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Algebraic foundations for applied topology and data analysis / / Hal Schenck
Algebraic foundations for applied topology and data analysis / / Hal Schenck
Autore Schenck Hal
Pubbl/distr/stampa Cham, Switzerland : , : Springer, , [2022]
Descrizione fisica 1 online resource (231 pages)
Disciplina 514.2
Collana Mathematics of Data
Soggetto topico Algebraic topology
Topological algebras
Topologia algebraica
Àlgebres topològiques
Soggetto genere / forma Llibres electrònics
ISBN 9783031066641
9783031066634
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Intro -- Preface -- Contents -- 1 Linear Algebra Tools for Data Analysis -- 1.1 Linear Equations, Gaussian Elimination, Matrix Algebra -- 1.2 Vector Spaces, Linear Transformations, Basis and Change of Basis -- 1.2.1 Basis of a Vector Space -- 1.2.2 Linear Transformations -- 1.2.3 Change of Basis -- 1.3 Diagonalization, Webpage Ranking, Data and Covariance -- 1.3.1 Eigenvalues and Eigenvectors -- 1.3.2 Diagonalization -- 1.3.3 Ranking Using Diagonalization -- 1.3.4 Data Application: Diagonalization of the Covariance Matrix -- 1.4 Orthogonality, Least Squares Fitting, Singular Value Decomposition -- 1.4.1 Least Squares -- 1.4.2 Subspaces and Orthogonality -- 1.4.3 Singular Value Decomposition -- 2 Basics of Algebra: Groups, Rings, Modules -- 2.1 Groups, Rings and Homomorphisms -- 2.1.1 Groups -- 2.1.2 Rings -- 2.2 Modules and Operations on Modules -- 2.2.1 Ideals -- 2.2.2 Tensor Product -- 2.2.3 Hom -- 2.3 Localization of Rings and Modules -- 2.4 Noetherian Rings, Hilbert Basis Theorem, Varieties -- 2.4.1 Noetherian Rings -- 2.4.2 Solutions to a Polynomial System: Varieties -- 3 Basics of Topology: Spaces and Sheaves -- 3.1 Topological Spaces -- 3.1.1 Set Theory and Equivalence Relations -- 3.1.2 Definition of a Topology -- 3.1.3 Discrete, Product, and Quotient Topologies -- 3.2 Vector Bundles -- 3.3 Sheaf Theory -- 3.3.1 Presheaves and Sheaves -- 3.3.2 Posets, Direct Limit, and Stalks -- 3.3.3 Morphisms of Sheaves and Exactness -- 3.4 From Graphs to Social Media to Sheaves -- 3.4.1 Spectral Graph Theory -- 3.4.2 Heat Diffusing on a Wire Graph -- 3.4.3 From Graphs to Cellular Sheaves -- 4 Homology I: Simplicial Complexes to Sensor Networks -- 4.1 Simplicial Complexes, Nerve of a Cover -- 4.1.1 The Nerve of a Cover -- 4.2 Simplicial and Singular Homology -- 4.2.1 Singular homology -- 4.3 Snake Lemma and Long Exact Sequence in Homology.
4.3.1 Maps of complexes, Snake Lemma -- 4.3.2 Chain Homotopy -- 4.4 Mayer-Vietoris, Rips and Čech Complex, Sensor Networks -- 4.4.1 Mayer-Vietoris Sequence -- 4.4.2 Relative Homology -- 4.4.3 Čech Complex and Rips Complex -- 5 Homology II: Cohomology to Ranking Problems -- 5.1 Cohomology: Simplicial, Čech, de Rham Theories -- 5.1.1 Simplicial Cohomology -- 5.1.2 Čech Cohomology -- 5.1.3 de Rham Cohomology -- 5.2 Ranking, the Netflix Problem, and Hodge Theory -- 5.2.1 Hodge Decomposition -- 5.2.2 Application to Ranking -- 5.3 CW Complexes and Cellular Homology -- 5.4 Poincaré and Alexander Duality: Sensor Networks Revisited -- 5.4.1 Statement of Theorems and Examples -- 5.4.2 Alexander Duality: Proof -- 5.4.3 Sensor Networks Revisited -- 5.4.4 Poincaré Duality -- 6 Persistent Algebra: Modules Over a PID -- 6.1 Principal Ideal Domains and Euclidean Domains -- 6.2 Rational Canonical Form of a Matrix -- 6.3 Linear Transformations, K[t]-Modules, Jordan Form -- 6.4 Structure of Abelian Groups and Persistent Homology -- 6.4.1 Z-Graded Rings -- 7 Persistent Homology -- 7.1 Barcodes, Persistence Diagrams, Bottleneck Distance -- 7.1.1 History -- 7.1.2 Persistent Homology and the Barcode -- 7.1.3 Computation of Persistent Homology -- 7.1.4 Alpha and Witness Complexes -- 7.1.5 Persistence Diagrams -- 7.1.6 Metrics on Diagrams -- 7.2 Morse Theory -- 7.3 The Stability Theorem -- 7.4 Interleaving and Categories -- 7.4.1 Categories and Functors -- 7.4.2 Interleaving -- 7.4.3 Interleaving Vignette: Merge Trees -- 7.4.4 Zigzag Persistence and Quivers -- 8 Multiparameter Persistent Homology -- 8.1 Definition and Examples -- 8.1.1 Multiparameter Persistence -- 8.2 Graded Algebra, Hilbert Function, Series, Polynomial -- 8.2.1 The Hilbert Function -- 8.2.2 The Hilbert Series -- 8.3 Associated Primes and Zn-Graded Modules -- 8.3.1 Geometry of Sheaves.
