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We construct a complex analytic version of an equivariant cohomology theory which appeared in a paper of Rezk, and which is roughly modelled on the Borel-equivariant cohomology of the double free loop space. The construction is defined on finite, torus-equivariant CW complexes and takes values in coherent holomorphic sheaves over the moduli stack of complex elliptic curves. Our methods involve an inverse limit construction over all finite-dimensional subcomplexes of the double free loop space, following an analogous construction of Kitchloo for single free loop spaces. We show that, for any given complex elliptic curve $\mathcal {C}$, the fiber of our construction over $\mathcal {C}$ is isomorphic to Grojnowski's equivariant elliptic cohomology theory associated to $\mathcal {C}$.

Loop-mediated isothermal amplification (LAMP) is a sensitive and widely used gene amplification technique. However, high amplification efficiency and amplification products containing multiple inverted repeats make the LAMP reaction extremely vulnerable to false-positive amplification caused by contamination. Herein, a contamination-free LAMP (CUT-LAMP) assisted by the CRISPR/Cas9 cleavage with superior reliability and durability has been reported. The core of CUT-LAMP is the engineering of the forward or backward inner primer in the target-independent region, which makes the LAMP products contain a protospacer adjacent motif (PAM) site for the CRISPR/Cas9 recognition. For the CUT-LAMP reaction, cross-contamination can be efficiently cleaved by the corresponding Cas9/sgRNA, but the target gene can get rid of digestion due to the lack of a PAM site near the recognition region. CUT-LAMP shows impressive contamination resistance but does not significantly increase procedure complexity; thus, it represents a simple and versatile toolkit facilitating the adoption by open- and closed-tube detection format.

We study when the derived intersection of two smooth subvarieties of a smooth variety is formal. As a consequence we obtain a derived base change theorem for non-transversal intersections. We also obtain applications to the study of the derived fixed locus of a finite group action and argue that for a global quotient orbifold the exponential map is an isomorphism between the Lie algebra of the free loop space and the loop space itself. This allows us to give new proofs of the HKR decomposition of orbifold Hochschild (co)homology into twisted sectors., 23 pages

We show that a torsion-free nilpotent loop (that is, a loop nilpotent with respect to the dimension filtration) has a torsion-free nilpotent left multiplication group of, at most, the same class. We also prove that a free loop is residually torsion-free nilpotent and that the same holds for any free commutative loop. Although this last result is much stronger than the usual residual nilpotence of the free loop proved by Higman, it is established, essentially, by the same method.

We construct an equivalence of $E_{2}$ algebras between two models for the Thom spectrum of the free loop space that are related by derived Koszul duality. To do this, we describe the functoriality and invariance properties of topological Hochschild cohomology.

In this paper, we propose SolarSLAM, a battery-free loop closure method for indoor localisation. Inertial Measurement Unit (IMU) based indoor localisation method has been widely used due to its ubiquity in mobile devices, such as mobile phones, smartwatches and wearable bands. However, it suffers from the unavoidable long term drift. To mitigate the localisation error, many loop closure solutions have been proposed using sophisticated sensors, such as cameras, laser, etc. Despite achieving high-precision localisation performance, these sensors consume a huge amount of energy. Different from those solutions, the proposed SolarSLAM takes advantage of an energy harvesting solar cell as a sensor and achieves effective battery-free loop closure method. The proposed method suggests the key-point dynamic time warping for detecting loops and uses robust simultaneous localisation and mapping (SLAM) as the optimiser to remove falsely recognised loop closures. Extensive evaluations in the real environments have been conducted to demonstrate the advantageous photocurrent characteristics for indoor localisation and good localisation accuracy of the proposed method.

We introduce a notion of Koszul A∞-algebra that generalizes Priddy's notion of a Koszul algebra and we use it to construct small A∞-algebra models for Hochschild cochains. As an application, this yields new techniques for computing free loop space homology algebras of manifolds that are either formal or coformal (over a field or over the integers). We illustrate these techniques in two examples.

