Search

Let X be a closed subspace of a metric space M. It is well known that, under mild hypotheses, one can estimate the Betti numbers of X from a finite set P ⊂ M of points approximating X. In this paper, we show that one can also use P to estimate much more detailed topological properties of X. We achieve this by proving the stability of A ∞ -persistent homology. In its most general case, this stability means that given a continuous function f : Y → R on a topological space Y, small perturbations in the function f imply at most small perturbations in the family of A ∞ -barcodes. This work can be viewed as a proof of the stability of cup-product and generalized-Massey-products persistence. The technical key of this paper consists of figuring out a setting which makes A ∞ -persistence functorial. [ABSTRACT FROM AUTHOR]

In this paper, considering the kth shape loop space ... p k (X, x), for an HPol∗-expansion p : (X, x) → ((Xλ, xλ), [pλλ′ ], Λ) of a pointed topological space (X, x), first we prove that ... k commutes with the product under some conditions and then we show that ... p k (X, x) ≅= lim← ... p k (Xi, xi), for a pro-discrete space (X, x) = lim← (Xi, xi) of compact polyhedra. Finally, we conclude that these spaces are metric, second countable and separable. [ABSTRACT FROM AUTHOR]

Abstract We study some Borsuk–Ulam type results for the loop space of an euclidean sphere without loops equal to their inverses. [ABSTRACT FROM AUTHOR]

We completely determine spherical classes in single, double and triple loop spaces of spheres. We also show that for l ⩾ 4 and n > 1 , there exists no spherical class in H ⁎ Ω l S l + n in dimensions more than 2 l n + 2 l − 1 ( l − 2 ) + 2 . This bound may not be optimal, but reduces the computation of spherical classes in H ⁎ Ω l S l + n to verifying finite number of cases. We also provide a table for possible spherical classes in H ⁎ Ω l S l + n with 3 < l < 9 . Our results, provide positive evidence for the truth of Eccles conjecture. [ABSTRACT FROM AUTHOR]

Our goal in this paper is to compute the integral free loop space homology of $$(n-1)$$ -connected 2 n-manifolds. We do this when $$n\ge 2$$ and $$n\ne 2,4,8$$ , though the techniques here should cover a much wider range of manifolds. We also give partial information concerning the action of the Batalin-Vilkovisky operator. [ABSTRACT FROM AUTHOR]

Let p be an odd prime, and let λ ∊ Z. Consider the loop space Yt = S2t−1 (p) for t ⩾ 1 with t∣(p−1). Then we first determine the condition for the powermap Φλ on Yt to be an Ap-map.We next assume that X is a simply connected Fp-finite Ap-space and that λ is a primitive (p−1)st root of unity mod p. Our results show that if the reduced power operations {Pi}i⩾1 act trivially on the indecomposable module QH*(X;Fp) and the power map Φλ on X is an An-map with n>(p −1)/2, then X is Fp-acyclic. [ABSTRACT FROM AUTHOR]

We prove the existence of spectral gaps of Ornstein–Uhlenbeck operators on loop spaces over a class of Riemannian manifolds which include hyperbolic spaces. This is an alternative proof and an extension of a result in Chen et al. [13] . Further, we study the asymptotic behavior of the spectral gap. [ABSTRACT FROM AUTHOR]

Associated to every connected, topological space $$X$$ there is a Hopf algebra-the Pontrjagin ring of the based loop space of the configuration space of two points in $$X$$ . We prove that this Hopf algebra is not a homotopy invariant of the space. We also exhibit interesting examples of $$H$$ -spaces, which are homotopy equivalent as spaces, which lead to isomorphic rational Hopf algebras or not, depending crucially on the existence of Whitehead products. Moreover, we investigate a (naturally motivated) twisted version of these Pontrjagin rings in the various aforementioned contexts. In all of these examples, Massey products abound and play a key role. [ABSTRACT FROM AUTHOR]

We consider two categorifications of the cohomology of a topological space X by taking coefficients in the category of differential graded categories. We consider both derived global sections of a constant presheaf and singular cohomology and find the resulting dg-categories are quasi-equivalent and moreover quasi-equivalent to representations in perfect complexes of chains on the loop space of X. [ABSTRACT FROM AUTHOR]

