This is an introductory textbook on general and algebraic topology, aimed at anyone with a basic knowledge of calculus and linear algebra. It provides full proofs and includes many examples and exercises. The covered topics include: set theory and cardinal arithmetic; axiom of choice and Zorn's lemma; topological spaces and continuous functions; con- nectedness and compactness; Alexandrov compactification; quotient topol- ogies; countability and separation axioms; prebasis and Alexander's theorem; the Tychonoff theorem and paracompactness; complete metric spaces and function spaces; Baire spaces; homotopy of maps; the fundamental group; the van Kampen theorem; covering spaces; Brouwer and Borsuk's theorems; free groups and free product of groups and basic category theory. While it is very concrete at the beginning, abstract concepts are gradually introduced.
This second edition contains a new chapter with a topological introduction to sheaf cohomology and applications. It also corrects some inaccuracies and some additional exercises are proposed.
The textbook is suitable for anyone needing a basic, comprehensive introduction to general and algebraic topology and its applications.
Author(s): Marco Manetti
Series: UNITEXT 153
Edition: 2
Publisher: Springer Nature Switzerland
Year: 2023
Language: English
Pages: 377
City: Cham
Tags: Topology, Manifolds, Homotopy, Fundamental Group, Covering Spaces, van Kampen's Theorem, Sheaf Cohomology, Algebraic Topology
Preface
To the Student
To the Lecturer
Preface to Second English Edition
Contents
1 Geometrical Introduction to Topology
1.1 A Bicycle Ride Through the Streets of Rome
1.2 Topological Sewing
1.3 The Notion of Continuity
1.4 Homeomorphisms
1.5 Facts Without Proof
2 Sets
2.1 Notations and Basic Concepts
2.2 Induction and Completeness
2.3 Cardinality
2.4 The Axiom of Choice
2.5 Zorn's Lemma
2.6 The Cardinality of the Product
3 Topological Structures
3.1 Topological Spaces
3.2 Interior of a Set, Closure and Neighbourhoods
3.3 Continuous Maps
3.4 Metric Spaces
3.5 Subspaces and Immersions
3.6 Topological Products
3.7 Hausdorff Spaces
4 Connectedness and Compactness
4.1 Connectedness
4.2 Connected Components
4.3 Covers
4.4 Compact Spaces
4.5 Wallace's Theorem
4.6 Topological Groups
4.7 Exhaustions by Compact Sets
5 Topological Quotients
5.1 Identifications
5.2 Quotient Topology
5.3 Quotients by Groups of Homeomorphisms
5.4 Projective Spaces
5.5 Locally Compact Spaces
5.6 The Fundamental Theorem of Algebra
6 Sequences
6.1 Countability Axioms
6.2 Sequences
6.3 Cauchy Sequences
6.4 Compact Metric Spaces
6.5 Baire's Theorem
6.6 Completions
6.7 Function Spaces and Ascoli–Arzelà Theorem
6.8 Directed Sets and Nets (Generalised Sequences)
7 Manifolds, Infinite Products and Paracompactness
7.1 Sub-bases and Alexander's Theorem
7.2 Infinite Products
7.3 Refinements and Paracompactness
7.4 Topological Manifolds
7.5 Normal Spaces
7.6 Separation Axioms
8 More Topics in General Topology
8.1 Russell's Paradox
8.2 The Axiom of Choice Implies Zorn's Lemma
8.3 Zermelo's Theorem
8.4 Ultrafilters
8.5 The Compact-Open Topology
8.6 Noetherian Spaces
8.7 A Long Exercise: Tietze's Extension Theorem
9 Intermezzo
9.1 Trees
9.2 Polybricks and Betti Numbers
9.3 What Algebraic Topology Is
10 Homotopy
10.1 Locally Connected Spaces and the Functor π0
10.2 Homotopy
10.3 Retractions and Deformations
10.4 Categories and Functors
10.5 A Detour s
11 The Fundamental Group
11.1 Path Homotopy
11.2 The Fundamental Group
11.3 The Functor π1
11.4 The Sphere Sn Is Simply Connected (n≥2)
11.5 Topological Monoids
12 Covering Spaces
12.1 Local Homeomorphisms and Sections
12.2 Covering Spaces
12.3 Quotients by Properly Discontinuous Actions
12.4 Lifting Homotopies
12.5 Brouwer's Theorem and Borsuk's Theorem
12.6 A Non-Abelian Fundamental Group
13 Monodromy
13.1 Monodromy of Covering Spaces
13.2 Group Actions on Sets
13.3 An Isomorphism Theorem
13.4 Lifting Arbitrary Maps
13.5 Regular Coverings
13.6 Universal Coverings
13.7 Coverings with Given Monodromy
14 van Kampen's Theorem
14.1 van Kampen's Theorem, Universal Version
14.2 Free Groups
14.3 Free Products of Groups
14.4 Free Products and van Kampen's Theorem
14.5 Attaching Spaces and Topological Graphs
14.6 Cell Complexes
Attaching 1-Cells
Attaching 2-Cells
Attaching n-Cells, n≥3
15 A Topological View of Sheaf Cohomology
15.1 Natural Transformations
15.2 Sheaves
15.3 Exact Sequences
15.4 Direct and Inverse Image Functors
15.5 Complexes of Abelian Groups
15.6 Cohomology of Sheaves
15.7 Cohomology and Continuous Maps
15.8 Homotopy Invariance
15.9 Cohomology of Spheres and Applications
16 Selected Topics in Algebraic Topology
16.1 Groupoids and Equivalence of Categories
16.2 Inner and Outer Automorphisms
16.3 The Cantor Set and Peano Curves
16.4 The Topology of SO(3,R)
16.5 The Hairy Ball Theorem
16.6 Complex Polynomial Functions
16.7 Grothendieck's Proof of van Kampen's Theorem
16.8 A Long Exercise: The Poincaré–Volterra Theorem
17 Hints and Solutions
Reference
Index
