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On the Betti Numbers of the Free Loop Space of a Coformal Space

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On the Betti Numbers of the Free Loop Space of a Coformal Space View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 161 (2001) 177–192 www.elsevier.com/locate/jpaa On the Betti numbers of the free loop space of a coformal space Pascal Lambrechts LaboGA de l’Universiteà d’Artois, Faculteà des Sciences Jean Perrin, rue Jean Souvraz SP18, F-62307 Lens Cedex, France Received 11 October 1999; received in revised form 17 February 2000 Communicated by C. Kassel Abstract Let X be a simply connected ÿnite CW-complex such that dim ∗(X ) ⊗ Q = ∞. We prove that if X is coformal (this is an hypothesis coming from the rational homotopy theory) then S1 the sequence of rational Betti numbers of its free loop space, (dim Hn(X ; Q))n≥1, has an exponential growth. Since the Betti numbers of the free loop space on a simply connected closed Riemannian manifold bound below the number of closed geodesics, we deduce from the inequality above that on hyperbolic coformal manifolds, the number of closed geodesics of length ≤ t grows exponentially with t. Our methods permit also to prove a dichotomy theorem for the growth of Hochschild homology of graded Lie algebras of ÿnite-dimensional cohomology. c 2001 Elsevier Science B.V. All rights reserved. MSC: 55P62; 55P35; 53C22; 17B56; 16E40; 16P90 1. Introduction For a simply connected topological space X , one denotes by LX or X S1 its free loop space, i.e. the space of continuous maps from the circle S1 to X endowed with the compact-open topology. In the present paper we prove that if dim ∗(X ) ⊗ Q = ∞ and if X is coformal (this is a condition coming from the rational homotopy theory, n see Section 2 for the deÿnition) then the numbers maxn≤k dim H (LX ; Q) grow ex- ponentially with k. Before to describe more precisely our result we ÿrst give some motivations in terms of closed geodesics. E-mail address: [email protected], [email protected] (P. Lambrechts). 0022-4049/01/$ - see front matter c 2001 Elsevier Science B.V. All rights reserved. PII: S0022-4049(00)00098-0 178 P. Lambrechts / Journal of Pure and Applied Algebra 161 (2001) 177–192 When X is a closed Riemannian manifold, the Betti numbers of LX give a lower bound for the number of closed geodesics on X . More precisely, if g is a Riemannian metric on X , denote by Ng(t) the number of geometrically distinct closed geodesics on X of length lesser than t. A Riemannian metric g on X is called bumpy [1] if the closed geodesics on X are non degenerate; this technical property on the metric is generic. We have the following minoration of Ng(t), ÿrst established by Gromov and then improved by Ballmann and Ziller: Theorem (Gromov [8], Ballman and Ziller [4]). Suppose that g is a bumpy Rieman- nian metric on a simply connected closed manifold X . Then there exist ¿0 and ÿ¿0 such that Ng(t) ≥ max rank(Hn(LX ; R)) n≤ÿt for any principal ideal domain R and for t su6ciently large. Gromov conjectures in [8] that for “almost all” closed manifolds X the numbers maxn≤ÿt rank Hn(LX ; R) grow exponentially with t. ViguÃe-Poirrier has investigated this conjecture in [19] with the tools of rational homotopy theory. To explain her results we recall the notions of exponential and polynomial growth. A sequence (bn)n≥0 of nonnegative numbers is said to have an exponential growth if there exist ¿1 and k C¿0 such that maxn≤k bn ≥ C: for k large enough (and the supremum of such ’s is called the rate of the exponential growth); the sequence (bn)n≥0 has a polynomial N growth if there exist N ≥ 0 and C¿0 such that bn ≤ C:n for each n ≥ 1. In [19], ViguÃe-Poirrier proved that if dim ∗(X ) ⊗ Q ¡ ∞ then the sequence (dim Hn(LX ; Q))n≥0 has a polynomial growth and she conjectures that if dim ∗(X )⊗ Q = ∞ and if X is a ÿnite simply connected CW-complex then the sequence (dim Hn(LX ; Q))n≥0 has an exponential growth. She proved this conjecture for wedges of spheres and for manifolds of Lusternik–Schnirelmann category less than 2. In the present paper we consider this conjecture on the class of coformal spaces [15]. Roughly speaking, a space is coformal if its rational homotopy type is determined in a formal way by its rational homotopy Lie algebra (see Section 2 for a precise deÿnition). We will prove: Corollary 9. Let X be a simply connected ÿnite CW-complex. Suppose that X is coformal. Then the rational Betti numbers of LX have an exponential growth if X is hyperbolic. Notice that we also prove in [13] the conjecture of ViguÃe for connected sums of manifolds