8.3.2 Associated Primes and Primary Decomposition -- 8.3.3 Additional Structure in the Zn-Graded Setting -- 8.4 Filtrations and Ext -- 9 Derived Functors and Spectral Sequences -- 9.1 Injective and Projective Objects, Resolutions -- 9.1.1 Projective and Injective Objects -- 9.1.2 Resolutions -- 9.2 Derived Functors -- 9.2.1 Categories and Functors -- 9.2.2 Constructing Derived Functors -- 9.2.3 Ext -- 9.2.4 The Global Sections Functor -- 9.2.5 Acyclic Objects -- 9.3 Spectral Sequences -- 9.3.1 Total Complex of Double Complex -- 9.3.2 The Vertical Filtration -- 9.3.3 Main Theorem -- 9.4 Pas de Deux: Spectral Sequences and Derived Functors -- 9.4.1 Resolution of a Complex -- 9.4.2 Grothendieck Spectral Sequence -- 9.4.3 Comparing Cohomology Theories -- 9.4.4 Cartan-Eilenberg Resolution -- A Examples of Software Packages -- A.1 Covariance and Spread of Data via R -- A.2 Persistent Homology via scikit-tda -- A.3 Computational Algebra via Macaulay2 -- A.4 Multiparameter Persistence via RIVET -- Bibliography -- Index.
Record Nr. UNISA-996499869303316
Schenck Hal  
Cham, Switzerland : , : Springer, , [2022]
Materiale a stampa
Lo trovi qui: Univ. di Salerno
Opac: Controlla la disponibilità qui
Algebraic foundations for applied topology and data analysis / / Hal Schenck
Algebraic foundations for applied topology and data analysis / / Hal Schenck
Autore Schenck Hal
Pubbl/distr/stampa Cham, Switzerland : , : Springer, , [2022]
Descrizione fisica 1 online resource (231 pages)
Disciplina 514.2
Collana Mathematics of Data
Soggetto topico Algebraic topology
Topological algebras
Topologia algebraica
Àlgebres topològiques
Soggetto genere / forma Llibres electrònics
ISBN 9783031066641
9783031066634
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Intro -- Preface -- Contents -- 1 Linear Algebra Tools for Data Analysis -- 1.1 Linear Equations, Gaussian Elimination, Matrix Algebra -- 1.2 Vector Spaces, Linear Transformations, Basis and Change of Basis -- 1.2.1 Basis of a Vector Space -- 1.2.2 Linear Transformations -- 1.2.3 Change of Basis -- 1.3 Diagonalization, Webpage Ranking, Data and Covariance -- 1.3.1 Eigenvalues and Eigenvectors -- 1.3.2 Diagonalization -- 1.3.3 Ranking Using Diagonalization -- 1.3.4 Data Application: Diagonalization of the Covariance Matrix -- 1.4 Orthogonality, Least Squares Fitting, Singular Value Decomposition -- 1.4.1 Least Squares -- 1.4.2 Subspaces and Orthogonality -- 1.4.3 Singular Value Decomposition -- 2 Basics of Algebra: Groups, Rings, Modules -- 2.1 Groups, Rings and Homomorphisms -- 2.1.1 Groups -- 2.1.2 Rings -- 2.2 Modules and Operations on Modules -- 2.2.1 Ideals -- 2.2.2 Tensor Product -- 2.2.3 Hom -- 2.3 Localization of Rings and Modules -- 2.4 Noetherian Rings, Hilbert