We introduce the abstract notion of a closed necklical set in order to describe a functorial combinatorial model of the free loop fibration $\Omega Y\rightarrow \Lambda Y\rightarrow Y$ over the geometric realization $Y=|X|$ of a path connected simplicial set $X.$ In particular, to any path connected simplicial set $X$ we associate a closed necklical set $\widehat{\mathbf{\Lambda}}X$ such that its geometric realization $|\widehat{\mathbf{\Lambda}}X|$, a space built out of gluing "freehedrical" and "cubical" cells, is homotopy equivalent to the free loop space $\Lambda Y$ and the differential graded module of chains $C_*(\widehat{\mathbf{\Lambda}}X)$ generalizes the coHochschild chain complex of the chain coalgebra $C_\ast(X).$

We prove that conformal nets of finite index are an instance of the notion of a factorization algebra. This result is an ingredient in our proof that, for $G=SU(n)$, the Drinfel'd center of the category of positive energy representations of the based loop group is equivalent to the category of positive energy representations of the free loop group., Comment: 11 pages

Discovering loop structures in a process model is an important research topic for business process mining. The event logs generated in a real-life business process may be incomplete because of the missing activities. A process discovery algorithm can construct a proper process model from incomplete event logs. In this paper, an $\alpha $ -FL algorithm is proposed to discover a free loop structure in a process model based on Petri nets from incomplete event logs. First, repeated activities are extracted and some relations of these activities are analyzed. Then, some algorithms are designed to obtain the free loop structures. Finally, the correctness and effectiveness of the proposed approach are verified via some case studies and experiments.

The aim of this paper is to define a chain level refinement of the Batalin-Vilkovisky (BV) algebra structure on the homology of the free loop space of a closed, oriented $C^\infty$-manifold. For this purpose, we define a (nonsymmetric) cyclic dg operad which consists of "de Rham chains" of free loops with marked points. A notion of de Rham chains, which is a certain hybrid of the notions of singular chains and differential forms, is a key ingredient in our construction. Combined with a generalization of cyclic Deligne's conjecture, this dg operad produces a chain model of the free loop space which admits an action of a chain model of the framed little disks operad, recovering the string topology BV algebra structure on the homology level.

In this paper, we prove that for every irreversible Finsler n-dimensional real projective space ( R P n , F ) with reversibility λ and flag curvature K satisfying 16 9 ( λ 1 + λ ) 2 K ≤ 1 with λ 3 , there exist at least n − 1 non-contractible closed geodesics. In addition, if the metric F is bumpy with 64 25 ( λ 1 + λ ) 2 K ≤ 1 and λ 5 3 , then there exist at least 2 [ n + 1 2 ] non-contractible closed geodesics, which is the optimal lower bound due to Katok's example. The main ingredients of the proofs are the Fadell–Rabinowitz index theory of non-contractible closed geodesics on ( R P n , F ) and the S 1 -equivariant Poincare series of the non-contractible component of the free loop space on R P n .

Using the $E_\infty-$structure on singular cochains, we construct a homotopy coherent map from the cyclic bar construction of the differential graded algebra of cochains on a space to a model for the cochains on its free loop space. This fills a gap in the paper "Cyclic homology and equivariant homology" by John D.S. Jones.

Let $R$ be an $E_\infty$-ring spectrum. Given a map $\zeta$ from a space $X$ to $BGL_1R$, one can construct a Thom spectrum, $X^\zeta$, which generalises the classical notion of Thom spectrum for spherical fibrations in the case $R=S^0$, the sphere spectrum. If $X$ is a loop space ($\simeq \Omega Y$) and $\zeta$ is homotopy equivalent to $\Omega f$ for a map $f$ from $Y$ to $B^2GL_1R$, then the Thom spectrum has an $A_\infty$-ring structure. The Topological Hochschild Homology of these $A_\infty$-ring spectra is equivalent to the Thom spectrum of a map out of the free loop space of $Y$. This paper considers the case $X=S^1$, $R=K_p^\wedge$, the p-adic $K$-theory spectrum, and $\zeta = 1-p \in \pi_1BGL_1K_p^\wedge$. The associated Thom spectrum $(S^1)^\zeta$ is equivalent to the mod p $K$-theory spectrum $K/p$. The map $\zeta$ is homotopy equivalent to a loop map, so the Thom spectrum has an $A_\infty$-ring structure. I will compute $\pi_*THH^{K_p^\wedge}(K/p)$ using its description as a Thom spectrum.

