Free loop spaces and cyclohedra

Markl, Martin

Displaying similar documents to “Free loop spaces and cyclohedra”

Rational string topology

Yves Félix, Jean-Claude Thomas, Micheline Vigué-Poirrier (2007)

Journal of the European Mathematical Society

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We use the computational power of rational homotopy theory to provide an explicit cochain model for the loop product and the string bracket of a simply connected closed manifold $M$ . We prove that the loop homology of $M$ is isomorphic to the Hochschild cohomology of the cochain algebra $C^{*} (M)$ with coefficients in $C^{*} (M)$ . Some explicit computations of the loop product and the string bracket are given.

G-functors, G-posets and homotopy decompositions of G-spaces

Stefan Jackowski, Jolanta Słomińska (2001)

Fundamenta Mathematicae

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We describe a unifying approach to a variety of homotopy decompositions of classifying spaces, mainly of finite groups. For a group G acting on a poset W and an isotropy presheaf d:W → (G) we construct a natural G-map $h o c o l i m_{_{d}} G / d (-) \to | W |$ which is a (non-equivariant) homotopy equivalence, hence $h o c o l i m_{_{d}} E G \times_{G} F_{d} \to E G \times_{G} | W |$ is a homotopy equivalence. Different choices of G-posets and isotropy presheaves on them lead to homotopy decompositions of classifying spaces. We analyze higher limits over the categories associated to isotropy...

On weak $i$ -homotopy equivalences of modules

Zheng-Xu He (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Si definisce il gruppo di $i$ —omotopia di un singolo modulo e si introduce la nozione di equivalenza $i$ -omotopica debole. Sotto determinate condizioni per l'anello di base $\Lambda$ oppure per i moduli considerati, le equivalenze $i$ -omotopiche deboli coincidono con le equivalenze $i$ -omotopiche (forti).

Minimal component numbers of fixed point sets

Xuezhi Zhao (2003)

Fundamenta Mathematicae

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Let f: (X,A) → (X,A) be a relative map of a pair of compact polyhedra. We introduce a new relative homotopy invariant $N^{C} (f; X, A)$ , which is a lower bound for the component numbers of fixed point sets of the self-maps in the relative homotopy class of f. Some properties of $N^{C} (f; X, A)$ are given, which are very similar to those of the relative Nielsen number N(f;X,A).

A decomposition of $E^{3}$ into straight arcs and singletons

S. Armentrout

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CONTENTS1. Introduction............................................................................................................................................. 52. Notation and. terminology................................................................................................................... 63. Description of G.................................................................................................................................... 64. Preliminaries to the...

The Salvetti complex and the little cubes

Dai Tamaki (2012)

Journal of the European Mathematical Society

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For a real central arrangement $𝒜$ , Salvetti introduced a construction of a finite complex Sal $(𝒜)$ which is homotopy equivalent to the complement of the complexified arrangement in [Sal87]. For the braid arrangement $𝒜_{k - 1}$ , the Salvetti complex Sal $(𝒜_{k - 1})$ serves as a good combinatorial model for the homotopy type of the configuration space $F (ℂ, k)$ of $k$ points in $C$ , which is homotopy equivalent to the space $C_{2} (k)$ of k little $2$ -cubes. Motivated by the importance of little cubes in homotopy theory, especially in...

Moufang loops of order coprime to three that cyclically extend groups of dihedral type

Aleš Drápal (2016)

Commentationes Mathematicae Universitatis Carolinae

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This paper completely solves the isomorphism problem for Moufang loops $Q = G C$ where $G ⊴ Q$ is a noncommutative group with cyclic subgroup of index two and $| Z (G) | \leq 2$ , $C$ is cyclic, $G \cap C = 1$ , and $Q$ is finite of order coprime to three.

Maps into the torus and minimal coincidence sets for homotopies

D. L. Goncalves, M. R. Kelly (2002)

Fundamenta Mathematicae

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Let X,Y be manifolds of the same dimension. Given continuous mappings $f_{i}, g_{i} : X \to Y$ , i = 0,1, we consider the 1-parameter coincidence problem of finding homotopies $f_{t}, g_{t}$ , 0 ≤ t ≤ 1, such that the number of coincidence points for the pair $f_{t}, g_{t}$ is independent of t. When Y is the torus and f₀,g₀ are coincidence free we produce coincidence free pairs f₁,g₁ such that no homotopy joining them is coincidence free at each level. When X is also the torus we characterize the solution of the problem in terms of the...