We extend the theory of Thom spectra and the associated obstruction theory for orientations in order to support the construction of the $E_{\infty }$ string orientation of $tmf$, the spectrum of topological modular forms. Specifically, we show that, for an $E_{\infty }$ ring spectrum $A$, the classical construction of $gl_{1} {A}$, the spectrum of units, is the right adjoint of the functor\[\Sigma ^{\infty }_{+ } \Omega ^{\infty } \colon \mathrm {ho} (\mbox {{connective spectra}}) \longrightarrow \mathrm {ho} ({\text {{$E_{\infty } $ ring spectra}}}).\]To a map of spectra\[f\colon b \longrightarrow bgl_{1} {A},\]we associate an $E_{\infty }$ $A$-algebra Thom spectrum $Mf$, which admits an $E_{\infty }$ $A$-algebra map to $R$ if and only if the composition\[b \longrightarrow bgl_{1} {A} \longrightarrow bgl_{1} {R}\]is null; the classical case developed by May, Quinn, Ray, and Tornehave arises when $A$ is the sphere spectrum. We develop the analogous theory for $A_{\infty }$ ring spectra: if $A$ is an $A_{\infty }$ ring spectrum, then to a map of spaces\[f\colon B \longrightarrow B{ GL}_{1} {A},\]we associate an $A$-module Thom spectrum $Mf,$ which admits an $R$-orientation if and only if\[B \longrightarrow B{ GL}_{1} {A} \longrightarrow B{ GL}_{1} {R}\]is null. Our work is based on a new model of the Thom spectrum as a derived smash product. [ABSTRACT FROM AUTHOR]

Copyright of Technical Transactions / Czasopismo Techniczne is the property of Sciendo and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we investigate the simplicial groups obtained from the link groups of naive cablings on any given framed link. Our main result states that the resulting simplicial groups have the homotopy type of the loop space of a wedge of 3-spheres. This gives simplicial group models for some loop spaces using link groups. [ABSTRACT FROM AUTHOR]

To any path connected topological space X, such that rk Hi(X) < ∞ for all i ⩾ 0, are associated the following two sequences of integers: bi = rk Hi(ΩX) and ri = rkπi+1(X). If X is simply connected, the Milnor-Moore theorem together with the Poincaré-Birkoff-Witt theorem provides an explicit relation between these two sequences. If we assume moreover that Hi(X;Q) = 0, for all i⪢0, it is a classical result that the sequence of Betti numbers (bi) grows polynomially or exponentially, depending on whether the sequence (ri) is eventually zero or not. The purpose of this note is to prove, in both cases, that the rth Betti number br is controlled by the immediately preceding ones. The proof of this result is based on a careful analysis of the Sullivan model of the free loop space XS1. [ABSTRACT FROM AUTHOR]

If X is a cosimplical En+1 space then Tot(X) is an En+1 space and its mod 2 homology H.(Tot(X)) has Dyer-Lashof and Browder operations. It's natural to ask if the spectral sequence converging to H.(Tot(X)) admits compatible operations. In this paper we give a positive answer to this question. [ABSTRACT FROM AUTHOR]

A special family of non-trivial loops on a surface called handle and tunnel loops associates closely to geometric features of "handles" and "tunnels" respectively in a 3D model. The identification of these handle and tunnel loops can benefit a broad range of applications from topology simplification / repair, and surface parameterization, to feature and shape recognition. Many of the existing efficient algorithms for computing non-trivial loops cannot be used to compute these special type of loops. The two algorithms known for computing handle and tunnel loops provably have a serious drawback that they both require a tessellation of the interior and exterior spaces bounded by the surface. Computing such a tessellation of three dimensional space around the surface is a non-trivial task and can be quite expensive. Furthermore, such a tessellation may need to refine the surface mesh, thus causing the undesirable side-effect of outputting the loops on an altered surface mesh. In this paper, we present an efficient algorithm to compute a basis for handle and tunnel loops without requiring any 3D tessellation. This saves time considerably for large meshes making the algorithm scalable while computing the loops on the original input mesh and not on some refined version of it. We use the concept of the Reeb graph which together with several key theoretical insights on linking number provide an initial set of loops that provably constitute a handle and a tunnel basis. We further develop a novel strategy to tighten these handle and tunnel basis loops to make them geometrically relevant. We demonstrate the efficiency and effectiveness of our algorithm as well as show its robustness against noise, and other anomalies in the input. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to introduce the concept of generalized continuous maps by using generalized closed sets and investigate some of their characterizations. [ABSTRACT FROM AUTHOR]

