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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. 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
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 / / 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
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
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
Opac: Controlla la disponibilità qui
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-9910828775103321
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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Homotopical topology / / by Anatoly Fomenko, Dmitry Fuchs
Homotopical topology / / by Anatoly Fomenko, Dmitry Fuchs
Autore Fomenko Anatoly
Edizione [Second edition]
Pubbl/distr/stampa Cham : , : Springer International Publishing, [2016]
Descrizione fisica 1 online resource (XI, 627 p. 210 illus.)
Disciplina 512.55
Collana Graduate Texts in Mathematics
Soggetto topico Topologia algebraica
Àlgebra homològica
Category theory (Mathematics)
Homological algebra
K-theory
Algebraic topology
Category Theory, Homological Algebra
K-Theory
Algebraic Topology
ISBN 3-319-23488-9
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Introduction -- Homotopy -- Homology -- Spectral Sequences of Fibrations -- Cohomology Operations -- The Adams Spectral Sequence -- K-Theory and Other Extraordinary Cohomology Theories.
Record Nr. UNINA-9910254096503321
Fomenko Anatoly  
Cham : , : Springer International Publishing, [2016]
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui