By Jean-Claude Hausmann
Cohomology and homology modulo 2 is helping the reader clutch extra without problems the fundamentals of a huge instrument in algebraic topology. in comparison to a extra normal method of (co)homology this clean strategy has many pedagogical advantages:
1. It leads extra fast to the necessities of the subject,
2. a scarcity of symptoms and orientation concerns simplifies the theory,
3. Computations and complicated functions will be provided at an prior level,
4. uncomplicated geometrical interpretations of (co)chains.
Mod 2 (co)homology was once constructed within the first sector of the 20 th century as a substitute to crucial homology, earlier than either grew to become specific circumstances of (co)homology with arbitrary coefficients.
The first chapters of this publication might function a foundation for a graduate-level introductory direction to (co)homology. Simplicial and singular mod 2 (co)homology are brought, with their items and Steenrod squares, in addition to equivariant cohomology. Classical purposes contain Brouwer's mounted element theorem, Poincaré duality, Borsuk-Ulam theorem, Hopf invariant, Smith concept, Kervaire invariant, and so forth. The cohomology of flag manifolds is taken care of intimately (without spectral sequences), together with the connection among Stiefel-Whitney sessions and Schubert calculus. more moderen advancements also are coated, together with topological complexity, face areas, equivariant Morse concept, conjugation areas, polygon areas, among others. each one bankruptcy ends with workouts, with a few tricks and solutions on the finish of the book.
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