Topological methods in euclidean spaces

Gregory L. Naber

256 pages, parution le 27/11/2003

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Résumé

Extensive development of a number of topics central to topology, including elementary combinatorial techniques, Sperner's Lemma, the Brouwer Fixed Point Theorem, homotopy theory and the fundamental group, simplicial homology theory, the Hopf Trace Theorem, the Lefschetz Fixed Point Theorem, the Stone-Weierstrass Theorem, and Morse functions. Includes new section of solutions to selected problems.

Point-set topology of Euclidean spaces
- Introduction
- Preliminaries
- Open sets, closed sets, and continuity
- Compact spaces
- Connectivity properties
- Real-valued continuous functions
- Retracts
- Topological dimension Supplementary exercises
Elementary combinatorial techniques
- Introduction
- Hyperplanes in Rⁿ
- Simplexes and complexes
- Sample triangulations
- Simplicial maps
- Barycentric subdivision
- The Simpiicial Approximation Theorem
- Sperner's Lemma
- The Brouwer Fixed Point Theorem
- Topological dimension of compact subsets of Rⁿ Supplementary exercises
Homotopy theory and the fundamental group
- Introduction
- The homotopy relation, nullhomotopic maps, and contractible spaces
- Maps of spheres
- The fundamental group
- Fundamental groups of the spheres Supplementary exercises
Simplicial homology theory
- Introduction
- Oriented complexes and chains
- Boundary operators
- Cycles, boundaries, and homology groups
- Elementary examples
- Cone complexes, augmented complexes, and the homology groups H_p (K(s_n+l))
- Incidence numbers and the homology groups H_p (K_n(S_n+1))
- Elementary homological algebra
- The homology complex of a geometric complex
- Acyclic carrier functions
- Invariance of homoiogy groups under barycentric subdivision
- Homomorphisms induced by continuous maps
- Homology groups of topological polyhedra
- The Hopf Trace Theorem
- The Lefschetz Fixed Point Theorem Supplementary exercises
Differential techniques
- Introduction
- Smooth maps
- The Stone-Weierstrass Theorem
- Derivatives as linear transformations
- Diffcrentiable manifolds
- Tangent spaces and derivatives
- Regular and critical values of smooth maps
- Measure zero and Sard's Theorem
- Morse functions
- Manifolds with boundary
- One-dimensional manifolds
- Topological characterization of S^k
- Smooth tangent vector fields Supplementary exercises
Solutions to Selected Exercises
Guide to further study
Bibliography
List of symbols andnotation
Index

Caractéristiques techniques

	PAPIER
Éditeur(s)	Dover
Auteur(s)	Gregory L. Naber
Parution	27/11/2003
Nb. de pages	256
Format	13,5 x 21,5
Couverture	Broché
Poids	275g
Intérieur	Noir et Blanc
EAN13	9780486414522
ISBN13	978-0-48-641452-2