Cohomology and homology modulo 2 helps the reader grasp more readily the basics of a major tool
in algebraic topology. Compared to a more general approach to (co)homology this refreshing
approach has many pedagogical advantages: 1. It leads more quickly to the essentials of the
subject 2. An absence of signs and orientation considerations simplifies the theory 3.
Computations and advanced applications can be presented at an earlier stage 4. Simple
geometrical interpretations of (co)chains. Mod 2 (co)homology was developed in the first
quarter of the twentieth century as an alternative to integral homology before both became
particular cases of (co)homology with arbitrary coefficients. The first chapters of this book
may serve as a basis for a graduate-level introductory course to (co)homology. Simplicial and
singular mod 2 (co)homology are introduced with their products and Steenrod squares as well
as equivariant cohomology. Classical applications include Brouwer's fixed point theorem
Poincaré duality Borsuk-Ulam theorem Hopf invariant Smith theory Kervaire invariant etc.
The cohomology of flag manifolds is treated in detail (without spectral sequences) including
the relationship between Stiefel-Whitney classes and Schubert calculus. More recent
developments are also covered including topological complexity face spaces equivariant Morse
theory conjugation spaces polygon spaces amongst others. Each chapter ends with exercises
with some hints and answers at the end of the book.