An equivalence $$\sim $$ upon a loop is said to be multiplicative if it satisfies $$x\sim y$$ , $$u \sim v \Rightarrow xu \sim yv$$ . Let X be a set with elements $$x\ne y$$ and let $$\sim $$ be the least multiplicative equivalence upon a free loop F(X) for which $$x\sim y$$ . If $$a,b\in F(X)$$ are such that $$a\ne b$$ and $$a\sim b$$ , then neither $$a\backslash c \sim b\backslash c$$ nor $$c/a \sim c/b$$ is true, for every $$c\in F(X)$$ .

When projectors are used to formulate the loop-star preconditioning, a sparse but singular graph-Laplacian matrix must be inverted. In this article, the charge neutrality condition is enforced to remove the zero singular value of this matrix and a sparse direct solver is used to invert the result. While the formulation is low-frequency stabilized and requires few extra operations per iteration, the number of iterations remains high. The iteration count is improved by combining it with an elementdiagonal sparse preconditioner.

Given a manifold $M$ and a proper sub-bundle $\Delta\subset TM$, we study homotopy properties of the horizontal base-point free loop space $\Lambda$, i.e. the space of absolutely continuous maps $\gamma:S^1\to M$ whose velocities are constrained to $\Delta$ (for example: legendrian knots in a contact manifold). A key technical ingredient for our study is the proof that the base-point map $F:\Lambda \to M$ (the map associating to every loop its base-point) is a Hurewicz fibration for the $W^{1,2}$ topology on $\Lambda$. Using this result we show that, even if the space $\Lambda$ might have deep singularities (for example: constant loops form a singular manifold homeomorphic to $M$), its homotopy can be controlled nicely. In particular we prove that $\Lambda$ (with the $W^{1,2}$ topology) has the homotopy type of a CW-complex, that its inclusion in the standard base-point free loop space (i.e. the space of loops with no non-holonomic constraint) is a homotopy equivalence, and consequently its homotopy groups can be computed as $\pi_k(\Lambda)\simeq \pi_k(M) \ltimes \pi_{k+1}(M)$ for all $k\geq 0.$ These topological results are applied, in the second part of the paper, to the problem of the existence of closed sub-riemannian geodesics. In the general case we prove that if $(M, \Delta)$ is a compact sub-riemannian manifold, each non trivial homotopy class in $\pi_1(M)$ can be represented by a closed sub-riemannian geodesic. In the contact case, we prove a min-max result generalizing the celebrated Lyusternik-Fet theorem: if $(M, \Delta)$ is a compact, contact manifold, then every sub-riemannian metric on $\Delta$ carries at least one closed sub-riemannian geodesic. This result is based on a combination of the above topological results with a delicate study of the Palais-Smale condition in the vicinity of abnormal loops (singular points of $\Lambda$)., Comment: 25 pages. Final version to appear in the Transactions of the American Math. Society, Series B

The notion of Hochschild homology of a dg algebra admits a natural dualization, the coHochschild homology of a dg coalgebra, introduced in arXiv:0711.1023 by Hess, Parent, and Scott as a tool to study free loop spaces. In this article we prove "agreement" for coHochschild homology, i.e., that the coHochschild homology of a dg coalgebra $C$ is isomorphic to the Hochschild homology of the dg category of appropriately compact $C$-comodules, from which Morita invariance of coHochschild homology follows. Generalizing the dg case, we define the topological coHochschild homology (coTHH) of coalgebra spectra, of which suspension spectra are the canonical examples, and show that coTHH of the suspension spectrum of a space $X$ is equivalent to the suspension spectrum of the free loop space on $X$, as long as $X$ is a nice enough space (for example, simply connected.) Based on this result and on a Quillen equivalence established by the authors in arXiv:1402.4719, we prove that "agreement" holds for coTHH as well., To appear in JPAA

