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Free Loop Spaces in Topology and Physics Free loop spaces in topology and physics Kathryn Hess What is the space of free loops? Free loop spaces in topology and physics Enumeration of geodesics Hochschild and cyclic homology Kathryn Hess Homological conformal field theories Institute of Geometry, Algebra and Topology Ecole Polytechnique Fédérale de Lausanne Meeting of the Edinburgh Mathematical Society Glasgow, 14 November 2008 Free loop spaces The goal of this lecture in topology and physics Kathryn Hess What is the space of free loops? Enumeration of geodesics Hochschild and cyclic homology An overview of a few of the many important roles played Homological conformal field by free loop spaces in topology and mathematical theories physics. Free loop spaces Outline in topology and physics Kathryn Hess What is the space 1 What is the space of free loops? of free loops? Enumeration of geodesics Hochschild and 2 Enumeration of geodesics cyclic homology Homological conformal field theories 3 Hochschild and cyclic homology 4 Homological conformal field theories Cobordism and CFT’s String topology Loop groups Free loop spaces The functional definition in topology and physics Kathryn Hess What is the space Let X be a topological space. of free loops? Enumeration of The space of free loops on X is geodesics Hochschild and cyclic homology 1 LX = Map(S ; X): Homological conformal field theories If M is a smooth manifold, then we take into account the smooth structure and set LM = C1(S1; M): Free loop spaces The pull-back definition in topology and physics Let X be a topological space. Let PX = Map [0; 1]; X . Kathryn Hess What is the space Let q : PX ! X × X denote the fibration given by of free loops? Enumeration of q(λ) = λ(0); λ(1): geodesics Hochschild and cyclic homology Homological conformal field Then LX fits into a pull-back square theories LX / PX e q ∆ X / X × X; where e(λ) = λ(1) for all free loops λ : S1 ! X. Note that the fiber of both e and q over a point x0 is ΩX, the space of loops on X that are based in x0. Free loop spaces Structure: the circle action in topology and physics Kathryn Hess What is the space of free loops? Enumeration of The free loop space LX admits an action of the circle geodesics group S1, given by rotating the loops. Hochschild and cyclic homology More precisely, there is an action map Homological conformal field theories κ : S1 × LX ! LX; where κ(z; λ): S1 ! X : z0 7! λ(z · z0): Free loop spaces Structure: the power maps in topology and physics Kathryn Hess What is the space of free loops? Enumeration of For any natural number r, the free loop space LX admits geodesics th an r -power map Hochschild and cyclic homology Homological `r : LX ! LX conformal field theories given by 1 r `r (λ): S ! X : z 7! λ(z ); i.e., the loop `r (λ) goes r times around the same path as λ, moving r times as fast. Free loop spaces A related construction in topology and physics Let U and V be subspaces of X. Kathryn Hess The space of open strings in X starting in U and ending What is the space of free loops? in V is Enumeration of n o geodesics PU;V X = λ :[0; 1] ! X j λ(0) 2 U; λ(1) 2 V ; Hochschild and cyclic homology Homological which fits into a pull-back diagram conformal field theories PU;V X / PX : q¯ q (prU ;prV ) U × V / X × X Both the free loop space and the space of open strings are special cases of the homotopy coincidence space of a pair of maps f : Y ! X and g : Y ! X. Free loop spaces The enumeration problem in topology and physics Kathryn Hess What is the space of free loops? Enumeration of geodesics Hochschild and cyclic homology Question Homological conformal field Let M be a closed, compact Riemannian manifold. theories How many distinct closed geodesics lie on M? Free loop spaces Betti numbers and geodesics in topology and physics Kathryn Hess What is the space of free loops? For any space X and any field |, let Enumeration of geodesics n bn(X; ) = dim H (X; ): Hochschild and | | | cyclic homology Homological conformal field theories Theorem (Gromoll & Meyer, 1969) If there is field | such that bn(LM; |) n≥0 is unbounded, then M admits infinitely many distinct prime geodesics. Proof by infinite-dimensional Morse-theoretic methods. Free loop spaces The rational case in topology and physics Kathryn Hess What is the space of free loops? Theorem (Sullivan & Vigué, 1975) Enumeration of geodesics If Hochschild and M is simply connected, and cyclic homology ∗ Homological the graded commutative algebra H (M; ) is not conformal field Q theories monogenic, then bn(LM; Q) n≥0 is unbounded, and therefore M admits infinitely many distinct prime geodesics. Proof using the Sullivan models of rational homotopy theory. Free loop spaces The case of homogeneous spaces I in