A twisted tale of cochains and connections

2010 ◽  
Vol 17 (1) ◽  
pp. 203-215
Author(s):  
Jim Stasheff
Keyword(s):  
Homotopy Theory ◽  
Graded Algebra ◽  
Massey Products ◽  
Differential Graded Algebra ◽  
Personal Tribute ◽  
History Of ◽  
Homotopy Algebra ◽  
Homotopy Invariants

Abstract Early in the history of higher homotopy algebra [Stasheff, Trans. Am. Math. Soc. 108: 293–312, 1963], it was realized that Massey products are homotopy invariants in a special sense, but it was the work of Tornike Kadeishvili that showed they were but a shadow of an 𝐴∞-structure on the homology of a differential graded algebra. Here we relate his work to that of Victor Gugenheim [J. Pure Appl. Algebra 25: 197–205, 1982] and K. T. (Chester) Chen [Ann. of Math. (2) 97: 217–246, 1973]. This paper is a personal tribute to Tornike and the Georgian school of homotopy theory as well as to Gugenheim and Chen, who unfortunately are not with us to appreciate this convergence.

scholarly journals A strictly commutative model for the cochain algebra of a space

2020 ◽  
Vol 156 (8) ◽  
pp. 1718-1743
Author(s):  
Birgit Richter ◽  
Steffen Sagave
Keyword(s):  
Homotopy Theory ◽  
Rational Homotopy ◽  
Graded Algebras ◽  
Differential Graded Algebra ◽  
Simplicial Sets ◽  
Differential Graded Algebras ◽  
Chain Complexes ◽  
Simplicial Set ◽  
New Construction ◽  
Integral Version

AbstractThe commutative differential graded algebra $A_{\mathrm {PL}}(X)$ of polynomial forms on a simplicial set $X$ is a crucial tool in rational homotopy theory. In this note, we construct an integral version $A^{\mathcal {I}}(X)$ of $A_{\mathrm {PL}}(X)$. Our approach uses diagrams of chain complexes indexed by the category of finite sets and injections $\mathcal {I}$ to model $E_{\infty }$ differential graded algebras (dga) by strictly commutative objects, called commutative $\mathcal {I}$-dgas. We define a functor $A^{\mathcal {I}}$ from simplicial sets to commutative $\mathcal {I}$-dgas and show that it is a commutative lift of the usual cochain algebra functor. In particular, it gives rise to a new construction of the $E_{\infty }$ dga of cochains. The functor $A^{\mathcal {I}}$ shares many properties of $A_{\mathrm {PL}}$, and can be viewed as a generalization of $A_{\mathrm {PL}}$ that works over arbitrary commutative ground rings. Working over the integers, a theorem by Mandell implies that $A^{\mathcal {I}}(X)$ determines the homotopy type of $X$ when $X$ is a nilpotent space of finite type.

scholarly journals An extention of Nomizu’s Theorem –A user’s guide–

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Hisashi Kasuya
Keyword(s):  
Lie Group ◽  
Simple Proof ◽  
Graded Algebra ◽  
De Rham Cohomology ◽  
Differential Graded Algebra ◽  
Finite Dimensional ◽  
De Rham ◽  
Simply Connected ◽  
Complex Valued ◽  
Solvable Lie Group

AbstractFor a simply connected solvable Lie group G with a lattice Γ, the author constructed an explicit finite-dimensional differential graded algebra A*Γ which computes the complex valued de Rham cohomology H*(Γ\G, C) of the solvmanifold Γ\G. In this note, we give a quick introduction to the construction of such A*Γ including a simple proof of H*(A*Γ) ≅ H*(Γ\G, C).

Russell McCormmach as a teacher

2007 ◽  
Vol 37 (2) ◽  
pp. 453-462
Author(s):  
Finn Aaserud
Keyword(s):  
History Of Science ◽  
Human Being ◽  
General Sense ◽  
Doctoral Research ◽  
Johns Hopkins University ◽  
Personal Tribute ◽  
History Of

The author gives a personal tribute of Russell McCormmach as a scholar and a person. From 1972 to 1976, McCormmach's writings, notably his introductions to the HSPS, served as unique inspiration for the author's .rst grapplings with the history of science in far-away Norway. From 1976 to 1984 the author was a student at Johns Hopkins University, with McCormmach as dissertation adviser until he left Hopkins in 1983. Because the doctoral research was carried out for the most part in Scandinavia, McCormmach's advice is to a great extent preserved in personal letters, which are quoted at some length. Ever since, the author and McCormmach have maintained a close, if sporadic, relationship. While his approach is personal, the author hopes to convey a general sense of McCormmach's unique qualities as a writer, editor and teacher, as well as a human being.

scholarly journals LINK MAPS IN ARBITRARY MANIFOLDS AND THEIR HOMOTOPY INVARIANTS

2003 ◽  
Vol 12 (01) ◽  
pp. 79-104 ◽  
Author(s):  
U. KOSCHORKE
Keyword(s):  
Homotopy Theory ◽  
Configuration Spaces ◽  
Higher Dimensional ◽  
Homotopy Invariants

