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]

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]

In this paper, the notion of a truncating twisting function from a cubical set to a permutahedral set and the corresponding notion of a twisted Cartesian product of these sets are introduced. The latter becomes a permutocubical set that models, in particular, the path fibration on a loop space. The chain complex of this twisted Cartesian product is in fact a comultiplicative twisted tensor product of cubical chains of base and permutahedral chains of fibre. This construction is formalized as a theory of twisted tensor products for Hirsch algebras. [ABSTRACT FROM AUTHOR]

We associate an algebra $$\Gamma ^\infty (\mathfrak {A})$$ to each bornological algebra $$\mathfrak {A}$$ . Each symmetric ideal $$S$$ of the algebra $$\ell ^\infty $$ of complex bounded sequences gives rise to an ideal $$I_{S(\mathfrak {A})}$$ of $$\Gamma ^\infty (\mathfrak {A})$$ . We show that all ideals arise in this way when $$\mathfrak {A}$$ is the algebra of complex numbers. We prove that for suitable $$S$$ , Weibel's $$K$$ -theory of $$I_{S(\mathfrak {A})}$$ is homotopy invariant, and show that the failure of the map from Quillen's to Weibel's $$K$$ -theory of $$I_{S(\mathfrak {A})}$$ to be an isomorphism is measured by cyclic homology. [ABSTRACT FROM AUTHOR]

In this paper, we use Chas–Sullivan theory on loop homology and Leray–Serre spectral sequence to investigate the topological structure of the non-contractible component of the free loop space on the real projective spaces with odd dimensions. Then we apply the result to get the resonance identity of non-contractible homologically visible prime closed geodesics on such spaces provided the total number of distinct prime closed geodesics is finite. [ABSTRACT FROM AUTHOR]

In this paper we will first construct Polyakov loops for the ABJ theory. Then we will construct the connection and curvature in this loop space. We will also analyse certain generalization of Polyakov loops and apply them to the ABJ theory. Finally, we will use this formalism for analysing topological defects like monopoles in the ABJ theory. [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]

Several improvements are made to the 2-primary homotopy theory of the double suspension. We refine Richter's factorization of the second-power map on through the loops on the double suspension, show that the 4th-power map on factors through the double suspension, and that the 4th-power map on is null homotopic, where is the fiber of the double suspension. The latter two results are improvements by one loop. [ABSTRACT FROM AUTHOR]

Abstract: We study the mod p cohomology of loop spaces and free loop spaces of irreducible compact Hermitian symmetric spaces of classical types. We also prove that the integral cohomology of loop spaces of them is torsion free, or has only 2-torsion of order 2. [Copyright &y& Elsevier]

The Infinite Time Turing Machine model [8] of Hamkins and Kidder is. in an essential sense. a "∑2-machine" in that it uses a ∑2 Liminf Rule to determine cell values at limit stages of time. We give a generalisation of these machines with an appropriate ∑n rule. Such machines either halt or enter an infinite loop by stage ζ(n) = df μζ(n)[∃∑(n) > ζ(n) Lζ(n) �ζn Lζ(n)], again generalising precisely the ITTM case. The collection of such machines taken together computes precisely those reals of the least model of analysis. [ABSTRACT FROM AUTHOR]

Abstract: Let p be an odd prime. For a kind of p-local co-H space Y of low-rank, it is shown in this paper that ΩY has as a product factor for a sequence of k which admits an exponential growth. The result appears as a result of the application of recently developed technique for functorial decompositions of looped co-H spaces via the functorial coalgebra decompositions of tensor algebras. [Copyright &y& Elsevier]

Abstract: We prove a coherence theorem for braided monoidal bicategories and relate it to the coherence theorem for monoidal bicategories. We show how coherence for these structures can be interpreted topologically using up-to-homotopy operad actions and the algebraic classification of surface braids. [Copyright &y& Elsevier]