Basis Theorem, Varieties -- 2.4.1 Noetherian Rings -- 2.4.2 Solutions to a Polynomial System: Varieties -- 3 Basics of Topology: Spaces and Sheaves -- 3.1 Topological Spaces -- 3.1.1 Set Theory and Equivalence Relations -- 3.1.2 Definition of a Topology -- 3.1.3 Discrete, Product, and Quotient Topologies -- 3.2 Vector Bundles -- 3.3 Sheaf Theory -- 3.3.1 Presheaves and Sheaves -- 3.3.2 Posets, Direct Limit, and Stalks -- 3.3.3 Morphisms of Sheaves and Exactness -- 3.4 From Graphs to Social Media to Sheaves -- 3.4.1 Spectral Graph Theory -- 3.4.2 Heat Diffusing on a Wire Graph -- 3.4.3 From Graphs to Cellular Sheaves -- 4 Homology I: Simplicial Complexes to Sensor Networks -- 4.1 Simplicial Complexes, Nerve of a Cover -- 4.1.1 The Nerve of a Cover -- 4.2 Simplicial and Singular Homology -- 4.2.1 Singular homology -- 4.3 Snake Lemma and Long Exact Sequence in Homology.
4.3.1 Maps of complexes, Snake Lemma -- 4.3.2 Chain Homotopy -- 4.4 Mayer-Vietoris, Rips and Čech Complex, Sensor Networks -- 4.4.1 Mayer-Vietoris Sequence -- 4.4.2 Relative Homology -- 4.4.3 Čech Complex and Rips Complex -- 5 Homology II: Cohomology to Ranking Problems -- 5.1 Cohomology: Simplicial, Čech, de Rham Theories -- 5.1.1 Simplicial Cohomology -- 5.1.2 Čech Cohomology -- 5.1.3 de Rham Cohomology -- 5.2 Ranking, the Netflix Problem, and Hodge Theory -- 5.2.1 Hodge Decomposition -- 5.2.2 Application to Ranking -- 5.3 CW Complexes and Cellular Homology -- 5.4 Poincaré and Alexander Duality: Sensor Networks Revisited -- 5.4.1 Statement of Theorems and Examples -- 5.4.2 Alexander Duality: Proof -- 5.4.3 Sensor Networks Revisited -- 5.4.4 Poincaré Duality -- 6 Persistent Algebra: Modules Over a PID -- 6.1 Principal Ideal Domains and Euclidean Domains -- 6.2 Rational Canonical Form of a Matrix -- 6.3 Linear Transformations, K[t]-Modules, Jordan Form -- 6.4 Structure of Abelian Groups and Persistent Homology -- 6.4.1 Z-Graded Rings -- 7 Persistent Homology -- 7.1 Barcodes, Persistence Diagrams, Bottleneck Distance -- 7.1.1 History -- 7.1.2 Persistent Homology and the Barcode -- 7.1.3 Computation of Persistent Homology -- 7.1.4 Alpha and Witness Complexes -- 7.1.5 Persistence Diagrams -- 7.1.6 Metrics on Diagrams -- 7.2 Morse Theory -- 7.3 The Stability Theorem -- 7.4 Interleaving and Categories -- 7.4.1 Categories and Functors -- 7.4.2 Interleaving -- 7.4.3 Interleaving Vignette: Merge Trees -- 7.4.4 Zigzag Persistence and Quivers -- 8 Multiparameter Persistent Homology -- 8.1 Definition and Examples -- 8.1.1 Multiparameter Persistence -- 8.2 Graded Algebra, Hilbert Function, Series, Polynomial -- 8.2.1 The Hilbert Function -- 8.2.2 The Hilbert Series -- 8.3 Associated Primes and Zn-Graded Modules -- 8.3.1 Geometry of Sheaves.
