Cohomology and homology modulo 2 is helping the reader grab extra with no trouble the fundamentals of a big software in algebraic topology. in comparison to a extra basic method of (co)homology this clean process has many pedagogical advantages:
1. It leads extra quick to the necessities of the subject,
2. a scarcity of symptoms and orientation concerns simplifies the theory,
3. Computations and complex purposes will be offered at an prior degree,
4. basic geometrical interpretations of (co)chains.
Mod 2 (co)homology used to be built within the first zone of the 20 th century instead to indispensable homology, prior to either grew to become specific situations of (co)homology with arbitrary coefficients.
The first chapters of this publication could function a foundation for a graduate-level introductory path to (co)homology. Simplicial and singular mod 2 (co)homology are brought, with their items and Steenrod squares, in addition to equivariant cohomology. Classical functions comprise Brouwer's fastened aspect theorem, Poincaré duality, Borsuk-Ulam theorem, Hopf invariant, Smith concept, Kervaire invariant, and so forth. The cohomology of flag manifolds is handled intimately (without spectral sequences), together with the connection among Stiefel-Whitney periods and Schubert calculus. newer advancements also are lined, together with topological complexity, face areas, equivariant Morse thought, conjugation areas, polygon areas, among others. each one bankruptcy ends with routines, with a few tricks and solutions on the finish of the book.