We calculate the homology of the free loop space of (n-1)-connected closed manifolds of dimension at most 3n-2 (n > 1), with the Chas-Sullivan loop product and loop bracket. Over a field of characteristic zero, we obtain an expression for the BV-operator. We also give explicit formulas for the Betti numbers, showing they grow exponentially. Our main tool is the connection between formality, coformality and Koszul algebras that was elucidated in earlier work by the first author., 35 pages

We construct new concrete examples of relative differential characters, which we call Cheeger–Chern–Simons characters. They combine the well-known Cheeger–Simons characters with Chern–Simons forms. In the same way as Cheeger–Simons characters generalize Chern–Simons invariants of oriented closed manifolds, Cheeger–Chern–Simons characters generalize Chern–Simons invariants of oriented manifolds with boundary. We study the differential cohomology of compact Lie groups G and their classifying spaces BG. We show that the even degree differential cohomology of BG canonically splits into Cheeger–Simons characters and topologically trivial characters. We discuss the transgression in principal G-bundles and in the universal bundle. We introduce two methods to lift the universal transgression to a differential cohomology valued map. They generalize the Dijkgraaf–Witten correspondence between 3-dimensional Chern–Simons theories and Wess–Zumino–Witten terms to fully extended higher-order Chern–Simons theories. Using these lifts, we also prove two versions of a differential Hopf theorem. Using Cheeger–Chern–Simons characters and transgression, we introduce the notion of differential trivializations of universal characteristic classes. It generalizes well-established notions of differential String classes to arbitrary degree. Specializing to the class $${\frac{1}{2} p_1 \in H^4(B{\rm Spin}_n;\mathbb{Z})}$$ , we recover isomorphism classes of geometric string structures on Spin n -bundles with connection and the corresponding spin structures on the free loop space. The Cheeger–Chern–Simons character associated with the class $${\frac{1}{2} p_1}$$ together with its transgressions to loop space and higher mapping spaces defines a Chern–Simons theory, extended down to points. Differential String classes provide trivializations of this extended Chern–Simons theory. This setting immediately generalizes to arbitrary degree: for any universal characteristic class of principal G-bundles, we have an associated Cheeger–Chern–Simons character and extended Chern–Simons theory. Differential trivialization classes yield trivializations of this extended Chern–Simons theory.

In this note, it is shown that the Hilbert–Poincare series for the rational homology of the free loop space on a moment-angle complex is a rational function if and only if the moment-angle complex is a product of odd spheres and a disk. A partial result is included for the Davis–Januszkiewicz spaces. The opportunity is taken to correct the result (Bahri et al., Proceedings of the Steklov Institute of Mathematics, Russian Academy of Sciences, vol. 286, pp. 219–223. doi:10.1134/S0081543814060121, 2014) which used a theorem from Berglund and Jollenbeck (J Algebra 315:249–273, 2007).

An iterative method is proposed for the explicit computation of discrete-time nonlinear filter networks containing delay-free loops. The method relies on a fixed-point search of the signal values at every temporal step. The formal as well as numerical properties of fixed-point solvers delimit its applicability: On the one hand, the method allows for a reliable prediction of the frequency rates where the simulation is stable, while, on the other hand, its straightforward applicability is counterbalanced by low speed of convergence. Especially in presence of specific nonlinear characteristics, the use of a fixed-point search is limited if the real-time constraint holds. For this reason, the method becomes useful especially during the digital model prototyping stage, as exemplified while revisiting a previous discrete-time realization of the voltage-controlled filter aboard the EMS VCS3 analog synthesizer. Further tests conducted on a digital ring modulator model support the above considerations.

