Displaying similar documents to “A faithful linear-categorical action of the mapping class group of a surface with boundary”

Noncommutative Hodge-to-de Rham spectral sequence and the Heegaard Floer homology of double covers

Robert Lipshitz, David Treumann (2016)

Journal of the European Mathematical Society

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Let A be a dg algebra over 𝔽 2 and let M be a dg A -bimodule. We show that under certain technical hypotheses on A , a noncommutative analog of the Hodge-to-de Rham spectral sequence starts at the Hochschild homology of the derived tensor product M A L M and converges to the Hochschild homology of M . We apply this result to bordered Heegaard Floer theory, giving spectral sequences associated to Heegaard Floer homology groups of certain branched and unbranched double covers.

Homological computations in the universal Steenrod algebra

A. Ciampella, L. A. Lomonaco (2004)

Fundamenta Mathematicae

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We study the (bigraded) homology of the universal Steenrod algebra Q over the prime field ₂, and we compute the groups H s , s ( Q ) , s ≥ 0, using some ideas and techniques of Koszul algebras developed by S. Priddy in [5], although we presently do not know whether or not Q is a Koszul algebra. We also provide an explicit formula for the coalgebra structure of the diagonal homology D ( Q ) = s 0 H s , s ( Q ) and show that D⁎(Q) is isomorphic to the coalgebra of invariants Γ introduced by W. Singer in [6].

On the geometry of polynomial mappings at infinity

Anna Valette, Guillaume Valette (2014)

Annales de l’institut Fourier

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We associate to a given polynomial map from 2 to itself with nonvanishing Jacobian a variety whose homology or intersection homology describes the geometry of singularities at infinity of this map.

Steenrod homology

Yu. T. Lisitsa, S. Mardešić (1986)

Banach Center Publications

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A note on product structures on Hochschild homology of schemes

Abhishek Banerjee (2011)

Colloquium Mathematicae

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We extend the definition of Hochschild and cyclic homologies of a scheme over a commutative ring k to define the Hochschild homologies HH⁎(X/S) and cyclic homologies HC⁎(X/S) of a scheme X with respect to an arbitrary base scheme S. Our main purpose is to study product structures on the Hochschild homology groups HH⁎(X/S). In particular, we show that H H ( X / S ) = n H H ( X / S ) carries the structure of a graded algebra.

Determinantal Barlow surfaces and phantom categories

Christian Böhning, Hans-Christian Graf von Bothmer, Ludmil Katzarkov, Pawel Sosna (2015)

Journal of the European Mathematical Society

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We prove that the bounded derived category of the surface S constructed by Barlow admits a length 11 exceptional sequence consisting of (explicit) line bundles. Moreover, we show that in a small neighbourhood of S in the moduli space of determinantal Barlow surfaces, the generic surface has a semiorthogonal decomposition of its derived category into a length 11 exceptional sequence of line bundles and a category with trivial Grothendieck group and Hochschild homology, called a phantom...

A 2-category of chronological cobordisms and odd Khovanov homology

Krzysztof K. Putyra (2014)

Banach Center Publications

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We create a framework for odd Khovanov homology in the spirit of Bar-Natan's construction for the ordinary Khovanov homology. Namely, we express the cube of resolutions of a link diagram as a diagram in a certain 2-category of chronological cobordisms and show that it is 2-commutative: the composition of 2-morphisms along any 3-dimensional subcube is trivial. This allows us to create a chain complex whose homotopy type modulo certain relations is a link invariant. Both the original and...

Torsion of Khovanov homology

Alexander N. Shumakovitch (2014)

Fundamenta Mathematicae

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Khovanov homology is a recently introduced invariant of oriented links in ℝ³. It categorifies the Jones polynomial in the sense that the (graded) Euler characteristic of Khovanov homology is a version of the Jones polynomial for links. In this paper we study torsion of Khovanov homology. Based on our calculations, we formulate several conjectures about the torsion and prove weaker versions of the first two of them. In particular, we prove that all non-split alternating links have their...