In this paper we generalize Milnor's μ-invariants of classical links to certain ("κ-Brunnian") higher dimensional link maps into fairly arbitrary manifolds. Our approach involves the homotopy theory of configuration spaces and of wedges of spheres. We discuss the strength of these invariants and their compatibilities e.g. with (Hilton decompositions of) linking coefficients. Our results suggest, in particular, a conjecture about possible new link homotopies.

scholarly journals A COMBINATORIAL DGA FOR LEGENDRIAN KNOTS FROM GENERATING FAMILIES

2013 ◽  
Vol 15 (02) ◽  
pp. 1250059 ◽  
Author(s):  
MICHAEL B. HENRY ◽  
DAN RUTHERFORD
Keyword(s):  
Standard Form ◽  
Geometric Interpretation ◽  
Graded Algebra ◽  
Legendrian Knots ◽  
Complex Sequence ◽  
Differential Graded Algebra ◽  
Legendrian Knot ◽  
Definition Of ◽  
Morse Complex ◽  
Differential Counts

For a Legendrian knot L ⊂ ℝ3, with a chosen Morse complex sequence (MCS), we construct a differential graded algebra (DGA) whose differential counts "chord paths" in the front projection of L. The definition of the DGA is motivated by considering Morse-theoretic data from generating families. In particular, when the MCS arises from a generating family F, we give a geometric interpretation of our chord paths as certain broken gradient trajectories which we call "gradient staircases". Given two equivalent MCS's we prove the corresponding linearized complexes of the DGA are isomorphic. If the MCS has a standard form, then we show that our DGA agrees with the Chekanov–Eliashberg DGA after changing coordinates by an augmentation.

scholarly journals Invariants of Legendrian Knots in Circle Bundles

2003 ◽  
Vol 05 (04) ◽  
pp. 569-627 ◽  
Author(s):  
Joshua M. Sabloff
Keyword(s):  
Contact Structure ◽  
Homology Class ◽  
Graded Algebra ◽  
Legendrian Knots ◽  
Contact Homology ◽  
Circle Bundle ◽  
Differential Graded Algebra ◽  
Circle Bundles ◽  
Standard Contact ◽  
Definition Of

Let M be a circle bundle over a Riemann surface that supports a contact structure transverse to the fibers. This paper presents a combinatorial definition of a differential graded algebra (DGA) that is an invariant of Legendrian knots in M. The invariant generalizes Chekanov's combinatorial DGA invariant of Legendrian knots in the standard contact 3-space using ideas from Eliashberg, Givental, and Hofer's contact homology. The main difficulty lies in dealing with what are ostensibly 1-parameter families of generators for the DGA; these are solved using "Morse–Bott" techniques. As an application, the invariant is used to distinguish two Legendrian knots that are smoothly isotopic, realize a nontrivial homology class, but are not Legendrian isotopic.

Differential Graded Algebras

2020 ◽  
pp. 145-150
Author(s):  
Loring W. Tu
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Algebraic Model ◽  
Lie Derivative ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Differential Graded Algebra ◽  
Differential Graded Algebras ◽  
Rham Complex ◽  
De Rham

This chapter investigates differential graded algebras. Throughout the chapter, G will be a Lie group with Lie algebra g. On a manifold M, the de Rham complex is a differential graded algebra, a graded algebra that is also a differential complex. If the Lie group G acts smoothly on M, then the de Rham complex Ω‎(M) is more than a differential graded algebra. It has in addition two actions of the Lie algebra: interior multiplication and the Lie derivative. A differential graded algebra Ω‎ with an interior multiplication and a Lie derivative satisfying Cartan's homotopy formula is called a g-differential graded algebra. To construct an algebraic model for equivariant cohomology, the chapter first constructs an algebraic model for the total space EG of the universal G-bundle. It is a g-differential graded algebra called the Weil algebra.

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

2002 ◽  
Vol 30 (11) ◽  
pp. 667-696 ◽  
Author(s):  
Luis Fernando Mejias
Keyword(s):  
Finite Type ◽  
Differential Forms ◽  
Natural Isomorphism ◽  
Graded Algebra ◽  
Differential Graded Algebra ◽  
Noncommutative Version ◽  
Simplicial Set ◽  
De Rham ◽  
Poincaré Lemma

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.

scholarly journals (q,σ,τ)-Differential Graded Algebras

Universe ◽  
2018 ◽  
Vol 4 (12) ◽  
pp. 138 ◽  
Author(s):  
Viktor Abramov ◽  
Olga Liivapuu ◽  
Abdenacer Makhlouf
Keyword(s):  
Clifford Algebra ◽  
Quantum Plane ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Differential Graded Algebra ◽  
Differential Graded Algebras ◽  
Generalized Clifford Algebra

We propose the notion of ( q , σ , τ ) -differential graded algebra, which generalizes the notions of ( σ , τ ) -differential graded algebra and q-differential graded algebra. We construct two examples of ( q , σ , τ ) -differential graded algebra, where the first one is constructed by means of the generalized Clifford algebra with two generators (reduced quantum plane), where we use a ( σ , τ ) -twisted graded q-commutator. In order to construct the second example, we introduce the notion of ( σ , τ ) -pre-cosimplicial algebra.

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