In this paper, we consider a Finsler sphere $(M,F)=(S^n,F)$ with the dimension $n>1$ and the flag curvature $K\equiv 1$. The action of the connected isometry group $G=I_o(M,F)$ on $M$, together with the action of $T=S^1$ shifting the parameter $t\in \mathbb{R}/\mathbb{Z}$ of the closed curve $c(t)$, define an action of $\hat{G}=G\times T$ on the free loop space $\lambda M$ of $M$. In particular, for each closed geodesic, we have a $\hat{G}$-orbit of closed geodesics. We assume the Finsler sphere $(M,F)$ described above has only finite orbits of prime closed geodesics. Our main theorem claims, if the subgroup $H$ of all isometries preserving each close geodesic has a dimension $m$, then there exists $m$ geometrically distinct orbits $\mathcal{B}_i$ of prime closed geodesics, such that for each $i$, the union ${B}_i$ of geodesics in $\mathcal{B}_i$ is a totally geodesic sub-manifold in $(M,F)$ with a non-trivial $H_o$-action. This theorem generalizes and slightly refines the one in a previous work, which only discussed the case of finite prime closed geodesics. At the end, we show that, assuming certain generic conditions, the Katok metrics, i.e. the Randers metrics on spheres with $K\equiv 1$, provide examples with the sharp estimate for our main theorem.

Let $M$ be any simply-connected Gorenstein space over any field. F\'elix and Thomas have extended to simply-connected Gorenstein spaces, the loop (co)products of Chas and Sullivan on the homology of the free loop space $H_*(LM)$. We describe these loop (co)products in terms of the torsion and extension functors by developing string topology in appropriate derived categories. As a consequence, we show that the Eilenberg-Moore spectral sequence converging to the loop homology of a Gorenstein space admits a multiplication and a comultiplication with shifted degree which are compatible with the loop product and the loop coproduct of its target, respectively. We also define a generalized cup product on the Hochschild cohomology $HH^*(A,A^\vee)$ of a commutative Gorenstein algebra $A$ and show that over $\mathbb{Q}$, $HH^*(A_{PL}(M),A_{PL}(M)^\vee)$ is isomorphic as algebras to $H_*(LM)$. Thus, when $M$ is a Poincar\'e duality space, we recover the isomorphism of algebras $\mathbb{H}_*(LM;\mathbb{Q})^\cong HH^*(A_{PL}(M),A_{PL}(M))$ of F\'elix and Thomas., Comment: 40 pages, this version is one of two preprints divided from the first version, an appendix on shriek maps is revised

In this paper we introduce an equivariant extension of the Chern-Simons form, associated to a path of connections on a bundle over a manifold M, to the free loop space LM, and show it determines an equivalence relation on the set of connections on a bundle. We use this to define a ring, loop differential K-theory of M, in much the same way that differential K-theory can be defined using the Chern-Simons form [SS]. We show loop differential K-theory yields a refinement of differential K-theory, and in particular incorporates holonomy information into its classes. Additionally, loop differential K-theory is shown to be strictly coarser than the Grothendieck group of bundles with connection up to gauge equivalence. Finally, we calculate loop differential K-theory of the circle., Comment: 30 pages; new last section, appendix. To appear in Annales Mathematiques Blaise Pascal

Firstly, we wish to motivate that Conley pairs, realized via Salamon's definition [17], are rather useful building blocks in geometry: Initially we met Conley pairs in an attempt to construct Morse filtrations of free loop spaces [21]. From this fell off quite naturally, firstly, an alternative proof [20] of the cell attachment theorem in Morse theory [13] and, secondly, some ideas [12] how to try to organize the closures of the unstable manifolds of a Morse-Smale gradient flow as a CW decomposition of the underlying manifold. Relaxing non-degeneracy of critical points to isolatedness we use these Conley pairs to implement the gradient flow proof of the Lusternik-Schnirelmann Theorem [10] proposed in Bott's survey [3]. Secondly, we shall use this opportunity to provide an exposition of Lusternik-Schnirelmann (LS) theory based on thickenings of unstable manifolds via Conley pairs. We shall cover the Lusternik-Schnirelmann Theorem [10], cuplength, subordination, the LS refined minimax principle, and a variant of the LS category called ambient category., exposition, 23 pages, 7 figures