We prove that the mapping stack Map(, ) of topological stacks  and  is again a topological stack if  admits a groupoid presentation [ Y1 ⇉ Y0] such that Y0 and Y1 are compact topological spaces. If Y0 and Y1 are only locally compact, we show that Map(, ) is a paratopological stack. In particular, it has a classifying space (hence, a natural weak homotopy type). We also show that the weak homotopy type of the mapping stack Map( Y, ) does not change if we replace  by its classifying space, provided that Y is a paracompact topological space. As an example, we describe the loop stack of the classifying stack ℬ G of a topological group G in terms of twisted loop groups of G. [ABSTRACT FROM AUTHOR]

Abstract: The importance of small loops in the covering space theory was pointed out by Brodskiy, Dydak, Labuz, and Mitra in and . A small loop is a loop which is homotopic to a loop contained in an arbitrarily small neighborhood of its base point and a small loop space is a topological space in which every loop is small. Small loops are the strongest obstruction to semi-locally simply connectedness. We construct a small loop space using the Harmonic Archipelago. Furthermore, we define the small loop group of a space and study its impact on covering spaces, in particular its contribution to the fundamental group of the universal covering space. [Copyright &y& Elsevier]

For a general time-varying system, we prove that existence of an "Output Robust Control Lyapunov Function" implies existence of continuous time-varying feedback stabilizer, which guarantees output asymptotic stability with respect to the resulting closed-loop system. The main results of the present work constitute generalizations of a well known result due to Coron and Rosier [J. Math. Syst. Estim. Control 4 (1994) 67-84] concerning stabilization of autonomous systems by means of time-varying periodic feedback. [ABSTRACT FROM AUTHOR]

In this paper we prove that the Steenrod operations act naturally on the negative cyclic homology of a differential graded algebra A over the prime field Fp satisfying some extra conditions. When A denotes the singular cochains with coefficients in Fp of a 1-connected space X, these extra conditions are satisfied. The Jones isomorphism identifies these Steenrod operations with the usual ones on the S¹-equivariant cohomology of the free loop space on X with coefficients in Fp. We conclude by performing some calculations on the negative cyclic homology. [ABSTRACT FROM AUTHOR]

Several properties of fuzzy semiclosure and fuzzy semi-interior of fuzzy sets defined by Yalvac (1988), have been established and supported by counterexamples. We also study the characterizations and properties of fuzzy semi-open and fuzzy semi-closed sets. Moreover, we define fuzzy s-open and fuzzy s-closed mappings and give some interesting characterizations. [ABSTRACT FROM AUTHOR]

We compute the structure relations in special A ∞-bialgebras whose operations are limited to those defining the underlying A ∞-(co)algebra substructure. Such bialgebras appear as the homology of certain loop spaces. Whereas structure relations in general A ∞-bialgebras depend upon the combinatorics of permutahedra, only Stasheff’s associahedra are required here. [ABSTRACT FROM AUTHOR]

Let Mn be a closed, oriented, n-manifold, and LM its free loop space. In [Chas and Sullivan, ‘String topology’, Ann. of Math., to appear] a commutative algebra structure in homology, H*(LM), and a Lie algebra structure in equivariant homology , were defined. In this paper, we prove that these structures are homotopy invariants in the following sense. Let f:M1 → M2 be a homotopy equivalence of closed, oriented n-manifolds. Then the induced equivalence, Lf:LM1 → LM2 induces a ring isomorphism in homology, and an isomorphism of Lie algebras in equivariant homology. The analogous statement also holds true for any generalized homology theory h* that supports an orientation of the Mi. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper, we will consider Laplace''s method for a class of heat processes on loop spaces. We will obtain the first term of the asymptotics under assumptions that the function under consideration attains its minimum at a unique point and that the Hessian at the point is non-degenerate. This kind of process was first introduced by P. Malliavin in 1990 for the loop group case and then gradually generalized by various authors. Our tool is the rough path theory of T. Lyons. This technique was pioneered by S. Aida for finite-dimensional processes in his unpublished paper. [Copyright &y& Elsevier]

We consider a bundle determined by a classifying map with skeleton smooth in the Chen — Souriau sense. We show that the stochastic classifying map is homotopic to a deterministic classifying map on the Holder loop space. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying some properties of the Courant algebroids: we explain the so-called "conducting bundle construction" and use it to attach the Courant algebroid to the Dixmier-Douady gerbe (following the ideas of P. Severa). We note that the WZNW-Poisson condition of Klimcik and Strobl (math.SG/0104189) is the same as the Dirac structure in some particular Courant algebroid. We propose the construction of the Lie algebroid on the loop space starting from the Lie algebroid on the manifold and conjecture that this construction applied to this Dirac structure should give the Lie algebroid of symmetries in the WZNW-Poisson σ-model and show that it is indeed true in the particular case of the Poisson σ-model. [ABSTRACT FROM AUTHOR]