Waldhausen’s Nil groups and continuously controlled K-theory

Hans Munkholm, Stratos Prassidis (1999)

Fundamenta Mathematicae

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Let Γ = Γ 1 * G Γ 2 be the pushout of two groups Γ i , i = 1,2, over a common subgroup G, and H be the double mapping cylinder of the corresponding diagram of classifying spaces B Γ 1 B G B Γ 2 . Denote by ξ the diagram I p H 1 X = H , where p is the natural map onto the unit interval. We show that the N i l groups which occur in Waldhausen’s description of K * ( Γ ) coincide with the continuously controlled groups * c c ( ξ ) , defined by Anderson and Munkholm. This also allows us to identify the continuously controlled groups * c c ( ξ + ) which are known to form...

Injective comodules and Landweber exact homology theories

Mark Hovey (2007)

Fundamenta Mathematicae

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We classify the indecomposable injective E(n)⁎E(n)-comodules, where E(n) is the Johnson-Wilson homology theory. They are suspensions of the J n , r = E ( n ) ( M r E ( r ) ) , where 0 ≤ r ≤ n, with the endomorphism ring of J n , r being E ( r ) ^ * E ( r ) ^ , where E ( r ) ^ denotes the completion of E(r).

Homology of representable sets

Marian Mrozek, Bogdan Batko (2010)

Annales Polonici Mathematici

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We generalize the notion of cubical homology to the class of locally compact representable sets in order to propose a new convenient method of reducing the complexity of a set while computing its homology.

Transverse Homology Groups

S. Dragotti, G. Magro, L. Parlato (2006)

Bollettino dell'Unione Matematica Italiana

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We give, here, a geometric treatment of intersection homology theory.

A computation in Khovanov-Rozansky homology

Daniel Krasner (2009)

Fundamenta Mathematicae

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We investigate the Khovanov-Rozansky invariant of a certain tangle and its compositions. Surprisingly the complexes we encounter reduce to ones that are very simple. Furthermore, we discuss a "local" algorithm for computing Khovanov-Rozansky homology and compare our results with those for the "foam" version of sl₃-homology.

Effective homology for homotopy colimit and cofibrant replacement

Marek Filakovský (2014)

Archivum Mathematicum

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We extend the notion of simplicial set with effective homology presented in [22] to diagrams of simplicial sets. Further, for a given finite diagram of simplicial sets X : sSet such that each simplicial set X ( i ) has effective homology, we present an algorithm computing the homotopy colimit hocolim X as a simplicial set with effective homology. We also give an algorithm computing the cofibrant replacement X cof of X as a diagram with effective homology. This is applied to computing of equivariant cohomology...

Introduction to the basics of Heegaard Floer homology

Bijan Sahamie (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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This paper provides an introduction to the basics of Heegaard Floer homology with some emphasis on the hat theory and to the contact geometric invariants in the theory. The exposition is designed to be comprehensible to people without any prior knowledge of the subject.

Reeb vector fields and open book decompositions

Vincent Colin, Ko Honda (2013)

Journal of the European Mathematical Society

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We determine parts of the contact homology of certain contact 3-manifolds in the framework of open book decompositions, due to Giroux.We study two cases: when the monodromy map of the compatible open book is periodic and when it is pseudo-Anosov. For an open book with periodic monodromy, we verify the Weinstein conjecture. In the case of an open book with pseudo-Anosov monodromy, suppose the boundary of a page of the open book is connected and the fractional Dehn twist coefficient c ...

A note on singular homology groups of infinite products of compacta

Kazuhiro Kawamura (2002)

Fundamenta Mathematicae

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Let n be an integer with n ≥ 2 and X i be an infinite collection of (n-1)-connected continua. We compare the homotopy groups of Σ ( i X i ) with those of i Σ X i (Σ denotes the unreduced suspension) via the Freudenthal Suspension Theorem. An application to homology groups of the countable product of the n(≥ 2)-sphere is given.