We study compact and simply-connected Riemannian manifolds with positive sectional curvature $K\ge 1.$ For a non-trivial homology class of lowest dimension in the space of loops based at a point $p$ or in the free loop space one can define a critical length ${\sf crl}_p\left(M,g\right)$ resp. ${\sf crl}\left(M,g\right).$ Then ${\sf crl}_p\left(M,g\right)$ equals the length of a geodesic loop and ${\sf crl}\left(M,g\right)$ equals the length of a closed geodesic. This is the idea of the proof of the existence of a closed geodesic of positive length presented by Birkhoff in case of a sphere and by Lusternik and Fet in the general case. It is the main result of the paper that the numbers ${\sf crl}_p\left(M,g\right)$ resp. ${\sf crl}\left(M,g\right)$ attain its maximal value $2\pi$ only for the round metric on the $n$-sphere. Under the additional assumption $K \le 4$ this result for ${\sf crl}\left(M,g\right)$ follows from results by Sugimoto in even dimensions and Ballmann, Thorbergsson and Ziller in odd dimensions., Comment: 14 pages, minor changes, section 6 added

We prove that certain involutions defined by Vogell and Burghelea-Fiedorowicz on the rational algebraic $K$-theory of spaces coincide. This gives a way to compute the positive and negative eigenspaces of the involution on rational homotopy groups of pseudoisotopy spaces from the involution on rational $S^{1}$-equivariant homology group of the free loop space of a simply-connected manifold. As an application, we give explicit dimensions of the open manifolds $V$ that appear in Belegradek-Farrell-Kapovitch's work for which the spaces of complete nonnegatively curved metrics on $V$ have nontrivial rational homotopy groups., 23 pages, to appear in Transactions of the AMS

In this paper, we introduce and study representation homology of topological spaces, which is a natural homological extension of representation varieties of fundamental groups. We give an elementary construction of representation homology parallel to the Loday-Pirashvili construction of higher Hochschild homology; in fact, we establish a direct geometric relation between the two theories by proving that the representation homology of the suspension of a (pointed connected) space is isomorphic to its higher Hochschild homology. We also construct some natural maps and spectral sequences relating representation homology to other homology theories associated with spaces (such as Pontryagin algebras, $S^1$-equivariant homology of the free loop space and stable homology of automorphism groups of f.g. free groups). We compute representation homology explicitly (in terms of known invariants) in a number of interesting cases, including spheres, suspensions, complex projective spaces, Riemann surfaces and some 3-dimensional manifolds, such as link complements in $\R^3$ and the lens spaces $ L(p,q) $. In the case of link complements, we identify the representation homology in terms of ordinary Hochschild homology, which gives a new algebraic invariant of links in $\R^3$., A substantially revised version. New results are added, including the existence of the derived representation adjunction and the commutativity of the derived representation functor with arbitrary homotopy colimits. We deduce these results from a version of (derived) adjunction theorem for categories with weak equivalences that extends formally Quillen's classical theorem for model categories

We propose a categorification of the Chern character that refines earlier work of To\"en and Vezzosi and of Ganter and Kapranov. If X is an algebraic stack, our categorified Chern character is a symmetric monoidal functor from a category of mixed noncommutative motives over X, which we introduce, to S1-equivariant perfect complexes on the derived free loop stack LX. As an application of the theory, we show that To\"en and Vezzosi's secondary Chern character factors through secondary K-theory. Our techniques depend on a careful investigation of the functoriality of traces in symmetric monoidal (infinity,n)-categories, which is of independent interest., Comment: Final version, to appear in Adv. Math

Let $X$ be a finite CW complex, and let $DX$ be its dual in the category of spectra. We demonstrate that the Poincar\'e/Koszul duality between $THH(DX)$ and the free loop space $\Sigma^\infty_+ LX$ is in fact a genuinely $S^1$-equivariant duality that preserves the $C_n$-fixed points. Our proof uses an elementary but surprisingly useful rigidity theorem for the geometric fixed point functor $\Phi^G$ of orthogonal $G$-spectra., Comment: Accepted version, 46 pages. Replaces the first half of the earlier preprint "On the topological Hochschild homology of $DX$." Part of the author's thesis

For any perfect fibration $E \rightarrow B$, there is a "free loop transfer map" $LB_+ \rightarrow LE_+$, defined using topological Hochschild homology. We prove that this transfer is compatible with the Becker-Gottlieb transfer, allowing us to extend a result of Dorabia\l{}a and Johnson on the transfer map in Waldhausen's $A$-theory. In the case where $E \rightarrow B$ is a smooth fiber bundle, we also give a concrete geometric model for the free loop transfer in terms of Pontryagin-Thom collapse maps. We recover the previously known computations of the free loop transfer due to Schlichtkrull, and make a few new computations as well., Comment: v. 2; new string diagrams added, 54 pages, 7 figures. To appear in Trans. AMS