topology and physics Kathryn Hess What is the space of free loops? Enumeration of Theorem (McCleary & Ziller, 1987) geodesics Hochschild and If M is a simply connected homogeneous space that is cyclic homology Homological not diffeomorphic to a symmetric space of rank 1, then conformal field theories bn(LM; F2) n≥0 is unbounded and therefore M admits infinitely many distinct prime geodesics. Proof by explicit spectral sequence calculation, given the classification of such M. Free loop spaces The case of homogeneous spaces II in topology and physics Kathryn Hess What is the space of free loops? Remark Enumeration of It is easy to show that if M is diffeomorphic to a geodesics Hochschild and symmetric space of rank 1, then bn(LM; |) n≥0 is cyclic homology bounded for all , but Homological | conformal field Hingston proved that a simply connected manifold theories with the rational homotopy type of a symmetric space of rank 1 generically admits infinitely many closed geodesics, and Franks and Bangert showed that S2 admits infinitely many geodesics, independently of the metric. Free loop spaces A suggestive result for based loop spaces in topology and physics Kathryn Hess What is the space of free loops? Enumeration of geodesics Theorem (McCleary, 1987) Hochschild and cyclic homology If X is a simply connected, finite CW-complex such that Homological H∗(X; ) is not monogenic, then b (ΩX; ) is conformal field Fp n Fp n≥0 theories unbounded. Proof via an algebraic argument based on the Bockstein spectral sequence. Free loop spaces A conjecture and its consequences in topology and physics Kathryn Hess What is the space of free loops? Conjecture Enumeration of geodesics If X is a simply connected, finite CW-complex such that Hochschild and ∗ cyclic homology H (X; Fp) is not monogenic, then bn(LX; Fp) is n≥0 Homological unbounded. conformal field theories Corollary ∗ If there is a prime p such that H (M; Fp) is not monogenic, then M admits infinitely many distinct closed geodesics. Free loop spaces Proof strategy in topology and physics Kathryn Hess (Joint work with J. Scott.) What is the space of free loops? Construct “small” algebraic model Enumeration of geodesics Hochschild and B / A cyclic homology Homological ' ' conformal field theories C∗LX / C∗ΩX of the inclusion of the based loops into the free loops. By careful analysis of McCleary’s argument, show that ∗ representatives in A of the classes in H (ΩX; Fp) giving rise to its unbounded Betti numbers lift to B, giving rise to unbounded Betti numbers for LX. Free loop spaces Hochschild (co)homology of algebras in topology and physics Kathryn Hess Let A be a (perhaps differential graded) associative What is the space of free loops? algebra over a field |. Enumeration of geodesics The Hochschild homology of A is Hochschild and cyclic homology A⊗Aop HH∗A = Tor (A; A) Homological ∗ conformal field theories and the Hochschild cohomology of A is ∗ ∗ ] HH A = ExtA⊗Aop (A; A ); ] where A = hom|(A; |). If A is a (differential graded) Hopf algebra, then HH∗A is naturally a graded algebra. Free loop spaces HH and free loop spaces in topology and physics Kathryn Hess Theorem (Burghelea & Fiedorowicz, Cohen, What is the space Goodwillie) of free loops? Enumeration of If X is a path-connected space, then there are |-linear geodesics Hochschild and isomorphisms cyclic homology Homological HH C (ΩX; ) =∼ H (LX; ) conformal field ∗ ∗ | ∗ | theories and ∗ ∼ ∗ HH C∗(ΩX; |) = H (LX; |): Theorem (Menichi) ∗ ∼ ∗ The isomorphism HH C∗(ΩX; |) = H (LX; |) respects multiplicative structure. Free loop spaces Power maps: the commutative algebra case in topology and physics Kathryn Hess What is the space Theorem (Loday, Vigué) of free loops? Enumeration of If A is a commutative (dg) algebra, then HH∗A admits a geodesics natural “r th-power map” that is topologically meaningful in Hochschild and cyclic homology the following sense. Homological conformal field theories If A is the commutative dg algebra of rational piecewise-linear forms on a simplicial complex X, then there is an isomorphism ∼ ∗ HH−∗A = H (LX; Q) that commutes with r th-power maps. Free loop spaces Power maps: the cocommutative Hopf in topology and physics algebra case Kathryn Hess What is the space of free loops? Theorem (H.-Rognes) Enumeration of geodesics If A is a cocommutative (dg) Hopf algebra, then HH∗A Hochschild and admits a natural “r th-power map” that is topologically cyclic homology Homological meaningful in the following sense. conformal field theories Let K be a simplicial set that is a double suspension. If A is the cocommutative dg Hopf algebra of normalized chains on GK (the Kan loop group on K ), then there is an isomorphism ∼ HH∗A = H∗(LjK j) that commutes with r th-power maps.
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