and some formal spaces, with a generalisation to ÿelds of characteristic p =0. Our methods permit also to prove a dichotomy theorem for the Hochschild homol- ogy of graded Lie algebras. Let L be a graded Lie algebra concentrated in positive P. Lambrechts / Journal of Pure and Applied Algebra 161 (2001) 177–192 179 degree over a ÿeld k of characteristic 0. Suppose that H ∗(L; k) is ÿnite-dimensional. A consequence of the dichotomy theorem of [5] is that either dim L¡∞ or the sequence (dim Ln)n≥1 has an exponential growth. The enveloping algebra UL is a graded algebra and we can consider its Hochschild homology HH∗(UL). We have: Theorem 11. With the hypothesis above; we have the dichotomy (i) either dim L¡∞ and (dim HHn(UL))n≥1 has a polynomial growth; (ii) or dim L = ∞ and (dim HHn(UL))n≥1 has an exponential growth. This paper is organised as follows. In the next section we review notations and results from rational homotopy theory and also on PoincarÃe series of loop spaces. In Section 3, we prove our main theorem on the exponential growth of the Betti numbers of free loop space for a very large class of coformal spaces. The last section is devoted to the applications stated in this introduction. 2. Rational homotopy theory and PoincarÃe series In this section we review some results and notations in rational homotopy theory. We review also notions on PoincarÃe series of a loop space X , and in particular the “analytic hypothesis” (AH). Henceforth by a space we mean a topological space X having the homotopy type of a CW-complex and with ÿnite rational Betti numbers: dim H k (X ; Q) ¡ ∞ for k ≥ 0. We review the theory of Sullivan models ([17,9,18] are standard references, a good summary is given in [5], and a comprehensive study will appear in [6]). The de Rham–Sullivan functor of PL-forms associates to each space X a commutative graded diLerential algebra over Q (a CGDA in short) denoted by APL(X ). If Z is a graded vector space over Q we denote by ∧Z the free graded commutative algebra generated by Z. A quasi-isomorphism is a CGDA-morphism that induces an isomorphism in homology. A Sullivan model of a space X is a quasi-isomorphism (∧Z; d) → APL(X ): Such a Sullivan model is called minimal if d(Z) ⊂∧≥2Z. Each nilpotent space (like a simply connected space, or the free loop space on a simply connected space) admits a minimal Sullivan model. Let (∧Z; d) be a minimal model of a nilpotent space X ; then ∼ ∗ ∼ H(∧Z; d) = H (X ; Q) and, if X is simply connected, Z = hom(∗(X ); Q). Moreover, n if we denote by ZN the graded vector space such that ZN ∼= Zn+1 for all n, then (∧Z;N 0) is a Sullivan model of X when X is simply connected. A simply connected space X is called coformal [15] if it admits a Sullivan model of the form (∧Z; d) with a quadratic diLerential: d(Z) ⊂∧2Z. This is equivalent to say that the Quillen model of X [16] is weakly equivalent to (∗(X ) ⊗ Q; 0), or that the Adams–Hilton model (TV; @) → C∗(X; Q) is quasi-isomorphic to (H∗(X ; Q); 0). 180 P. Lambrechts / Journal of Pure and Applied Algebra 161 (2001) 177–192 Examples of coformal spaces are: spheres; connected sums of products of two spheres: Sk1 × Sn−k1 # ···#Skr × Sn−kr ; wedges and products of coformal spaces. The LS-category of a space X , denoted by cat X , is the smallest integer m (or ∞) such that there exists a covering of X by m + 1 open subsets, each contractible in X . Each ÿnite CW-complex has ÿnite LS-category. For a simply connected space X of ÿnite LS-category we say that X is elliptic if dim ∗(X ) ⊗ Q ¡ ∞, and otherwise that X is hyperbolic. In the latter case the sequence (dim Hn(X ; Q))n≥0 has an exponential growth: this is the dichotomy theorem proved in [5]. The proofs of our theorems are heavily based on properties of PoincarÃe series of a loop space. The PoincarÃe series of X is ∞ n X (z):= dim Hn(X ; Q)z n=0 and its radius of convergence is denoted by !X . When X is a simply connected space of ÿnite LS-category, then !X ¡ 1ifX is hyperbolic, and otherwise !X = 1 (unless ˜ ∗ H (X ; Q) = 0 in which case !X =+∞) (for a proof see [7]). When X is hyperbolic the rate of the exponential growth of the Betti numbers of X is 1=!X . In [11] we have deÿned the following analytic hypothesis (AH) which will be an important ingredient in our proof (more developments on this notion can be found in [12]): We say that the (AH) condition holds for the space X if (i) 0 ¡!X ¡ 1 and (ii) there exist r¿!X and a polynomial Q ∈ C[z] such that Q(z):X (z) is the Tay- lor series at the origin of a function holomorphic and with no zero on the disk Dr = {z ∈ C: |z| ¡r}.
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