Differential forms and torsion in the fundamental group

Cenkl, Bohumil; Porter, Richard D.

doi:10.1016/0001-8708(83)90089-0

Cited by 9 publications

(6 citation statements)

References 11 publications

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“…Building on this work, Dwyer [13] related Massey products in the cohomology of a group to properties of the quotients in the lower central series and also to a mod p central series different than the one used by Stallings. In [5,6,7], Cenkl and Porter used a commutative algebra of differential forms to model tame homotopy theory and the Lazard Lie algebra completion of the fundamental group.…”

Section: 3mentioning

confidence: 99%

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Homology, lower central series, and hyperplane arrangements

Porter

Suciu

2019

European Journal of Mathematics

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We explore finitely generated groups by studying the nilpotent towers and the various Lie algebras attached to such groups. Our main goal is to relate an isomorphism extension problem in the Postnikov tower to the existence of certain commuting diagrams. This recasts a result of G. Rybnikov in a more general framework and leads to an application to hyperplane arrangements, whereby we show that all the nilpotent quotients of a decomposable arrangement group are combinatorially determined.

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Section: 3mentioning

confidence: 99%

“…Hence, by Corollary 5.2, the extension (5.6) is classified by the k-invariant χ 2 = id ∈ Hom( 2 Z k , 2 Z k ). 6. Generalizations of Rybnikov's Theorem 6.1.…”

Section: Second Cohomology Of Nilpotent Groups and Associated K-invarmentioning

confidence: 99%

Homology, lower central series, and hyperplane arrangements

Porter

Suciu

2019

European Journal of Mathematics

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“…Dwyer [81,1979] proved that a version of Quillen's rational homotopy theory extends to the homotopy theory of tame spaces X which are (r − 1)-connected CW complexes, r ≥ 3, with π r+k (X) uniquely p-divisible for all primes p with 2p − 3 ≤ k. He proved the homotopy category of tame spaces endowed with an appropriate model structure is equivalent to a homotopy category of (r − 1)reduced integral DGLAs. Tame homotopy theory admits a version in the spirit of Sullivan's approach to rational homotopy theory by Cenkl-Porter [49,1983]. Scheerer-Tanré [239,1988] identified homotopy invariants, e.g., homology and the homotopy Lie algebra, in Dwyer's framework.…”

Section: Rational Homotopy Theory Of Function Spaces Quillenmentioning

confidence: 99%

The homotopy theory of function spaces: a survey

Smith¹

2010

Contemporary Mathematics

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We survey research on the homotopy theory of the space map(X, Y ) consisting of all continuous functions between two topological spaces. We summarize progress on various classification problems for the homotopy types represented by the path-components of map(X, Y ). We also discuss work on the homotopy theory of the monoid of self-equivalences aut (X) and of the free loop space LX. We consider these topics in both ordinary homotopy theory as well as after localization. In the latter case, we discuss algebraic models for the localization of function spaces and their applications.2010 Mathematics Subject Classification.

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“…Cenkl and Porter [6,7] constructed the so-called tame de Rham complex of polynomial forms T * (X, Z) with filtration T * ,q (X) ⊂ T * ,q+1 (X), depending on the degree of the polynomials and the forms, such that T * ,q (X) is a Q q -module (Q 0 = Q 1 = Z and Q q = Z[1/2,...,1/q] for q > 1). They proved that there exists an isomorphism I : H i (T * ,q (X; Q q )) → H i (X; Q q ) which is induced by integration of forms for all q ≥ 1 and for all i.…”

Section: Introduction Sullivanmentioning

confidence: 99%

“…Karoubi defined a noncommutative de Rham complex Ω(X) and proved the noncommutative de Rham theorem for a simplicial space X. A slightly more general version of the noncommutative de Rham theorem was proved by Cenkl in [4,6]. Both proofs are functorial and in principle are based on the idea of Cartan.…”

Section: Introduction Sullivanmentioning

confidence: 99%

The de Rham theorem for the noncommutative complex of Cenkl and Porter

Mejias

2002

International Journal of Mathematics and Mathematical Sciences

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We use noncommutative differential forms (which were first introduced by Connes) to construct a noncommutative version of the complex of Cenkl and PorterΩ∗,∗(X)for a simplicial setX. The algebraΩ∗,∗(X)is a differential graded algebra with a filtrationΩ∗,q(X)⊂Ω∗,q+1(X), such thatΩ∗,q(X)is aℚq-module, whereℚ0=ℚ1=ℤandℚq=ℤ[1/2,…,1/q]forq>1. Then we use noncommutative versions of the Poincaré lemma and Stokes' theorem to prove the noncommutative tame de Rham theorem: ifXis a simplicial set of finite type, then for eachq≥1and anyℚq-moduleM, integration of forms induces a natural isomorphism ofℚq-modulesI:Hi(Ω∗,q(X),M)→Hi(X;M)for alli≥0. Next, we introduce a complex of noncommutative tame de Rham currentsΩ∗,∗(X)and we prove the noncommutative tame de Rham theorem for homology: ifXis a simplicial set of finite type, then for eachq≥1and anyℚq-moduleM, there is a natural isomorphism ofℚq-modulesI:Hi(X;M)→Hi(Ω∗,q(X),M)for alli≥0.

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Differential forms and torsion in the fundamental group

Cited by 9 publications

References 11 publications

Homology, lower central series, and hyperplane arrangements

Homology, lower central series, and hyperplane arrangements

The homotopy theory of function spaces: a survey

The de Rham theorem for the noncommutative complex of Cenkl and Porter

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