, (2(S) is space of the loops at a fixed point. Homology and cohomology are with rational coefficients. 1.9. We denote by (AZ, d) the bigraded model (in the sense of HalperinStasheff) of the space S ([14]). It is a Tate-Josefiak resolution of the graded algebra H* (S). A space is said to be formal, if its minimal model and its bigraded model are isomorphic ([14]). This definition is equivalent to the definition given in 1.3.
II. Radius of Convergence of the Poincar6 Series II.1. Recall that if H*(S) is of finite type (i.e. d i m H i ( S ) < +o% V i > 2 ) then for each i > 2 , d i m l I i ( S ) | and so for each j > l , dimHJ(~2(S))< +oo. In this case, we put: bi=dimHi(S), fli=dimHi((2(S)), ai=dim(Hi+l(S)| +o9
P(S; t)= ~' b~t~
is the Poincar6 series of S.
i=0
P(~2(S); t)= ~ flit i
is the Poincar6 series of Q(S).
i=0
Q(S; t)= ~ air ~
is the Poincar6-Hurewicz series of O(S).
i=1
II.2. Let R s denote the radius of convergence of P(O(S); t). II.3. Since H,(O(S))~-U(II,(~?(S))| rem ([19]), one obtains
from the Poincar6-Birkoff-Witt theo+oO
l) ..... P(Q(S); 0 =r~ (l+t2k l)l= (1 - t2k) "2~ In particular, if S is elliptic, then P(~?(S); t) is a rational function, all of whose zeros and poles are roots of unity. II.4. By a skillful computation with power series, I.K. Babenko proved the following result.
Theorem [-4]. If S is a hyperbolic space, then the power series P(~?(S); t) and
Q(S; t) have the same radius of convergence. II.5. We have seen in II.3, that Rs> 1 if S is an elliptic space. In the following theorem we prove that the elliptic spaces are the only spaces with noetherian cohomology, having this property.
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Y. Felix and J.C. T h o m a s
II.6 Theorem. If H*(S) is noetherian, S is hyperbolic if and only if R s < 1. II.7. Proof. (a) If S is elliptic, then R s > 1 (II.3). (b) Suppose S hyperbolic, since It(S) is noetherian there exists a vector space Z such that
AZ------ H(S) and d i m Z < + oo. Let (Vi)i~ I denote a basis of Z .... and (AX, d) is a minimal model of S. Let ~ i e A X l c ~ k e r d be such that [(/)/]=v i. And define (ui)i~I by Dui=eb 2 with U be the vector space generated by (ui)~~ ( A X | D) is the minimal KS complex with DIx=d. The spectral sequence associated with the filtration
Fp=AX| satisfies EzP'q = Hq-P(H(A X, d) | A U, D*). Since H(AX, d ) | is a finitely generated H(AX, d)-module, we deduce that H(H(AX, d)| D*) is a finitely generated H(AX, d)-module. Thus, since the image of H(AX, d) in H(H(AX, d)| D*) is of finite dimension over (~, so is E 2 and H ( A X | U, D). Now Theorem 8.1 of [10] implies; There is an infinite sequence of integers ql < qq < . . - , with qi 1 such that dim (X | U)q' > C q',
i = 1, 2, ....
Since dim U < + ~ , the same relation holds for dim X q', and from II.4.,
Rs=limsup(ai)
_L
1
i__P. Mo-
The Radius of Convergence of Poincar6 Series of Loop Spaces
reover, there exists an infinite sequence Px < P 2 m) and a constant C > 1 such that
"'"
267
with pi+l dimZp 1.
First Case. d i m Z < +oo. Then from [9], the bigraded model (AZ, d) is twostage (Zp=0, p>2). In this case, if z~ necessarily Z 1 = 0 and AZ=AZ~o yen = H , so that R n = + o o . If z~ H = A Z o / ( d Z 1 ) . A Z o is a complete intersection ([9]) and R n = 1. Second Case. d i m Z = +oo. with d induced from d.
Let
(AZ>__l,d) be the quotient algebra A Z / A + Z o
Lemma 1. Cato(AZ=> l, d) < m (some m < oo). Proof Let (zi)i~I be a basis of Z~ve" and set dui= z 2
U = @ IKu i. iei
Using the spectral sequence associated with the filtration Fv=AZ| we obtain, as in II.7 (b), that
dimH(AZ|
d)< + oo.
This implies [10] that Cat0(AZ | A U, d) < m. There is an obvious projection of ( A Z | d) onto (AZ>=l,cl) and so from ([10], 5.1) we obtain Cato(AZ>- 0 < m .
Lemma 2. Let (AZ, d) be any bigraded model with H(AZ, d) connected. I f Zp+ l =0, then every linear map Zp--*IK extends to a derivation 0 of (AZ, d) of lower degree - p which factors to give a derivation of (AZ>>=1,d). In particular if 0#-0 it is an element of the Gottlieb group G(AZ, d) of (AZ, d). Thus, by [10]; 6.12 since cato(AZ>_l,d)< + oo, Zv:4=O, for any p > P . The next lemma (which completes the proof of the theorem) follows from L e m m a 2 in exactly the same way that Theorem 8.1 (ii) follows from Theorem 8.1 (i) in [10]. L e m m a 3. There exists a constant C > 1 and an infinite sequence (Pl) such that Pi=e'Pl>=C P ' > K P - 1 1
i=0
with K = C g > 1. IV.9. Remarks. a) T h e o r e m IV.7. proves in the graded case the conjecture of G o l o d and Gulliksen [1] and the conjecture of Avramov C 2 in [2, 3].
The Radius of Convergence of Poincar6 Series of Loop Spaces
269
b) Theorem IV.5. proves a weaker form of the conjecture C 3 of Avramov [3] which states; " H is a complete intersection if and only if one of the e~ is z&o for i > 2". IV.10. An upper bound for R H can be found in the same way as in III.5.
Theorem. I f H = @ H p is a commutative graded algebra of finite dimension with p>=O
R H1 +c(~
§
s ( E d i m Z f l t P = ( 1 - t ) -1 E dimZp tp' p--O r < p
p--1
F r o m III.3, we deduce lerl==- ~ dimZr r 2, dim Vo.i - dim V~., 4 i = dim I:ti(Sz) - dim I2Ii(SO we obtain for this weight decomposition 1 - ~ ( - 1)a' t D'= 1 + P ( S , ; t ) - P(S2, t) iel
and corollary results now from V.4. V.12. Example 1. We consider a continuous map:
f : S ~ v S 7 -*S.3 vSb3 v S c3 admitting the following homotopy cofiber:
Cf:IS' vS vS' )Uex , 8
q,
~8
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Y. Felix and J.C. Thomas
with ~ and 7j representing the h o m o t o p y classes EES~, Sb], 3 Sa] 3
and
3 3 EESa,Sb],Sc],
respectively.
Then C I is not a coformal space and C a t o ( C i ) = 2 . - P(S 2 ; t) = 1 - 3 t 2 + 2 t 6, from V.9, we obtain
Since,
I+P(S1;t )
Rc =l,
p>=O.
The Quillen model of S/s(r), (IL(V/V
