Algebraic Topology A Toolkit 1st Edition by Kevin Knudson ISBN 3111014869 9783111014869

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ISBN 10: 3111014869
ISBN 13: 9783111014869
Author: Kevin Knudson

This book is ideal as an introduction to algebraic topology and applied algebraic topology featuring a streamlined approach including coverage of basic categorical notions, simplicial, cellular, and singular homology, persistent homology, cohomology groups, cup products, Poincare Duality, homotopy theory, and spectral sequences. The focus is on examples and computations, and there are many end of chapter exercises and extensive student projects.

Algebraic Topology A Toolkit 1st Table of contents:

1 Preliminaries

1.1 Topological spaces

1.1.1 Some notions from general topology

Exercises

1.1.2 Simplicial complexes

Exercises

1.1.3 Cell complexes

1.1.4 Manifolds

1.1.5 Dramatis personæ

Exercises

1.2 Continuous maps and homotopy

1.2.1 Homotopy of maps

Exercises

1.2.2 Homotopy equivalence of spaces

1.2.3 Homotopy extension

Exercises

1.3 Lifting properties and fibrations

1.3.1 Liftings of continuous maps

1.3.2 Locally trivial fiber bundles

Exercises

1.4 Basic category theory

Exercises

1.5 The fundamental group

The induced homomorphism

Applications of the fundamental group

Exercises

1.6 The Seifert–Van Kampen Theorem

Attaching cells

Exercises

Project: covering spaces and the fundamental group

Project: SL2(ℤ) as an amalgamated free product

ℤ2∗ℤ2

ℤn∗ℤm

SL2(ℤ)

Bibliographic notes

2 Homology

2.1 What is homology?

2.1.1 Chain complexes

2.1.2 Homology of chain complexes

Exercises

2.2 Simplicial homology

2.2.1 Simplical chains

Exercises

2.2.2 Calculations

Exercises

2.3 Singular homology

2.3.1 Singular chains

Exercises

2.3.2 Relative homology

Exercises

2.3.3 Excision

Exercises

2.3.4 Homology of spheres revisited

Exercises

2.3.5 The Mayer–Vietoris sequence

Exercises

2.3.6 Homology with coefficients

Exercises

2.4 Cellular homology

Exercises

2.4.1 Cellular chains

Exercises

2.4.2 Calculations

Exercises

2.5 Comparison of homology theories

2.5.1 Simplicial vs. singular

Exercises

2.5.2 Cellular vs. singular

Exercises

2.5.3 Eilenberg–Steenrod axioms

2.6 Homology of manifolds

2.6.1 Orientability

2.6.2 The top homology group and the fundamental class

Exercises

2.6.3 Euler characteristic

Exercises

2.7 Persistent homology and topological data analysis

2.7.1 Filtrations, barcodes, and persistence diagrams

Exercises

2.7.2 Stability

Exercises

2.7.3 Persistence modules

Exercises

2.7.4 Vietoris–Rips complexes

Exercises

Project: the Jordan–Brouwer Separation Theorem

Project: the Nerve Theorem

Project: example persistent homology calculations

Bibliographic notes

3 Cohomology and duality

3.1 Cohomology of spaces

3.1.1 Motivation from calculus

Exercises

3.1.2 Cochains

Exercises

3.1.3 Calculations

Exercises

3.2 Cup products and the cohomology ring

3.2.1 Definition of the cup product

Exercises

3.2.2 Calculations of cohomology rings

Exercises

3.3 Cohomology of manifolds and Poincaré duality

3.3.1 Cap products and the duality theorem

Exercises

3.3.2 Interaction with cup products

Exercises

3.3.3 Consequences of duality

Exercises

3.4 Universal coefficient and Künneth theorems

3.4.1 Universal coefficient theorems

Exercises

3.4.2 The Künneth formula

Exercises

Project: the geometry of Poincaré Duality

Bibliographic notes

4 Homotopy and spectral sequences

4.1 Homotopy groups

4.1.1 Definitions and basic calculations

Exercises

4.1.2 Whitehead’s theorem

Exercises

4.1.3 Hurewicz theorem

Exercises

4.1.4 The long exact sequence of a fibration

Exercises

4.2 The Leray–Serre spectral sequence

4.2.1 Cohomology of fibrations

4.2.2 Calculations

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Tags: Kevin Knudson, Algebraic Topology