We give a description of the negative bundles for the energy integral on the free loop space $L{\mathbb{C}\mathrm{P}}^n$ in terms of circle vector bundles over projective Stiefel manifolds. We compute the mod $p$ Chern classes of the associated homotopy orbit bundles., To appear in Math. Scand

We prove that any complete (and possibly non-compact) Riemannian manifold $M$ possesses infinitely many closed geodesics provided its free loop space has unbounded Betti numbers in degrees larger than the dimension of $M$, and there are no close conjugate points at infinity. Our argument builds on an existence result due to Benci and Giannoni, and generalizes the celebrated theorem of Gromoll and Meyer for closed manifolds., Comment: 9 pages; Version 2: minor corrections

We show that the category of abelian gerbes over a smooth manifold is equivalent to a certain category of principal bundles over the free loop space. These principal bundles are equipped with fusion products and are equivariant with respect to thin homotopies between loops. The equivalence is established by a functor called regression, and complements a similar equivalence for bundles and gerbes equipped with connections, derived previously in Part II of this series of papers. The two equivalences provide a complete loop space formulation of the geometry of gerbes; functorial, monoidal, natural in the base manifold, and consistent with passing from the setting "with connections" to the one "without connections". We discuss an application to lifting problems, which provides in particular loop space formulations of spin structures, complex spin structures, and spin connections., 37 pages. v2: proof of Lemma 5.4 improved. v3: typos corrected, more comments about the thin loop stack; v3 is the final and published version

The purpose of this paper is to calculate the rational cohomology \({H^{\ast}(X^{{S}^{1}} ; \mathbb{Q})}\) of the free loop space for a simply connected closed 4-manifold X. We use minimal models, so the starting point is the cohomology algebra \({H^{\ast}(X; \mathbb{Q})}\) which depends only on the second Betti number b2 and the signature of X itself. Calculations of \({H^{\ast}(X^{{S}^{1}} ; \mathbb{Q})}\) for b2 ≤ 2 are known. We study the case b2 > 2. We obtain an explicit formula for Poincare series of the space \({X^{{S}^{1}}}\), with the second Betti number b2 as a parameter.

We show that after rationalization there is a homotopy fiber sequence BBU -> K(ku) -> K(Z). We interpret this as a correspondence between the virtual 2-vector bundles over a space X and their associated anomaly bundles over the free loop space LX. We also rationally compute K(KU) by using the localization sequence, and K(MU) by a method that applies to all connective S-algebras., 22 pages

A chain complex model for the free loop space of a connected, closed and oriented manifold is presented, and on its homology, the Gerstenhaber and Batalin-Vilkovisky algebra structures are defined and identified with the string topology structures. The gravity algebra on the equivariant homology of the free loop space is also modeled. The construction includes non simply-connected case, and therefore gives an algebraic and chain level model of Chas-Sullivan's String Topology., 30 pages; revised version with errors corrected

Unstable manifolds of critical points at infinity in the variational problems relating to periodic orbits of Reeb vector-fields in Contact Form Geometry are viewed in this paper as part of the attaching maps along which these variational problems attach themselves to natural generalizations that they have. The specific periodic orbit problem for the Reeb vector-field $\xi_0$ of the standard contact structure/form of $S^3$ is studied; the extended variational problem is the closed geodesics problem on $S^2$. The attaching maps are studied for low-dimensional (at most $4$) cells. Some circle and ''loop" actions on the loop space of $S^3$, that are lifts (via Hopf-fibration map) of the standard $S^1$-action on the free loop space of $S^2$, are also defined. ''Conjugacy" relations relating these actions are established.

We present some results concerning the Morse Theory of the energy function on the free loop space of the three sphere for metrics all of whose geodesics are closed. We also explain how these results relate to the Berger conjecture in dimension three.

