Search results for "Mathematics::Algebraic Topology"
showing 10 items of 65 documents
The homotopy Leray spectral sequence
2018
In this work, we build a spectral sequence in motivic homotopy that is analogous to both the Serre spectral sequence in algebraic topology and the Leray spectral sequence in algebraic geometry. Here, we focus on laying the foundations necessary to build the spectral sequence and give a convenient description of its $E_2$-page. Our description of the $E_2$-page is in terms of homology of the local system of fibers, which is given using a theory similar to Rost's cycle modules. We close by providing some sample applications of the spectral sequence and some hints at future work.
Hochschild Cohomology Theories in White Noise Analysis
2008
We show that the continuous Hochschild cohomology and the differential Hochschild cohomology of the Hida test algebra endowed with the normalized Wick product are the same.
Homeomorphic graph manifolds: A contribution to the μ constant problem
1999
Abstract We give a characterization, in terms of homological data in covering spaces, of those maps between (3-dimensional) graph manifolds which are homotopic to homeomorphisms. As an application we give a condition on a cobordism between graph manifolds that guarantees that they are homeomorphic. This in turn is applied to give a partial result on the μ -constant problem in (complex) dimension three.
Beilinson Motives and Algebraic K-Theory
2019
Section 12 is a recollection on the basic results of stable homotopy theory of schemes, after Morel and Voevodsky. In particular, we recall the theory of orientations in a motivic cohomology theory. Section 13 is a recollection of the fundamental results on algebraic K-theory which we translate into results within stable homotopy theory of schemes. In particular, Quillen’s localization theorem is seen as an absolute purity theory for the K-theory spectrum. In Section 14, we introduce the fibred category of Beilinson motives as an appropriate Verdier quotient of the motivic stable homotopy category. Using the Adams filtration on K-theory, we prove that Beilinson motives have the properties o…
Multiple Zeta Values
2017
We study in some detail the very important class of periods called multiple zeta values (MZV). These are periods of mixed Tate motives, which we discussed in Sect. 6.4. Multiple zeta values are in fact periods of unramified mixed Tate motives, a full subcategory of all mixed Tate motives.
Vassiliev invariants for braids on surfaces
2000
We show that Vassiliev invariants separate braids on a closed oriented surface, and we exhibit an universal Vassiliev invariant for these braids in terms of chord diagrams labeled by elements of the fundamental group of the considered surface.
Automorphisms of $mathbb{A}^{1}$-fibered affine surfaces
2011
We develop technics of birational geometry to study automorphisms of affine surfaces admitting many distinct rational fibrations, with a particular focus on the interactions between automorphisms and these fibrations. In particular, we associate to each surface S of this type a graph encoding equivalence classes of rational fibrations from which it is possible to decide for instance if the automorphism group of S is generated by automorphisms preserving these fibrations.
Data structures and algorithms for topological analysis
2014
International audience; One of the steps of geometric modeling is to know the topology and/or the geometry of the objects considered. This paper presents different data structures and algorithms used in this study. We are particularly interested by algebraic structures, eg homotopy and homology groups, the Betti numbers, the Euler characteristic, or the Morse-Smale complex. We have to be able to compute these data structures, and for (homotopy and homology) groups, we also want to compute their generators. We are also interested in algorithms CIA and HIA presented in the thesis of Nicolas DELANOUE, which respectively compute the connected components and the homotopy type of a set defined by…
THE HOMOLOGY OF DIGRAPHS AS A GENERALIZATION OF HOCHSCHILD HOMOLOGY
2010
J. Przytycki has established a connection between the Hochschild homology of an algebra $A$ and the chromatic graph homology of a polygon graph with coefficients in $A$. In general the chromatic graph homology is not defined in the case where the coefficient ring is a non-commutative algebra. In this paper we define a new homology theory for directed graphs which takes coefficients in an arbitrary $A-A$ bimodule, for $A$ possibly non-commutative, which on polygons agrees with Hochschild homology through a range of dimensions.
Stable motivic homotopy theory at infinity
2021
In this paper, we initiate a study of motivic homotopy theory at infinity. We use the six functor formalism to give an intrinsic definition of the stable motivic homotopy type at infinity of an algebraic variety. Our main computational tools include cdh-descent for normal crossing divisors, Euler classes, Gysin maps, and homotopy purity. Under $\ell$-adic realization, the motive at infinity recovers a formula for vanishing cycles due to Rapoport-Zink; similar results hold for Steenbrink's limiting Hodge structures and Wildeshaus' boundary motives. Under the topological Betti realization, the stable motivic homotopy type at infinity of an algebraic variety recovers the singular complex at in…