A formula for topology/deformations and its significance

Ruth Lawrence, Dennis Sullivan (2014)

Fundamenta Mathematicae

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The formula is $\partial e = (a d_{e}) b + \sum_{i = 0}^{\infty} (B_{i}) / i! {(a d_{e})}^{i} (b - a)$ , with ∂a + 1/2 [a,a] = 0 and ∂b + 1/2 [b,b] = 0, where a, b and e in degrees -1, -1 and 0 are the free generators of a completed free graded Lie algebra L[a,b,e]. The coefficients are defined by $x / (e^{x} - 1) = \sum_{n = 0}^{\infty} B ₙ / n! x ⁿ$ . The theorem is that ∙ this formula for ∂ on generators extends to a derivation of square zero on L[a,b,e]; ∙ the formula for ∂e is unique satisfying the first property, once given the formulae for ∂a and ∂b, along with the condition that the “flow” generated by e moves a to...

Interplay between strongly universal spaces and pairs

Taras Banakh, Robert Cauty

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Given a pair (M,X) of spaces we investigate the connections between the (strong) universality of (M,X) and that of the space X. We apply this to prove Enlarging, Deleting, and Strong Negligibility Theorems for strongly universal and absorbing spaces. Given an absorbing space Ω we also study the question of topological uniqueness of the pair (M,X), where $M = {[0, 1]}^{ω}$ or $M = {(0, 1)}^{ω}$ and X is a copy of Ω in M having a locally homotopy negligible complement in M.

Extending the structural homomorphism of LCC loops

Piroska Csörgö (2005)

Commentationes Mathematicae Universitatis Carolinae

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A loop $Q$ is said to be left conjugacy closed if the set $A = {L_{x} / x \in Q}$ is closed under conjugation. Let $Q$ be an LCC loop, let $ℒ$ and $ℛ$ be the left and right multiplication groups of $Q$ respectively, and let $I (Q)$ be its inner mapping group, $M (Q)$ its multiplication group. By Drápal’s theorem [3, Theorem 2.8] there exists a homomorphism $Λ : ℒ \to I (Q)$ determined by $L_{x} \to R_{x}^{- 1} L_{x}$ . In this short note we examine different possible extensions of this $Λ$ and the uniqueness of these extensions.

Homotopy decompositions of orbit spaces and the Webb conjecture

Jolanta Słomińska (2001)

Fundamenta Mathematicae

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Let p be a prime number. We prove that if G is a compact Lie group with a non-trivial p-subgroup, then the orbit space $(B_{p} (G)) / G$ of the classifying space of the category associated to the G-poset $_{p} (G)$ of all non-trivial elementary abelian p-subgroups of G is contractible. This gives, for every G-CW-complex X each of whose isotropy groups contains a non-trivial p-subgroup, a decomposition of X/G as a homotopy colimit of the functor $X^{E ₙ} / (N E ₀ \cap . . . \cap N E ₙ)$ defined over the poset $(s d_{p} (G)) / G$ , where sd is the barycentric subdivision....

Strong density for higher order Sobolev spaces into compact manifolds

Pierre Bousquet, Augusto C. Ponce, Jean Van Schaftingen (2015)

Journal of the European Mathematical Society

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Given a compact manifold $N^{n}$ , an integer $k \in ℕ_{*}$ and an exponent $1 \leq p < \infty$ , we prove that the class $C^{\infty} ({\bar{Q}}^{m}; N^{n})$ of smooth maps on the cube with values into $N^{n}$ is dense with respect to the strong topology in the Sobolev space $W^{k, p} (Q^{m}; N^{n})$ when the homotopy group $π_{⌊ k p ⌋} (N^{n})$ of order $⌊ k p ⌋$ is trivial. We also prove density of maps that are smooth except for a set of dimension $m - ⌊ k p ⌋ - 1$ , without any restriction on the homotopy group of $N^{n}$ .

Real algebraic actions on projective spaces - A survey

Ted Petrie (1973)

Annales de l'institut Fourier

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Let $G$ be a compact lie group. We introduce the set $S_{G} (Y)$ for every smooth $G$ manifold $Y$ . It consists of equivalence classes of pair $(X, f)$ where $f : X \to Y$ is a $G$ map which defines a homotopy equivalence from $X$ to $Y$ . Two pairs $(X_{i}, f_{i})$ , for $i = 0, 1$ , are equivalent if there is a $G$ homotopy equivalence $φ : X_{0} \to X_{1}$ such that $f_{0}$ is $G$ homotopic to $f_{1} \circ φ$ . Properties of the set $S_{G} (Y)$ and related to the representation of $G$ on the tangent spaces of $X$ and $Y$ at the fixed points. For the case $G = S^{1}$ and $Y$ is the $S^{1}$ manifold defined by a “linear”...

On finite commutative loops which are centrally nilpotent

Emma Leppälä, Markku Niemenmaa (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let $Q$ be a finite commutative loop and let the inner mapping group $I (Q) ≅ C_{p^{n}} \times C_{p^{n}}$ , where $p$ is an odd prime number and $n \geq 1$ . We show that $Q$ is centrally nilpotent of class two.