8.3.2 Associated Primes and Primary Decomposition -- 8.3.3 Additional Structure in the Zn-Graded Setting -- 8.4 Filtrations and Ext -- 9 Derived Functors and Spectral Sequences -- 9.1 Injective and Projective Objects, Resolutions -- 9.1.1 Projective and Injective Objects -- 9.1.2 Resolutions -- 9.2 Derived Functors -- 9.2.1 Categories and Functors -- 9.2.2 Constructing Derived Functors -- 9.2.3 Ext -- 9.2.4 The Global Sections Functor -- 9.2.5 Acyclic Objects -- 9.3 Spectral Sequences -- 9.3.1 Total Complex of Double Complex -- 9.3.2 The Vertical Filtration -- 9.3.3 Main Theorem -- 9.4 Pas de Deux: Spectral Sequences and Derived Functors -- 9.4.1 Resolution of a Complex -- 9.4.2 Grothendieck Spectral Sequence -- 9.4.3 Comparing Cohomology Theories -- 9.4.4 Cartan-Eilenberg Resolution -- A Examples of Software Packages -- A.1 Covariance and Spread of Data via R -- A.2 Persistent Homology via scikit-tda -- A.3 Computational Algebra via Macaulay2 -- A.4 Multiparameter Persistence via RIVET -- Bibliography -- Index.
Record Nr. UNINA-9910631079903321
Schenck Hal  
Cham, Switzerland : , : Springer, , [2022]
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Algebraic Topology
Algebraic Topology
Autore Bray Clark
Pubbl/distr/stampa Cham : , : Springer International Publishing AG, , 2021
Descrizione fisica 1 online resource (216 pages)
Altri autori (Persone) ButscherAdrian
Rubinstein-SalzedoSimon
Soggetto topico Topologia algebraica
Soggetto genere / forma Llibres electrònics
ISBN 3-030-70608-7
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Intro -- Foreword -- Introduction -- Contents -- 1 Surface Preliminaries -- 1.1 Surfaces -- 1.2 Euclidean Space -- 1.3 Open Sets -- 1.4 Functions and Their Properties -- 1.5 Continuity -- 1.6 Problems -- 2 Surfaces -- 2.1 The Definition of a Surface -- 2.2 Examples of Surfaces -- 2.3 Spheres as Surfaces -- 2.4 Surfaces with Boundary -- 2.5 Closed, Bounded, and Compact Surfaces -- 2.6 Equivalence Relations and Topological Equivalence -- 2.7 Homeomorphic Spaces -- 2.8 Invariants -- 2.9 Problems -- 3 The Euler Characteristic and Identification Spaces -- 3.1 Triangulations and the Euler Characteristic -- 3.2 Invariance of the Euler Characteristic -- 3.3 Identification Spaces -- 3.4 ID Spaces as Surfaces -- 3.5 Abstract Topological Spaces -- 3.6 The Quotient Topology -- 3.7 Further Examples of ID Spaces -- 3.8 Triangulations of ID Spaces -- 3.9 The Connected Sum -- 3.10 The Euler Characteristic of a Compact Surface with Boundary -- 3.11 Problems -- 4 Classification Theorem of Compact Surfaces -- 4.1 The Geometry of the Projective Plane and the Klein Bottle -- 4.2 Orientable and Nonorientable Surfaces -- 4.3 The Classification Theorem for Compact Surfaces -- 4.4 Compact Surfaces Have Finite Triangulations -- 4.5 Proof of the Classification Theorem -- 4.6 Problems -- 5 Introduction to Group Theory -- 5.1 Why Use Groups? -- 5.2 A Motivating Example -- 5.3 Definition of a Group -- 5.4 Examples of Groups -- 5.5 Free Groups, Generators, and Relations -- 5.6 Free Products -- 5.7 Problems -- 6 Structure of Groups -- 6.1 Subgroups -- 6.2 Direct Products of Groups -- 6.3 Homomorphisms -- 6.4 Isomorphisms -- 6.5 Existence of Homomorphisms -- 6.6 Finitely Generated Abelian Groups -- 6.7 Problems -- 7 Cosets, Normal Subgroups, and Quotient Groups -- 7.1 Cosets -- 7.2 Lagrange's Theorem and Its Consequences -- 7.3 Coset Spaces and Quotient Groups.
