Introduction
Algebraic topology turns qualitative, flexible spaces into rigorous algebraic objects, enabling deep insights into shapes, holes, connectivity, and more—and impacts fields from geometry to data science to physics.
Contents
Fundamental Groups and Covering Spaces
Homotopy Types
The Fundamental Group
van Kampen’s Theorem
CW Complexes
Covering Spaces
Fundamental Theorem of Covering Spaces
The Galois Correspondence Theorem
Deck Transformations and Normal Covers
Covering Space Actions
Homology
Delta Complexes
Singular Homology
Reduced Homology
Exact Sequences and Relative Homology
Long Exact Sequence of a Good Pair
Degree and Local Homology
Cellular Homology
Euler Characteristics and Mayer-Vietoris Sequence
Simplicial Sets
Cohomology
Formalism of Homology and Cohomology
Singular Cohomology
Projective Resolution of Modules
Universal Coefficient Theorem
Cup Product and Ring Structure
The Künneth formula
Cap Product and Poincaré Duality