scholarly journals GENERATORS, RELATIONS, AND HOMOLOGY FOR OZSVÁTH–SZABÓ’S KAUFFMAN-STATES ALGEBRAS

2020 ◽  
pp. 1-59
Author(s):  
ANDREW MANION ◽  
MARCO MARENGON ◽  
MICHAEL WILLIS
Keyword(s):  
Floer Homology ◽  
Graded Algebras ◽  
Generators And Relations ◽  
Knot Floer Homology ◽  
Differential Graded Algebras

We give a generators-and-relations description of differential graded algebras recently introduced by Ozsváth and Szabó for the computation of knot Floer homology. We also compute the homology of these algebras and determine when they are formal.

scholarly journals Homological epimorphisms, homotopy epimorphisms and acyclic maps

2020 ◽  
Vol 32 (6) ◽  
pp. 1395-1406
Author(s):  
Joseph Chuang ◽  
Andrey Lazarev
Keyword(s):  
Wide Class ◽  
Topological Spaces ◽  
Loop Spaces ◽  
Graded Algebras ◽  
Differential Graded Algebras ◽  
Algebraic Description ◽  
Acyclic Map ◽  
Induced Maps

AbstractWe show that the notions of homotopy epimorphism and homological epimorphism in the category of differential graded algebras are equivalent. As an application we obtain a characterization of acyclic maps of topological spaces in terms of induced maps of their chain algebras of based loop spaces. In the case of a universal acyclic map we obtain, for a wide class of spaces, an explicit algebraic description for these induced maps in terms of derived localization.

scholarly journals COMBINATORIAL KNOT FLOER HOMOLOGY AND DOUBLE BRANCHED COVERS

2013 ◽  
Vol 22 (06) ◽  
pp. 1350014
Author(s):  
FATEMEH DOUROUDIAN
Keyword(s):  
Floer Homology ◽  
Combinatorial Proof ◽  
Branched Covers ◽  
Knot Floer Homology ◽  
Branched Cover ◽  
Heegaard Diagram

Using a Heegaard diagram for the pullback of a knot K ⊂ S3 in its double branched cover Σ2(K), we give a combinatorial proof for the invariance of the associated knot Floer homology over ℤ.

scholarly journals An unoriented skein exact triangle for knot Floer homology

2007 ◽  
Vol 14 (5) ◽  
pp. 839-852 ◽  
Author(s):  
Ciprian Manolescu
Keyword(s):  
Floer Homology ◽  
Knot Floer Homology ◽  
Exact Triangle

Circle Actions

2020 ◽  
pp. 161-166
Author(s):  
Loring W. Tu
Keyword(s):  
Differential Form ◽  
Circle Action ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Circle Actions ◽  
Algebra Isomorphism ◽  
Weil Algebra ◽  
Differential Graded Algebras ◽  
Cartan Model

This chapter focuses on circle actions. Specifically, it specializes the Weil algebra and the Weil model to a circle action. In this case, all the formulas simplify. The chapter derives a simpler complex, called the Cartan model, which is isomorphic to the Weil model as differential graded algebras. It considers the theorem that for a circle action, there is a graded-algebra isomorphism. Under the isomorphism F, the Weil differential δ‎ corresponds to a differential called the Cartan differential. An element of the Cartan model is called an equivariant differential form or equivariant form for a circle action on the manifold M.

scholarly journals Topological equivalences of E-infinity differential graded algebras

2018 ◽  
Vol 18 (2) ◽  
pp. 1115-1146
Author(s):  
Haldun Özgür Bayındır
Keyword(s):  
Graded Algebras ◽  
Differential Graded Algebras

scholarly journals Knots which admit a surgery with simple knot Floer homology groups

2011 ◽  
Vol 11 (3) ◽  
pp. 1243-1256 ◽  
Author(s):  
Eaman Eftekhary
Keyword(s):  
Floer Homology ◽  
Homology Groups ◽  
Knot Floer Homology

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 introduction to knot Floer homology and curved bordered algebras

2020 ◽  
Vol 80 (2) ◽  
pp. 211-236
Author(s):  
Antonio Alfieri ◽  
Jackson Van Dyke
Keyword(s):  
Floer Homology ◽  
Knot Floer Homology
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