Let M be a smooth, simply-connected, closed oriented manifold, and LM the free loop space of M. Using a Poincaré duality model for M, we show that the reduced equivariant homology of LM has the structure of a Lie bialgebra, and we construct a Hopf algebra which quantizes the Lie bialgebra. [ABSTRACT FROM AUTHOR]

Let f : X → Z and g : Y → Z be maps between connected pointed CW-complexes. Recall the definition of pairing with axes f and g due to N.Oda [16]. In this paper, we introduce (n)-pairing, which is a generalization of H(n)-space due to Y.Félix and D.Tanré [5] and define a family of subsets of the homotopy set of maps. We give some rational characterizations of it and illustrate some examples in Sullivan models. Also we consider about the G(n)-sequence of a fibration which is a generalization of G-sequence [11],[13]. [ABSTRACT FROM AUTHOR]

A novel battery recharging system for an all-electric driveline comprising a flywheel with a permanent magnet double wound synchronous machine (motor/generator) is presented. The double winding enables two voltage levels and two different power levels. This topology supersedes other all-electric drivelines. The battery operates in a low-power regime supplying the average power whereas the flywheel delivers and absorbs power peaks, which are up to a higher order of magnitude. The topology presents new challenges for the power conversion system, which is the focus of this investigation. The main challenge is the control of the power flow to the battery when the vehicle is parked despite the decay of the flywheel machine voltage; which is dependent on its charge state, that is, rotational speed. The design and simulation of an unidirectional DC/DC buck/boost converter for a variable rotational speed flywheel is presented. Conventional power electronic converters are used in a new application, which can maintain a constant current or voltage on the battery side. Successful PI current control has been implemented and simulated, together with the complete closed loop system. [ABSTRACT FROM AUTHOR]

Abstract: The formula (see Goodwillie [Cyclic homology, derivations, and the free loopspace, Topology 24 (2) (1985) 187–215]) on the normalized Hochschild complex is the standard replacement in noncommutative geometry for the classical Cartan homotopy formula. Our purpose is to extend this formula to the normalized bivariant Hochschild complex. To cite this article: A. Banerjee, C. R. Acad. Sci. Paris, Ser. I 347 (2009). [Copyright &y& Elsevier]

Abstract: In this paper, we study the decompositions of the loop spaces of triple wedge by using combinatorial methods and give some special decompositions of the loop spaces of triple wedge. [Copyright &y& Elsevier]

The state space of a machine admits the structure of time. For example, the geometric realization of a precubical set, a generalization of an unlabeled asyn- chronous transition system, admits a "local preorder" encoding control ow. In the case where time does not loop, the "locally preordered" state space splits into causally distinct components. The set of such components often gives a computable invariant of machine behavior. In the general case, no such meaningful partition could exist. However, as we show in this note, the locally preordered geometric realization of a precubical set admits a "locally monotone" covering from a state space in which time does not loop. Thus we hope to extend geometric techniques in static program analysis to looping processes. [ABSTRACT FROM AUTHOR]

We present a novel data structure to evaluate and render Loop subdivision surfaces. The algorithm exploits the fact that the Loop subdivision scheme is supported compactly and uses a straightforward technique to evaluate the vertices of new level efficiently. The developed data structure reserves the key topological information for the subdivision connotatively; it requires only a small amount of computer memory and enables local parameterization of the subdivision surface also. [ABSTRACT FROM AUTHOR]

We relate the brace products of a fibration with section to the differentials in its Serre spectral sequence. In the particular case of free loop fibrations, we establish a link between these differentials and Browder operations in the fiber. Applications and several calculations (for the particular cases of spheres and wedges of spheres) are given.2000 Mathematical Subject Classification: 55P35, 55Q15, 55R05, 55R20. [ABSTRACT FROM AUTHOR]

We show that the loop spaces on real projective spaces are topologically approximated by the spaces of rational maps $\mathbf{RP}^{1}\rightarrow \mathbf{RP}^{n}$. As a byproduct of our constructions we obtain an interpretation of the Kronecker characteristic (degree) of an ornament via particle spaces. [ABSTRACT FROM AUTHOR]