Situations analogous to some classical characterization are investigated, of topological spaces X for which Cp(X) belongs to a given coreflective class C of locally convex spaces. For instance, if C contains all strong Mazur spaces and is contained in the class of weak Mazur spaces, then Cp(X) belongs to C iff X is realcompact. If C is the coreflective hull of Rω and X is a P-space, then Cp(X) belongs to C iff X is realcompact. [Copyright &y& Elsevier]

Correcting an infinite loop in Douglas M. Priest's renormalization algorithm, the theory proved here supports streamlined algorithms to resolve the tablemaker's dilemma for the floating-point computation of real and complex sums and dot-products, properly rounded to the ultimate digit. Applications include computations of areas, volumes, and intersections. [ABSTRACT FROM AUTHOR]

We prove the existence of breathers (spatially localized and time-periodic oscillations) in diatomic Fermi–Pasta–Ulam (FPU) chains with arbitrary mass ratio. This completes an existence result by Livi, Spicci and MacKay valid for large mass ratio. The problem is formulated as a mapping in a loop space and analyzed via a discrete spatial centre manifold reduction. Plane wave solutions of the linearized system have frequencies in a higher “optic” band or a lower “acoustic” band. For frequencies close to band edges, all small amplitude solutions of the nonlinear system lie on a finite-dimensional centre manifold, which reduces the problem locally to the study of a finite-dimensional mapping. For good parameter values, the map admits homoclinic orbits to corresponding to discrete breathers. When the FPU interaction potential satisfies a hardening condition, we find breathers with frequencies slightly above the optic band, or in the gap slightly above the acoustic band. For a potential satisfying the opposite softening condition, we obtain breathers with frequencies in the gap slightly below the optic band. [Copyright &y& Elsevier]

Focuses on arbitrary complex separable connected metrizable manifolds M and N. Importance of loop groups in differential geometry, algebraic topology and theoretical physics; Definition of a group structure on the quotient space of a free loop space; Consideration of loop groups of different classes.

In this note we show that the positivity property of the equivariant signature of the loop space, first observed in [MS1] in the case of the even-dimensional projective spaces, is valid for Picard number 2 toric varieties. A new formula for the equivariant signature of the loop space in the case of a toric spin variety is derived. [ABSTRACT FROM AUTHOR]

The chain complex of the iterated loop space is expressed through the chain complex over an initial space in terms of the cobar construction. The corresponding spectral sequence is constructed and its first term is calculated. Applications to the calculation of the homology of the iterated loop spaces of the stunted real and complex projective spaces arc given. The origin of this work is in some machine computations undertaken by Sergeraert; a brief report about them is given in the appendix. [ABSTRACT FROM AUTHOR]

The k-invariants k[subn] in the Postnikov decomposition of a finite type connected H-space X are known to be torsion elements, that is, there exist numbers N[subn](X) with N[subn](X) ⋅ k[subn] = 0 for n ≥ 2. We describe classes of H-spaces for which N[subn](X) = N[subn] can be chosen to be the same number for all X in the class. Dually the boundary invariants β[subn], of a finite type simply connected co-H-space Y satisfy M[subn](Y) ⋅ β[subn] = 0. We describe classes of co-H-spaces for which M[subn](Y) = M[subn] can be chosen to be the same number for all Y in the class. [ABSTRACT FROM AUTHOR]

For a smooth, finite-dimensional manifold $M$ with a submanifold $S$ we study the topology of the straight loop space $\Omega^{st}_SM$, the space of loops whose intersections with $S$ are subject to a certain transversality condition. Our main tool is Rourke and Sanderson's compression theorem. We prove that the homotopy type of the straight loop space of a link in $S^3$ depends only on the number of link components. [ABSTRACT FROM AUTHOR]

We consider the general problem of constructing the structure of a smooth manifold on a given space of loops in a smooth finite-dimensional manifold. By generalising the standard construction for smooth loops, we derive a list of conditions for the model space which, if satisfied, mean that a smooth structure exists. We also show how various desired properties can be derived from the model space; for example, topological properties such as paracompactness. We pay particular attention to the fact that the loop spaces that can be defined in this way are all homotopy equivalent; and also to the action of the circle by rigid rotations. [ABSTRACT FROM AUTHOR]