Search results for "Singular homology"

showing 4 items of 4 documents

Real quadrics in C n , complex manifolds and convex polytopes

2006

In this paper, we investigate the topology of a class of non-Kähler compact complex manifolds generalizing that of Hopf and Calabi-Eckmann manifolds. These manifolds are diffeomorphic to special systems of real quadrics Cn which are invariant with respect to the natural action of the real torus (S1)n onto Cn. The quotient space is a simple convex polytope. The problem reduces thus to the study of the topology of certain real algebraic sets and can be handled using combinatorial results on convex polytopes. We prove that the homology groups of these compact complex manifolds can have arbitrary amount of torsion so that their topology is extremely rich. We also resolve an associated wall-cros…

General MathematicsHolomorphic functionSubspace arrangementsPolytope52C35Combinatorics52B05Ricci-flat manifoldTheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITYConvex polytopeComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATIONMathematics::Symplectic Geometry32Q55Mathematics32M17Equivariant surgeryTopology of non-Kähler compact complex manifoldsMathematics::Geometric TopologyManifoldAffine complex manifoldsMathematics::Differential GeometryDiffeomorphismComplex manifoldCombinatorics of convex polytopesSingular homologyReal quadrics
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European Congress of Mathematics Kraków, 2 – 7 July, 2012

2013

Persistent homology is a recent grandchild of homology that has found use in science and engineering as well as in mathematics. This paper surveys the method as well as the applications, neglecting completeness in favor of highlighting ideas and directions. 2010 Mathematics Subject Classification. Primary 55N99; Secondary 68W30.

GrandchildPersistent homologyMathematics Subject ClassificationScience and engineeringHomology (mathematics)EpistemologySingular homology
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Algebraic models of the Euclidean plane

2018

We introduce a new invariant, the real (logarithmic)-Kodaira dimension, that allows to distinguish smooth real algebraic surfaces up to birational diffeomorphism. As an application, we construct infinite families of smooth rational real algebraic surfaces with trivial homology groups, whose real loci are diffeomorphic to $\mathbb{R}^2$, but which are pairwise not birationally diffeomorphic. There are thus infinitely many non-trivial models of the euclidean plane, contrary to the compact case.

Mathematics - Differential GeometryPure mathematicsaffine complexificationLogarithmReal algebraic model01 natural sciencesMathematics - Algebraic GeometryMathematics::Algebraic Geometry0103 physical sciencesEuclidean geometryAlgebraic surfaceaffine surfaceFOS: Mathematics0101 mathematicsInvariant (mathematics)Algebraic numberMathematics::Symplectic GeometryAlgebraic Geometry (math.AG)MathematicsAlgebra and Number Theory010102 general mathematics[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG]q-homology planesbirational diffeomorphismDifferential Geometry (math.DG)[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG]rational fibrationPairwise comparison010307 mathematical physicsGeometry and TopologyDiffeomorphism[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]14R05 14R25 14E05 14P25 14J26[MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG]Singular homology
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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…

[ INFO ] Computer Science [cs]CIA and HIA algorithmsComputer scienceHomotopyCellular homologyHomology (mathematics)[INFO] Computer Science [cs]TopologyMathematics::Algebraic TopologyRegular homotopyn-connectedHomotopy sphereTheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITYComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATIONMoore space (algebraic topology)[INFO]Computer Science [cs]Betti numbersEuler characteristicSingular homology
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