Path Fibration Research Articles

A combinatorial model for the path fibration

We introduce the abstract notion of a necklical set in order to describe a functorial combinatorial model of the path fibration over the geometric realization of a path connected simplicial set. In particular, to any path connected simplicial set X we associate a necklical set \({\widehat{{\varvec{\Omega }}}}X\) such that its geometric realization \(|{\widehat{{\varvec{\Omega }}}}X|\), a space built out of gluing cubical cells, is homotopy equivalent to the based loop space on |X| and the differential graded module of chains \(C_*({\widehat{{\varvec{\Omega }}}}X)\) is a differential graded associative algebra generalizing Adams’ cobar construction.

The twisted Cartesian model for the double path fibration

AbstractIn 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.

Rational topological complexity

Motion planning is a rapidly growing area of research in robotics where topological methods have been applied [14]. From the topological point of view, a motion planning algorithm has as input two states of the motion, i.e., two arbitrary points in the configuration space X . As output it has to provide a continuous path between the two chosen states. In other words, such an algorithm consists of a section (not necessarily continuous) s : X × X −→ XI of the unpointed path fibration p : XI −→ X × X, p(σ) = ( σ(0),σ(1) ) , where XI denotes the space of free paths on X .

The Atiyah Algebroid of the Path Fibration over a Lie Group

Let G be a connected Lie group, LG its loop group, and π : PG → G the principal LG-bundle defined by quasi-periodic paths in G. This paper is devoted to differential geometry of the Atiyah algebroid A = T (PG)/LG of this bundle. Given a symmetric bilinear form on \({\mathfrak{g}}\) and the corresponding central extension of \({L\mathfrak{g}}\) , we consider the lifting problem for A, and show how the cohomology class of the Cartan 3-form \({\eta \in \Omega^3(G)}\) arises as an obstruction. This involves the construction of a 2-form \({\varpi \in \Omega^{2}({\rm PG})^{\rm LG}= \Gamma(\wedge^2 A^*)}\) with \({{\rm d}\varpi=\pi^*\eta}\) . In the second part of this paper we obtain similar LG-invariant primitives for the higher degree analogues of the form η, and for their G-equivariant extensions.

An explicit classification of three-stage Postnikov towers

The problems of classifying Hurewicz fibrations whose fibres have just two non-zero homotopy groups and classifying 3-stage Postnikov towers are substantially equivalent. We investigate the case where the fibres have the homotopy type of K(G,m)×K(H,n), for 1 < m < n. Our solution uses a classifying space M∞, i.e. a mapping space whose underlying set consists of all null homotopic maps from individual fibres of the path fibration PK(G,m+ 1) → K(G,m+ 1) to the space K(H,n+ 1), and the group E(K(G,m)×K(H,n)) of based homotopy classes of based self-homotopy equivalences of K(G,m)×K(H,n). If B is a given space, then a group action E(K(G,m)×K(H,n))× [B,M∞] → [B,M∞] is defined, and the orbit set [B,M∞] / E(K(G,m)×K(H,n)) is shown to classify the above fibrations over B up to fibrewise homotopy type. Our explicit definitions of the classifying spaces, together with our computationally effective group actions, are advantageous for computations and further developments. Two stable range simplifications are given here, together with a classification result for cases where B is a product of spheres. Dedicated to L. Gaunce Lewis, Jr (1949–2006) The author would like to express his appreciation of the advice, concerning this paper, that was given by Gaunce Lewis. It included detailed comments concerning the presentation and restructuring of the material, and a helpful suggestion concerning future developments. This was done at a time when Gaunce was well aware of the seriousness and immediacy of the health problems that he faced. It is but one example of his helpfulness, kindness and generosity. Received July 20, 2005, revised August 22, 2006; published on September 27, 2006. 2000 Mathematics Subject Classification: 55R15, 55R35, 55S45, 55P20.

The Predifferential of a Path Fibration

Abstract For a simply connected space 𝐵 the Hirsch model of path fibration is constructed in terms of 𝐵. In particular this means the calculation of loop space cohomology. As an application, the Hirsch model of the fiber of any fibration over 𝐵 is given.

A quick proof of the rational Hurewicz theorem and a computation of the rational homotopy groups of spheres

We give an elementary proof of the rational Hurewicz theorem and compute the rational cohomology groups of Eilenberg–MacLane spaces and the rational homotopy groups of spheres. Instead of using the Serre spectral sequence, we only assume the classical Hurewicz theorem, and give a short proof of the rational Gysin and Wang long exact sequences, which are applied inductively to the path fibration of Eilenberg–MacLane spaces.

