Cohomology $C_{∞}$-algebra and rational homotopy type

Tornike Kadeishvili

Displaying similar documents to “Cohomology $C_{\infty}$ -algebra and rational homotopy type”

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...

Unstable homotopy invariance for finite fields

Kevin P. Knudson (2002)

Fundamenta Mathematicae

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We prove that if k is a finite field with $p^{d}$ elements, then the natural map $H_{i} (G L ₙ (k), ℤ) \to H_{i} (G L ₙ (k [t]), ℤ)$ is an isomorphism for 0 ≤ i < d(p-1) and for all n.

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).

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...

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.

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...

A note on Sierpiński's problem related to triangular numbers

Maciej Ulas (2009)

Colloquium Mathematicae

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We show that the system of equations $t_{x} + t_{y} = t_{p}, t_{y} + t_{z} = t_{q}, t_{x} + t_{z} = t_{r}$ , where $t_{x} = x (x + 1) / 2$ is a triangular number, has infinitely many solutions in integers. Moreover, we show that this system has a rational three-parameter solution. Using this result we show that the system $t_{x} + t_{y} = t_{p}, t_{y} + t_{z} = t_{q}, t_{x} + t_{z} = t_{r}, t_{x} + t_{y} + t_{z} = t_{s}$ has infinitely many rational two-parameter solutions.

Various subsemirings of the field $ℚ$ of rational numbers

Vítězslav Kala, Tomáš Kepka, Jon D. Phillips (2009)

Acta Universitatis Carolinae. Mathematica et Physica

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Overconvergent de Rham-Witt cohomology

Christopher Davis, Andreas Langer, Thomas Zink (2011)

Annales scientifiques de l'École Normale Supérieure

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The goal of this work is to construct, for a smooth variety $X$ over a perfect field k of finite characteristic $p > 0$ , an overconvergent de Rham-Witt complex $W^{†} Ω_{X / k}$ as a suitable subcomplex of the de Rham-Witt complex of Deligne-Illusie. This complex, which is functorial in $X$ , is a complex of étale sheaves and a differential graded algebra over the ring $W^{†} (𝒪_{X})$ of overconvergent Witt-vectors. If $X$ is affine one proves that there is an isomorphism between Monsky-Washnitzer cohomology and (rational) overconvergent...

$L^{2}$ -cohomology

J. Eichhorn (1986)

Banach Center Publications

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Rational points on curves

Michael Stoll (2011)

Journal de Théorie des Nombres de Bordeaux

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This is an extended version of an invited lecture I gave at the Journées Arithmétiques in St. Étienne in July 2009. We discuss the state of the art regarding the problem of finding the set of rational points on a (smooth projective) geometrically integral curve $C$ over $ℚ$ . The focus is on practical aspects of this problem in the case that the genus of $C$ is at least $2$ , and therefore the set of rational points is finite.

Composite rational functions expressible with few terms

Clemens Fuchs, Umberto Zannier (2012)

Journal of the European Mathematical Society

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We consider a rational function $f$ which is ‘lacunary’ in the sense that it can be expressed as the ratio of two polynomials (not necessarily coprime) having each at most a given number $ℓ$ of terms. Then we look at the possible decompositions $f (x) = g (h (x))$ , where $g, h$ are rational functions of degree larger than 1. We prove that, apart from certain exceptional cases which we completely describe, the degree of $g$ is bounded only in terms of $ℓ$ (and we provide explicit bounds). This supports and quantifies...

Noetherian loop spaces

Natàlia Castellana, Juan Crespo, Jérôme Scherer (2011)

Journal of the European Mathematical Society

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The class of loop spaces of which the mod $p$ cohomology is Noetherian is much larger than the class of $p$ -compact groups (for which the mod $p$ cohomology is required to be finite). It contains Eilenberg–Mac Lane spaces such as $ℂ P^{\infty}$ and 3-connected covers of compact Lie groups. We study the cohomology of the classifying space $B X$ of such an object and prove it is as small as expected, that is, comparable to that of $B ℂ P^{\infty}$ . We also show that $B$ X differs basically from the classifying space of a $p$ -compact...

Algebraic homotopy classes of rational functions

Christophe Cazanave (2012)

Annales scientifiques de l'École Normale Supérieure

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Let $k$ be a field. We compute the set ${[𝐏^{1}, 𝐏^{1}]}^{N}$ ofhomotopy classes of pointed $k$ -scheme endomorphisms of the projective line $𝐏^{1}$ . Our result compares well with Morel’s computation in [11] of the ${[𝐏^{1}, 𝐏^{1}]}^{𝐀^{1}}$ of $𝐀^{1}$ -homotopy classes of pointed endomorphisms of $𝐏^{1}$ : the set ${[𝐏^{1}, 𝐏^{1}]}^{N}$ admits an a priori monoid structure such that the canonical map ${[𝐏^{1}, 𝐏^{1}]}^{N} \to {[𝐏^{1}, 𝐏^{1}]}^{𝐀^{1}}$ is a group completion.

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.

Characteristic Exponents of Rational Functions

Anna Zdunik (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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We consider two characteristic exponents of a rational function f:ℂ̂ → ℂ̂ of degree d ≥ 2. The exponent $χ_{a} (f)$ is the average of log∥f’∥ with respect to the measure of maximal entropy. The exponent $χ_{m} (f)$ can be defined as the maximal characteristic exponent over all periodic orbits of f. We prove that $χ_{a} (f) = χ_{m} (f)$ if and only if f(z) is conformally conjugate to $z \mapsto z^{\pm d}$ .

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 $N^{n}$ .

Rational BV-algebra in string topology

Yves Félix, Jean-Claude Thomas (2008)

Bulletin de la Société Mathématique de France

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Let $M$ be a 1-connected closed manifold of dimension $m$ and $L M$ be the space of free loops on $M$ . M.Chas and D.Sullivan defined a structure of BV-algebra on the singular homology of $L M$ , $H_{*} (L M; k)$ . When the ring of coefficients is a field of characteristic zero, we prove that there exists a BV-algebra structure on the Hochschild cohomology $H H^{*} (C^{*} (M); C^{*} (M))$ which extends the canonical structure of Gerstenhaber algebra. We construct then an isomorphism of BV-algebras between $H H^{*} (C^{*} (M); C^{*} (M))$ and the shifted homology $H_{* + m} (L M; k)$ . We also prove...

Manin’s and Peyre’s conjectures on rational points and adelic mixing

Alex Gorodnik, François Maucourant, Hee Oh (2008)

Annales scientifiques de l'École Normale Supérieure

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Let $X$ be the wonderful compactification of a connected adjoint semisimple group $G$ defined over a number field $K$ . We prove Manin’s conjecture on the asymptotic (as $T \to \infty$ ) of the number of $K$ -rational points of $X$ of height less than $T$ , and give an explicit construction of a measure on $X (𝔸)$ , generalizing Peyre’s measure, which describes the asymptotic distribution of the rational points $𝐆 (K)$ on $X (𝔸)$ . Our approach is based on the mixing property of $L^{2} (𝐆 (K) ∖ 𝐆 (𝔸))$ which we obtain with a rate of convergence. ...

Displaying similar documents to “Cohomology C ∞ -algebra and rational homotopy type”

Displaying similar documents to “Cohomology $C_{\infty}$ -algebra and rational homotopy type”