7.4 Properties and Examples of Normal Subgroups -- 7.5 Coset Representatives -- 7.6 A Quotient of a Dihedral Group -- 7.7 Building up Finite Groups -- 7.8 An Isomorphism Theorem -- 7.9 Problems -- 8 The Fundamental Group -- 8.1 Paths and Loops on a Surface -- 8.2 Equivalence of Paths and Loops -- 8.3 Equivalence Classes of Paths and Loops -- 8.4 Multiplication of Path and Loop Classes -- 8.5 Definition of the Fundamental Group -- 8.6 Problems -- 9 Computing the Fundamental Group -- 9.1 Homotopies of Maps and Spaces -- 9.2 Computing the Fundamental Group of a Circle -- 9.3 Problems -- 10 Tools for Fundamental Groups -- 10.1 More Fundamental Groups -- 10.2 The Degree of a Loop -- 10.3 Fundamental Group of a Circle-Redux -- 10.4 The Induced Homomorphism on Fundamental Groups -- 10.5 Retracts -- 10.6 Problems -- 11 Applications of Fundamental Groups -- 11.1 The Fundamental Theorem of Algebra -- 11.2 Further Applications of the Fundamental Group -- 11.3 Problems -- 12 The Seifert-Van Kampen Theorem -- 12.1 Wedges of circles -- 12.2 The Seifert-Van Kampen Theorem: First Version -- 12.3 More Fundamental Groups -- 12.4 The Seifert-Van Kampen Theorem: Second Version -- 12.5 The Fundamental Group of a Compact Surface -- 12.6 Even More Fundamental Groups -- 12.7 Proof of the Second Version of the Seifert-Van Kampen Theorem -- 12.8 General Seifert-Van Kampen Theorem -- 12.9 Groups as Fundamental Groups -- 12.10 Problems -- 13 Introduction to Homology -- 13.1 The Idea of Homology -- 13.2 Chains -- 13.3 The Boundary Map -- 13.4 Homology -- 13.5 The Zeroth Homology Group -- 13.6 Homology of the Klein Bottle -- 13.7 Homology and Euler Characteristic -- 13.8 Homology and Orientability -- 13.9 Smith Normal Form -- 13.10 The Induced Map on Homology -- 13.11 Problems -- 14 The Mayer-Vietoris Sequence -- 14.1 Exact Sequences -- 14.2 The Mayer-Vietoris Sequence.
14.3 Homology of Orientable Surfaces -- 14.4 The Jordan Curve Theorem -- 14.5 The Hurewicz Map -- 14.6 Problems -- Correction to: The Seifert-Van Kampen Theorem -- Correction to: Chapter 12 in: C. Bray et al., Algebraic Topology, https://doi.org/10.1007/978-3-030-70608-112 -- Appendix A Topological Notions -- A.1 Compactness Results -- A.2 Technical Conditions for Abstract Surfaces -- Appendix B A Brief Look at Singular Homology -- Appendix C Hints for Selected Problems -- Appendix References -- -- Index.
Record Nr. UNISA-996466400803316
Bray Clark  
Cham : , : Springer International Publishing AG, , 2021
Materiale a stampa
Lo trovi qui: Univ. di Salerno
Opac: Controlla la disponibilità qui
Algebraic Topology
Algebraic Topology
Autore Bray Clark
Pubbl/distr/stampa Cham : , : Springer International Publishing AG, , 2021
Descrizione fisica 1 online resource (216 pages)
Altri autori (Persone) ButscherAdrian
Rubinstein-SalzedoSimon
Soggetto topico Topologia algebraica
Soggetto genere / forma Llibres electrònics
ISBN 3-030-70608-7
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Intro -- Foreword -- Introduction -- Contents -- 1 Surface Preliminaries -- 1.1 Surfaces -- 1.2 Euclidean Space -- 1.3 Open Sets -- 1.4 Functions and Their Properties -- 1.5 Continuity -- 1.6 Problems -- 2 Surfaces -- 2.1 The Definition of a Surface -- 2.2 Examples of Surfaces -- 2.3 Spheres as Surfaces -- 2.4 Surfaces with Boundary -- 2.5 Closed, Bounded, and Compact Surfaces -- 2.6 Equivalence Relations and Topological Equivalence -- 2.7 Homeomorphic Spaces -- 2.8 Invariants -- 2.9 Problems -- 3 The Euler Characteristic and Identification Spaces -- 3.1 Triangulations and the Euler Characteristic -- 3.2 Invariance of the Euler Characteristic -- 3.3 Identification Spaces -- 3.4 ID Spaces as Surfaces -- 3.5 Abstract Topological Spaces -- 3.6 The Quotient Topology -- 3.7 Further Examples of ID Spaces -- 3.8 Triangulations of ID Spaces -- 3.9 The Connected Sum -- 3.10 The Euler Characteristic of a Compact Surface with Boundary -- 3.11 Problems -- 4 Classification Theorem of Compact Surfaces -- 4.1 The Geometry of the Projective Plane and the Klein Bottle -- 4.2 Orientable and Nonorientable Surfaces -- 4.3 The Classification Theorem for Compact Surfaces -- 4.4 Compact Surfaces Have Finite Triangulations -- 4.5 Proof of the Classification Theorem -- 4.6 Problems -- 5 Introduction to Group Theory -- 5.1 Why Use Groups? -- 5.2 A Motivating Example -- 5.3 Definition of a Group -- 5.4 Examples of Groups -- 5.5 Free Groups, Generators, and Relations -- 5.6 Free Products -- 5.7 Problems -- 6 Structure of Groups -- 6.1 Subgroups -- 6.2 Direct Products of Groups -- 6.3 Homomorphisms -- 6.4 Isomorphisms -- 6.5 Existence of Homomorphisms -- 6.6 Finitely Generated Abelian Groups -- 6.7 Problems -- 7 Cosets, Normal Subgroups, and Quotient Groups -- 7.1 Cosets -- 7.2 Lagrange's Theorem and Its Consequences -- 7.3 Coset Spaces and Quotient Groups.