Let \({\mathbb{K}}\) be a field of characteristic p > 0 and S1 the unit circle. We construct a model for the negative cylic homology of a commutative cochain algebra with two stages Sullivan minimal model. Using the notion of shc-formality introduced in Bitjong and Thomas (Topology 41:85–106), the main result of Bitjong and El Haouari (Math Ann 338:347–354) and techniques of Vigue-Poirrier (J Pure Appl Algebra 91:347–354) we compute the S1-equivariant cohomology algebras of the free loop spaces of the infinite complex projective space \({\mathbb{CP}(\infty)}\) and the odd spheres S2q+1.

The loop homology ring of an oriented closed manifold, defined by Chas and Sullivan, is interpreted as a fibrewise homology Pontrjagin ring. The basic structure, particularly the commutativity of the loop multiplication and the homotopy invariance, is explained from the viewpoint of the fibrewise theory, and the definition is extended to arbitrary compact manifolds. In (3) Chas and Sullivan defined the loop product on the (rational) homology, H∗+d(LM ), of the free loop space of an oriented closed d-manifold M , with a dimension shift d, and showed that the product gave H∗+d(LM ) the structure of a (graded) commutative associative H ∗ (M )-algebra. The free loop space LM , defined as the space of continuous loops α : R/Z → M , fibres over M by evaluation at 0: α ∈L M � α(0) ∈ M , and the fibre at x ∈ M is naturally identified with the space of based loops Ω(M, x) in the space M with basepoint x. In this framework the loop product can be interpreted as a fibrewise Pontrjagin product in the fibrewise homology of p : LM → M over M. This paper gives an account of loop homology from this point of view, with some examples. Although the elementary definitions can be made for any compact Euclidean neighbourhood retract M , the proof of the commutativity of the loop product given in § 5 requires M to be a smooth, but not necessarily orientable, closed manifold. The results are extended to a compact manifold with non-empty boundary ∂M by looking at the relative fibrewise homology over (M, ∂M). Granted some familiarity with fibrewise homology theory, the definition of loop homol- ogy as a fibrewise theory and the verification of its basic properties are straightforward exercises. In § 2 we provide a brief review of fibrewise homology theory. For more details the reader is referred to (9, Part II, § 15); the account of fibrewise stable homotopy given there was written primarily as a source for applications to geometry, and this is one such application (see also (8)). Section 3 describes the basic definition and structure of loop homology as a fibrewise homology theory. In § 4 we relate the construction to the

Let \({\mathbb{K}}\) be a field of characteristic \({p\geq 0}\) and S1 the unit circle. We prove that the shc-structure on a cochain algebra (A,dA) induces an associative product on the negative cyclic homology HC*−A. When the cochain algebra (A,dA) is the algebra of normalized cochains of the simply connected topological space X with coefficients in \({\mathbb{K}}\) , then HC*−A is isomorphic as a graded algebra to \({H^{-*}_{S^1}(LX;\mathbb{K})}\) the S1-equivariant cohomology algebra of LX, the free loop space of X. We use the notion of shc-formality introduced in Topology 41, 85–106 (2002) to compute the S1-equivariant cohomology algebras of the free loop space of the complex projective space \({\mathbb{C}P(n)}\) when n + 1 = 0 [p] and of the even spheres S2n when p = 2.

Let $G \to P \to M$ be a flat principal bundle over a closed and oriented manifold $M$ of dimension $m=2d$. We construct a map of Lie algebras $\Psi: \H_{2\ast} (L M) \to {\o}(\Mc)$, where $\H_{2\ast} (LM)$ is the even dimensional part of the equivariant homology of $LM$, the free loop space of $M$, and $\Mc$ is the Maurer-Cartan moduli space of the graded differential Lie algebra $\Omega^\ast (M, \adp)$, the differential forms with values in the associated adjoint bundle of $P$. For a 2-dimensional manifold $M$, our Lie algebra map reduces to that constructed by Goldman in \cite{G2}. We treat different Lie algebra structures on $\H_{2\ast}(LM)$ depending on the choice of the linear reductive Lie group $G$ in our discussion., Comment: 28 pages. This is the final version