Spectral sequences associated to deformations

Herscovich, Estanislao

doi:10.1007/s40062-016-0137-z

Cited by 3 publications

(2 citation statements)

References 18 publications

(33 reference statements)

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“…First we recall the definition of filtered A ∞ -algebras and their morphisms. Filtered A ∞ -algebras and their associated spectral sequences have been previously studied in [Lap03], [Lap08] and [Her16].…”

Section: 5mentioning

confidence: 99%

Derived A-infinity algebras and their homotopies

Cirici

Santander

Livernet

et al. 2018

Topology and its Applications

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The notion of a derived A-infinity algebra, considered by Sagave, is a generalization of the classical notion of A-infinity algebra, relevant to the case where one works over a commutative ring rather than a field. We initiate a study of the homotopy theory of these algebras, by introducing a hierarchy of notions of homotopy between the morphisms of such algebras. We define r-homotopy, for non-negative integers r, in such a way that r-homotopy equivalences underlie Er-quasi-isomorphisms, defined via an associated spectral sequence. We study the special case of twisted complexes (also known as multicomplexes) first since it is of independent interest and this simpler case clearly exemplifies the structure we study. We also give two new interpretations of derived A-infinity algebras as A-infinity algebras in twisted complexes and as A-infinity algebras in split filtered cochain complexes.

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Section: 5mentioning

confidence: 99%

Derived A-infinity algebras and their homotopies

Cirici

Santander

Livernet

et al. 2018

Topology and its Applications

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“…The interaction between spectral sequences and A ∞ -structures was previously studied by Lapin (see [14] and related works) and Herscovich [11] via formal deformation theory. Here we propose a different strategy, mimicking the approach of Markl [17].…”

Section: Introductionmentioning

confidence: 97%

Filtered 𝐴-infinity structures in complex geometry

Cirici

Sopena

2022

Proc. Amer. Math. Soc.

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A higher homotopic extension of persistent (co)homology

Herscovich

2017

J. Homotopy Relat. Struct.

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Our objective in this article is to show a possibly interesting structure of homotopic nature appearing in persistent (co)homology. Assuming that the filtration of the (say) simplicial set embedded in R n induces a multiplicative filtration (which would not be a so harsh hypothesis in our setting) on the dg algebra given by the complex of simplicial cochains, we may use a result by T. Kadeishvili to get a unique (up to noncanonical equivalence) A∞-algebra structure on the complete persistent cohomology of the filtered simplicial (or topological) set. We then provide a construction of a (pseudo)metric on the set of all (generalized) barcodes (that is, of all cohomological degrees) enriched with the A∞-algebra structure stated before, refining the usual bottleneck metric, and which is also independent of the particular A∞-algebra structure chosen (among those equivalent to each other). We think that this distance might deserve some attention for topological data analysis, for it in particular can recognize different linking or foldings patterns, as in the Borromean rings. As an aside, we give a simple proof of a result relating the barcode structure between persistent homology and cohomology. This result was observed in [9] under some restricted assumptions, which we do not suppose.Mathematics subject classification 2010: 16E45, 16W70, 18G55, 55U10, 68U05.

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Spectral sequences associated to deformations

Cited by 3 publications

References 18 publications

Derived A-infinity algebras and their homotopies

Derived A-infinity algebras and their homotopies

Filtered 𝐴-infinity structures in complex geometry

A higher homotopic extension of persistent (co)homology

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