CW complex

Abstract: I Preliminaries on Categories, Abelian Groups, and Homotopy.- x1 Categories and Functors.- x2 Abelian Groups (Exactness, Direct Sums, Free Abelian Groups).- x3 Homotopy.- II Homology of Complexes.- x1 Complexes.- x2 Connecting Homomorphism, Exact Homology Sequence.- x3 Chain-Homotopy.- x4 Free Complexes.- III Singular Homology.- x1 Standard Simplices and Their Linear Maps.- x2 The Singular Complex.- x3 Singular Homology.- x4 Special Cases.- x5 Invariance under Homotopy.- x6 Barycentric Subdivision.- x7 Small Simplices. Excision.- x8 Mayer-Vietoris Sequences.- IV Applications to Euclidean Space.- x1 Standard Maps between Cells and Spheres.- x2 Homology of Cells and Spheres.- x3 Local Homology.- x4 The Degree of a Map.- x5 Local Degrees.- x6 Homology Properties of Neighborhood Retracts in ?n.- x7 Jordan Theorem, Invariance of Domain.- x8 Euclidean Neighborhood Retracts (ENRs).- V Cellular Decomposition and Cellular Homology.- x1 Cellular Spaces.- x2 CW-Spaces.- x3 Examples.- x4 Homology Properties of CW-Spaces.- x5 The Euler-Poincare Characteristic.- x6 Description of Cellular Chain Maps and of the Cellular Boundary Homomorphism.- x7 Simplicial Spaces.- x8 Simplicial Homology.- VI Functors of Complexes.- x1 Modules.- x2 Additive Functors.- x3 Derived Functors.- x4 Universal Coefficient Formula.- x5 Tensor and Torsion Products.- x6 Horn and Ext.- x7 Singular Homology and Cohomology with General Coefficient Groups.- x8 Tensorproduct and Bilinearity.- x9 Tensorproduct of Complexes. Kunneth Formula.- x10 Horn of Complexes. Homotopy Classification of Chain Maps.- x11 Acyclic Models.- x12 The Eilenberg-Zilber Theorem. Kunneth Formulas for Spaces.- VII Products.- x1 The Scalar Product.- x2 The Exterior Homology Product.- x 3 The Interior Homology Product (Pontijagin Product).- x 4 Intersection Numbers in ?n.- x5 The Fixed Point Index.- x6 The Lefschetz-Hopf Fixed Point Theorem.- x7 The Exterior Cohomology Product.- x 8 The Interior Cohomology Product (?-Product).- x 9 ?-Products in Projective Spaces. Hopf Maps and Hopf Invariant.- x10 Hopf Algebras.- x11 The Cohomology Slant Product.- x12 The Cap-Product (?-Product).- x 13 The Homology Slant Product, and the Pontijagin Slant Product.- VIII Manifolds.- x1 Elementary Properties of Manifolds.- x2 The Orientation Bundle of a Manifold.- x3 Homology of Dimensions ? n in n-Manifolds.- x4 Fundamental Class and Degree.- x5 Limits.- x6 ?ech Cohomology of Locally Compact Subsets of ?n.- x7 Poincare-Lefschetz Duality.- x8 Examples, Applications.- x9 Duality in ?-Manifolds.- x10 Transfer.- x11 Thom Class, Thom Isomorphism.- x12 The Gysin Sequence. Examples.- x13 Intersection of Homology Classes.- Appendix: Kan- and ?ech-Extensions of Functors.- x1 Limits of Functors.- x2 Polyhedrons under a Space, and Partitions of Unity.- x3 Extending Functors from Polyhedrons to More General Spaces.

Abstract: o. Some Facts from General Topology 1. Categories, Functors and Natural Transformations 2. Homotopy Sets and Groups 3. Properties of the Homotopy Groups 4. Fibrations 5. CW-Complexes 6. Homotopy Properties of CW-Complexes 7. Homology and Cohomology Theories 8. Spectra 9. Representation Theorems 10. Ordinary Homology Theory 11. Vector Bundles and K-Theory 12. Manifolds and Bordism 13. Products 14. Orientation and Duality 15. Spectral Swquences 16. Characteristic Classes 17. Cohomology Operations and Homology Cooperations 18. The Steenrod Algebra and its Dual 19. The Adams Spectral Sequence and the e-Invariant 20. Calculation of the Corbordism Groups Bibliography Subject Index

Abstract: We study the homology of a filtered d-dimensional simplicial complex K as a single algebraic entity and establish a correspondence that provides a simple description over fields. Our analysis enables us to derive a natural algorithm for computing persistent homology over an arbitrary field in any dimension. Our study also implies the lack of a simple classification over non-fields. Instead, we give an algorithm for computing individual persistent homology groups over an arbitrary PIDs in any dimension.

Abstract: Corollary 1.2. Let Γ be a finitely generated group. If Γ, as a metric space with a word-length metric, admits a uniform embedding into Hilbert space, and its classifying space BΓ has the homotopy type of a finite CW complex, then the strong Novikov conjecture holds for Γ, i.e. the index map from K∗(BΓ) to K∗(C∗ r (Γ)) is injective. Corollary 1.2 follows from Theorem 1.1 and the descent principle [23]. By index theory, the strong Novikov conjecture implies the Novikov conjecture on the homotopy invariance of higher signatures (cf. [8] for an excellent