7.4 Properties and Examples of Normal Subgroups -- 7.5 Coset Representatives -- 7.6 A Quotient of a Dihedral Group -- 7.7 Building up Finite Groups -- 7.8 An Isomorphism Theorem -- 7.9 Problems -- 8 The Fundamental Group -- 8.1 Paths and Loops on a Surface -- 8.2 Equivalence of Paths and Loops -- 8.3 Equivalence Classes of Paths and Loops -- 8.4 Multiplication of Path and Loop Classes -- 8.5 Definition of the Fundamental Group -- 8.6 Problems -- 9 Computing the Fundamental Group -- 9.1 Homotopies of Maps and Spaces -- 9.2 Computing the Fundamental Group of a Circle -- 9.3 Problems -- 10 Tools for Fundamental Groups -- 10.1 More Fundamental Groups -- 10.2 The Degree of a Loop -- 10.3 Fundamental Group of a Circle-Redux -- 10.4 The Induced Homomorphism on Fundamental Groups -- 10.5 Retracts -- 10.6 Problems -- 11 Applications of Fundamental Groups -- 11.1 The Fundamental Theorem of Algebra -- 11.2 Further Applications of the Fundamental Group -- 11.3 Problems -- 12 The Seifert-Van Kampen Theorem -- 12.1 Wedges of circles -- 12.2 The Seifert-Van Kampen Theorem: First Version -- 12.3 More Fundamental Groups -- 12.4 The Seifert-Van Kampen Theorem: Second Version -- 12.5 The Fundamental Group of a Compact Surface -- 12.6 Even More Fundamental Groups -- 12.7 Proof of the Second Version of the Seifert-Van Kampen Theorem -- 12.8 General Seifert-Van Kampen Theorem -- 12.9 Groups as Fundamental Groups -- 12.10 Problems -- 13 Introduction to Homology -- 13.1 The Idea of Homology -- 13.2 Chains -- 13.3 The Boundary Map -- 13.4 Homology -- 13.5 The Zeroth Homology Group -- 13.6 Homology of the Klein Bottle -- 13.7 Homology and Euler Characteristic -- 13.8 Homology and Orientability -- 13.9 Smith Normal Form -- 13.10 The Induced Map on Homology -- 13.11 Problems -- 14 The Mayer-Vietoris Sequence -- 14.1 Exact Sequences -- 14.2 The Mayer-Vietoris Sequence.
14.3 Homology of Orientable Surfaces -- 14.4 The Jordan Curve Theorem -- 14.5 The Hurewicz Map -- 14.6 Problems -- Correction to: The Seifert-Van Kampen Theorem -- Correction to: Chapter 12 in: C. Bray et al., Algebraic Topology, https://doi.org/10.1007/978-3-030-70608-112 -- Appendix A Topological Notions -- A.1 Compactness Results -- A.2 Technical Conditions for Abstract Surfaces -- Appendix B A Brief Look at Singular Homology -- Appendix C Hints for Selected Problems -- Appendix References -- -- Index.
Record Nr. UNINA-9910485588903321
Bray Clark  
Cham : , : Springer International Publishing AG, , 2021
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Algebraic topology / / Tammo tom Dieck
Algebraic topology / / Tammo tom Dieck
Autore Dieck Tammo tom
Pubbl/distr/stampa Zürich, Switzerland : , : European Mathematical Society, , [2008]
Descrizione fisica 1 online resource (578 pages)
Collana EMS Textbooks in Mathematics
Soggetto topico Topologia algebraica
Algebraic topology
Manifolds and cell complexes
ISBN 9783037190487
3-03719-548-7
Classificazione 55-xx57-xx
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Record Nr. UNINA-9910151933803321
Dieck Tammo tom  
Zürich, Switzerland : , : European Mathematical Society, , [2008]
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
A Basic Course in Topology / / by Gerd Laures, Markus Szymik
A Basic Course in Topology / / by Gerd Laures, Markus Szymik
Autore Laures Gerd
Edizione [1st ed. 2025.]
Pubbl/distr/stampa Berlin, Heidelberg : , : Springer Berlin Heidelberg : , : Imprint : Birkhäuser, , 2025
Descrizione fisica 1 online resource (355 pages)
Disciplina 516.35
Altri autori (Persone) SzymikMarkus
Collana Compact Textbooks in Mathematics
Soggetto topico Geometry, Algebraic
Algebraic topology
Algebraic Geometry
Algebraic Topology
Geometria algebraica
Topologia algebraica
Soggetto genere / forma Llibres electrònics.
ISBN 9783662706022
3662706024
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto - Basic Concepts of Topology -- Universal Constructions -- Connectivity and Separation -- Compactness and Mapping Spaces -- Transformation Groups -- Paths and Loops -- The Fundamental Group -- Covering Spaces -- Bundles and Fibrations -- Sheaves -- Simplicial Sets.
Record Nr. UNINA-9910983297903321
Laures Gerd  
Berlin, Heidelberg : , : Springer Berlin Heidelberg : , : Imprint : Birkhäuser, , 2025
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Basic Topology 3 [[electronic resource] ] : Algebraic Topology and Topology of Fiber Bundles / / by Mahima Ranjan Adhikari
Basic Topology 3 [[electronic resource] ] : Algebraic Topology and Topology of Fiber Bundles / / by Mahima Ranjan Adhikari
Autore Adhikari Mahima Ranjan
Edizione [1st ed. 2022.]
Pubbl/distr/stampa Singapore : , : Springer Nature Singapore : , : Imprint : Springer, , 2022
Descrizione fisica 1 online resource (XXV, 468 p. 99 illus., 5 illus. in color.)
Disciplina 514
Soggetto topico Topology
Mathematical analysis
Algebra
Analysis
Topologia algebraica
Soggetto genere / forma Llibres electrònics
ISBN 981-16-6550-8
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto 1. Prerequisite Concepts of Topology, Algebra and Category Theory -- 2. Homotopy Theory: Fundamental and Higher Homotopy Groups -- 3. Homology and Cohomology Theories: An Axiomatic Approach with Consequences -- 4. Topology of Fiber Bundles -- 5. Homotopy Theory of Bundles -- 6. Some Applications of Algebraic Topology -- 7. Brief History on Algebraic Topology and Fiber Bundles.
Record Nr. UNISA-996518463303316
Adhikari Mahima Ranjan  
Singapore : , : Springer Nature Singapore : , : Imprint : Springer, , 2022
Materiale a stampa
Lo trovi qui: Univ. di Salerno
Opac: Controlla la disponibilità qui
Basic Topology 3 [[electronic resource] ] : Algebraic Topology and Topology of Fiber Bundles / / by Mahima Ranjan Adhikari
Basic Topology 3 [[electronic resource] ] : Algebraic Topology and Topology of Fiber Bundles / / by Mahima Ranjan Adhikari
Autore Adhikari Mahima Ranjan
Edizione [1st ed. 2022.]
Pubbl/distr/stampa Singapore : , : Springer Nature Singapore : , : Imprint : Springer, , 2022
Descrizione fisica 1 online resource (XXV, 468 p. 99 illus., 5 illus. in color.)
Disciplina 514
Soggetto topico Topology
Mathematical analysis
Algebra
Analysis
Topologia algebraica
Soggetto genere / forma Llibres electrònics
ISBN 981-16-6550-8
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto 1. Prerequisite Concepts of Topology, Algebra and Category Theory -- 2. Homotopy Theory: Fundamental and Higher Homotopy Groups -- 3. Homology and Cohomology Theories: An Axiomatic Approach with Consequences -- 4. Topology of Fiber Bundles -- 5. Homotopy Theory of Bundles -- 6. Some Applications of Algebraic Topology -- 7. Brief History on Algebraic Topology and Fiber Bundles.
