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Algebraic Topology

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Author(s): H. Wilton, ed. Dexter Chua
Series: Cambridge Mathematical Tripos Part II Lecture Notes
Publisher: University of Cambridge
Year: 2015

Language: English
City: Cambridge
Tags: maths; mathematics; math; advanced; college; university; higher; further; pure; applied; topology; algebra; abstract algebra; modern algebra

Introduction
Definitions
Some recollections and conventions
Cell complexes
Homotopy and the fundamental group
Motivation
Homotopy
Paths
The fundamental group
Covering spaces
Covering space
The fundamental group of the circle and its applications
Universal covers
The Galois correspondence
Some group theory
Free groups and presentations
Another view of free groups
Free products with amalgamation
Seifert-van Kampen theorem
Seifert-van Kampen theorem
The effect on pi1 of attaching cells
A refinement of the Seifert-van Kampen theorem
The fundamental group of all surfaces
Simplicial complexes
Simplicial complexes
Simplicial approximation
Simplicial homology
Simplicial homology
Some homological algebra
Homology calculations
Mayer-Vietoris sequence
Continuous maps and homotopy invariance
Homology of spheres and applications
Homology of surfaces
Rational homology, Euler and Lefschetz numbers