A contribution to the closed geodesic problem. (English) Zbl 1195.55009

Ndombol, Bitjong; Thomas, Jean-Claude

J. Pure Appl. Algebra 214, No. 6, 937-949 (2010).

It is conjectured that if \(X\) is a simply connected finite complex such that its cohomology algebra over a field \(\mathbb F\) is generated by at least two generators, then \(\{\dim H^i(LX,\mathbb F )\}_{i\geq 0}\) grows unbounded. Here \(LX\) is the space of free loops on \(X\). This conjecture is known to hold for \(\mathbb F=\mathbb Q\) by work of D. Sullivan and M. Vigué-Poirrier [J. Differ. Geom. 11, 633–644 (1976; Zbl 0361.53058)].
The authors in this paper give a sufficient algebraic condition for when this conjecture holds. With \(X\) as in the hypothesis of the conjecture, they first exhibit a commutative differential graded algebra \(\Gamma\) such that \(H^0(\Gamma)=0=H^1(\Gamma)\) and \(H(\Gamma )\) is generated by two generators, with a surjective homomorphism of DGA \(p : T\to\Gamma\), where \(T\) is a free DGA quasi-isomorphic to the singular cochains \(C^*(X,\mathbb F )\). The existence of \(T\), with certain additional properties, can be deduced from work of Halperin-Lemaire, Lambrechts and McCleary. Then they show that if \(p\) admits an \(A_{\infty}\)-map \({\mathbf\sigma}:=\{\sigma_i\}_{i\geq 1}: \Gamma\to T\) with \(p\circ\sigma_1=id_\Gamma\), then the conjecture holds for \(X\). An \(A_{\infty}\)-map between two DGAs is what is needed to construct a homomorphism of graded coalgebras between their bar constructions.
The authors go on to identify a finite set of obstructions to the existence of such an \(A_{\infty}\)-section \(\mathbf\sigma\), and then use this to deduce that if \(M\) is a simply connected closed smooth manifold of dimension \(n\) such that for some field coefficients, the graded algebra \(H^*(M)\) is not monogenic and vanishes in degrees \(2\leq i\leq r-1\) with \(n<3(2r-1)\), then \(M\) has infinitely many geometrically distinct closed geodesics for any choice of Riemannian metric.

Reviewer: Sadok Kallel (Villeneuve d’Asq)

MSC:

55P35	Loop spaces
53C99	Global differential geometry
55U35	Abstract and axiomatic homotopy theory in algebraic topology
57T30	Bar and cobar constructions
53C22	Geodesics in global differential geometry

Keywords:

free loop space; bar construction; Hochschild homology; commutative model

Citations:

Zbl 0361.53058

References:

[1]	Mac Cleary, J., On the mod \(p\) Betti numbers of loop spaces, Invent. math., 87, 643-654 (1987) · Zbl 0611.57024
[2]	Gromoll, D.; Meyer, W., Periodic geodesic on compact Riemannian manifold, J. Differential Geom., 3, 493-510 (1969) · Zbl 0203.54401
[3]	Halperin, S.; Vigué-Poirrier, M., The homology of a free loop space, Pacific J. Math., 147, 311-324 (1991) · Zbl 0666.55011
[4]	Vigué-Poirrier, M.; Sullivan, D., The homology theory of the closed geodesic problem, J. Differential Geom., 11, 633-644 (1976) · Zbl 0361.53058
[5]	Smith, L., The EMSS and the mod 2 cohomology of certain free loop spaces, Illinois J. Math., 28, 516-522 (1984) · Zbl 0538.57025
[6]	Mac Cleary, J.; Ziller, W., On the free loop space of homogeneous spaces, Amer. J. Math., 109, 765-781 (1987) · Zbl 0635.57026
[7]	Ziller, W., The free loop space on globally symmetric spaces, Invent. Math., 41, 1-22 (1977) · Zbl 0338.58007
[9]	Jones, J. D.S., Cyclic homology and equivariant homology, Invent. math., 87, 403-423 (1987) · Zbl 0644.55005
[10]	Munkholm, H. J., The Eilenberg-Moore spectral sequence and strongly homotopy multiplicative maps, J. Pure Appl. Algebra, 5, 1-50 (1974) · Zbl 0294.55011
[11]	Ndombol, B.; Thomas, J.-C., On the cohomology algebra of free loop spaces, Topology, 41, 85-106 (2002) · Zbl 1011.16008
[13]	Floyd, E. E., The number of cells in a non bounding manifold, Ann. Math., 98, 210-225 (1973) · Zbl 0272.57019
[14]	Kadeshvili, T. V., On the homology theory of fiber spaces, Russian Math. Surveys, 35, 3, 231-238 (1980) · Zbl 0521.55015
[16]	Félix, Y.; Halperin, S.; Thomas, J.-C., Adams’ cobar equivalence, Trans. Amer. Math. Soc., 329, 531-549 (1992) · Zbl 0765.55005
[17]	Lambrechts, P., The Betti numbers of the free loop space of a connected sum, J. London Math. Soc., 64, 205-228 (2001) · Zbl 1018.55006
[18]	Halperin, Y. S.; Lemaire, J.-M., Notion of category in differential graded algebra, (Lectures Notes in Math., vol. 1318 (1988)), 138-153
[19]	El Haouari, \(p\)-formalité des espaces, J. Pure Appl. Algebra, 78, 245-257 (1992) · Zbl 0744.55007
[20]	Félix, Y.; Halperin, S.; Thomas, J.-C., (Rational Homotopy Theory. Rational Homotopy Theory, Graduate Texts in Math., vol. 205 (2000), Springer-Verlag)
[21]	Menichi, L., The cohomology ring of free loop spaces, Homology, Homotopy Appl., 3, 193-224 (2001) · Zbl 0974.55005
[22]	Anick, D., Homotopy exponents for spaces of category two, (Lectures Notes in Math., vol. 1370 (1989), Springer Verlag), 24-52 · Zbl 0671.55009
[23]	Quillen, D., Rational homotopy theory, Ann. Math., 81, 211-264 (1965)
[24]	Halperin, S., Universal enveloping algebras and loop space homology, J. Pure Appl. Algebra, 83, 237-282 (1992) · Zbl 0769.57025
[25]	Sullivan, D., Infinitesimal computations in topology, Publ. Math. Inst. Hautes Études Scientifiques, 47, 269-331 (1977) · Zbl 0374.57002
[26]	Anick, D., Hopf algebra up to homotopy, J. Amer. Math. Soc., 2, 417-453 (1989) · Zbl 0681.55006
[27]	Adams, F., On the non-existence of elements of Hopf invariant one, Ann. of Math., 72, 20-104 (1960) · Zbl 0096.17404
[28]	Goodwillie, T., Cyclic homology, derivations and the free loop space, Topology, 24, 187-215 (1985) · Zbl 0569.16021