Record Nr. UNINA-9910682548203321
Adhikari Mahima Ranjan  
Singapore : , : Springer Nature Singapore : , : Imprint : Springer, , 2022
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Handbook of Geometry and Topology of Singularities VII / / edited by José Luis Cisneros-Molina, Lê Dũng Tráng, José Seade
Handbook of Geometry and Topology of Singularities VII / / edited by José Luis Cisneros-Molina, Lê Dũng Tráng, José Seade
Autore Cisneros-Molina José Luis
Edizione [1st ed. 2025.]
Pubbl/distr/stampa Cham : , : Springer Nature Switzerland : , : Imprint : Springer, , 2025
Descrizione fisica 1 online resource (1700 pages)
Disciplina 516.35
Altri autori (Persone) Dũng TrángLê
SeadeJosé
Soggetto topico Geometry, Algebraic
Algebraic topology
Topological groups
Lie groups
Global analysis (Mathematics)
Manifolds (Mathematics)
Algebraic Geometry
Algebraic Topology
Topological Groups and Lie Groups
Global Analysis and Analysis on Manifolds
Geometria algebraica
Topologia algebraica
Grups topològics
Anàlisi global (Matemàtica)
Soggetto genere / forma Llibres electrònics
ISBN 9783031687112
9783031687105
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto 1 Bill Bruce, Peter Giblin, David Mond, Stephen Pizer and Les Wilson, Jim Damon’s Contributions to Singularity Theory and Its Applications -- 2 Viktor A. Vassiliev, Real Function Singularities and Their Bifurcation Sets -- 3 Adam Parusinski and Armin Rainer, Perturbation Theory of Polynomials and Linear Operators -- 4 Goo Ishikawa, Frontal Singularities and Related Problems -- 5 Osamu Saeki, Introduction to Global Singularity Theory of Differentiable Maps -- 6 Claude Sabbah, Singularities of Functions: A Global Point of View -- 7 Mark McLean, Floer Theory, Arc Spaces and Singularities -- 8 Stephen S.-T. Yau and Huaiqing Zuo, Various Derivation Lie Algebras of Isolated Singularities -- 9 Bingyi Chen, Stephen S.-T. Yau and Huaiqing Zuo, Three-Dimensional Rational Isolated Complete Intersection Singularities -- 10 Ziquan Zhuang, Stability of klt Singularities -- 11 Qianyu Chen, Bradley Dirks and Mircea Mustata, An introduction to V-Filtration -- 12 Kiyoshi Takeuchi, Geometric Monodromies, Mixed Hodge Numbers of Motivic Milnor Fibers and Newton Polyhedra -- 13 Willem Veys, Introduction to the Monodromy Conjecture -- 14 Laurentiu G. Maxim, Jose Israel Rodriguez and Botong Wang, Applications of Singularity Theory in Applied Algebraic Geometry and Algebraic Statistics.
Record Nr. UNINA-9910984592903321
Cisneros-Molina José Luis  
Cham : , : Springer Nature Switzerland : , : Imprint : Springer, , 2025
Materiale a stampa
Lo trovi qui: Univ. Federico II
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Homology Theory [[electronic resource] ] : An Introduction to Algebraic Topology / / by James W. Vick
Homology Theory [[electronic resource] ] : An Introduction to Algebraic Topology / / by James W. Vick
Autore Vick James W
Edizione [2nd ed. 1994.]
Pubbl/distr/stampa New York, NY : , : Springer New York : , : Imprint : Springer, , 1994
Descrizione fisica 1 online resource (XIV, 245 p.)
Disciplina 514.2
Collana Graduate Texts in Mathematics
Soggetto topico Algebraic topology
Topology
Algebraic Topology
Topologia algebraica
Soggetto genere / forma Llibres electrònics
ISBN 1-4612-0881-5
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto 1 Singular Homology Theory -- 2 Attaching Spaces with Maps -- 3 The Eilenberg-Steenrod Axioms -- 4 Covering Spaces -- 5 Products -- 6 Manifolds and Poincaré Duality -- 7 Fixed-Point Theory -- Appendix I -- Appendix II -- References -- Books and Historical Articles Since 1973 -- Books and Notes -- Survey and Expository Articles.
Record Nr. UNINA-9910789341303321
Vick James W  
New York, NY : , : Springer New York : , : Imprint : Springer, , 1994
Materiale a stampa
Lo trovi qui: Univ. Federico II
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