String structures and the path fibration of a group

We use the realisation of the universal bundle for the loop group as the path fibration of the group to investigate the string class, that is the obstruction to a loop group bundle lifting to a Kac-Moody group bundle. In the case that the loop group bundle is constructed by taking loops into a principal bundle we show that the classifying map is the holonomy around loops and give an explicit formula for the string class relating it to the Pontrjangin class of the principal bunble.

Circular bar construction

Let A be a CDG (commutative differential graded) algebra over a field of characteristic 0, and B(A), the bar construction of A. If C is a DG (differential graded) A @ Amodule, define B(A; C) = C BAOR B(A). If C’ and C” are DG A-modules, then B(A; C’ @ C”) coincides with the usual two sided bar construction B(C’, A, C”). (See [12].) In this sense, B(A; C) closes the two sides of the bar construction and is therefore “circular”. The geometric significance of B(A; C) lies in the case where A = A(M) and C = A(N) are the Rham complexes and the A @ A-module structure of C is induced by a differentiable map f : N-t M x M. Theorem 0.1[6] (Theorem 4.3.1[5]) asserts that, if M and N are differentiable manifolds and if M is simply connected with finite Betti numbers, then H(B(A; C)) is the real cohomology of the pullback space Ef of the following pullback diagram of the free path fibration of M:

Pullback De Rham Cohomology of the Free Path Fibration

Let M and N be smooth manifolds and let $\bar B (A)$ be the reduced bar construction on the de Rham complex $\Lambda (M)$ or a suitable subcomplex A of M. For every smooth map $f:N \to M \times M$, the tensor product $\Lambda (N) \otimes \bar B(A)$, equipped with a suitable differential, will yield the correct cohomology for the pullback of the free path fibration $P(M) \to M \times M$ via the smooth map F. Moreover, $\Lambda (N) \otimes \bar B(A)$ can be taken as a de Rham subcomplex of the pullback space.

Pullback de Rham cohomology of the free path fibration

Let M and N be smooth manifolds and let B ¯ ( A ) \bar B (A) be the reduced bar construction on the de Rham complex Λ ( M ) \Lambda (M) or a suitable subcomplex A of M. For every smooth map f : N → M × M f:N \to M \times M , the tensor product Λ ( N ) ⊗ B ¯ ( A ) \Lambda (N) \otimes \bar B(A) , equipped with a suitable differential, will yield the correct cohomology for the pullback of the free path fibration P ( M ) → M × M P(M) \to M \times M via the smooth map F. Moreover, Λ ( N ) ⊗ B ¯ ( A ) \Lambda (N) \otimes \bar B(A) can be taken as a de Rham subcomplex of the pullback space.

Reducing towers of principal fibrations

Consider a tower of principal fibrationsThat is, Ei+1 is the pullback of Ei→Ri and the path fibration PRi → Ri. The question arises as to whether or not the tower can be shortened, that is, whether or not En+1→ B is fiber homotopically equivalent to a nice fibration E → B. If “nice” is taken to mean “principal” then sufficient conditions are known. They involve connectivity assumptions on the Ei. In this paper “nice” is taken to mean “D-relatively principal” for some space D.

A two-stage Postnikov system where $E\sb{2}\not=E\sb{\infty }$ in the Eilenberg-Moore spectral sequence

Let Q.B -* PB -* B be the path fibration over the simply-connected space B, let Q.B—> E-+ X be the induced fibration via the map /: X —» B, and let X and B be generalized Eilenberg-MacLane spaces. G. Hirsch has conjectured that H*E is additively isomorphic to TorH.£ (Z2, H*X), where cohomology is with Z2 coefficients. Since the Eilenberg-Moore spectral sequence which converges to H*E has E2 = TorH.B(Z2, H*X), the conjecture is equivalent to saying E2 = E^. In the present paper we set X=K{Z2 + Z2, 2), B = K(Z2, 4) and /*i' = the product of the two fundamental classes, and we prove that E2J=E3, disproving Hirsch's conjecture. The proof involves the use of homology isomorphisms C*X X C(H*QX) X H*X developed by J. P. May, where C is the reduced cobar construction. The map g commutes with cup-1 products. Since the cup-1 product in C{H*Q.X) is well known, and since differentials in the spectral sequence correspond to certain cup-1 products, we may compute d2 on specific elements of E2. Introduction. A two-stage Postnikov system 3P is a diagram Q.B —> U.B