By Jean-Claude Hausmann
Cohomology and homology modulo 2 is helping the reader clutch extra with no trouble the fundamentals of a big device in algebraic topology. in comparison to a extra common method of (co)homology this fresh process has many pedagogical advantages:
1. It leads extra quick to the necessities of the subject,
2. a scarcity of indicators and orientation concerns simplifies the theory,
3. Computations and complicated functions could be provided at an past degree,
4. easy geometrical interpretations of (co)chains.
Mod 2 (co)homology was once built within the first zone of the 20th century in its place to crucial homology, sooner than either turned specific circumstances of (co)homology with arbitrary coefficients.
The first chapters of this publication may well 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 comprise Brouwer's mounted aspect theorem, Poincaré duality, Borsuk-Ulam theorem, Hopf invariant, Smith conception, 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 periods and Schubert calculus. more moderen advancements also are lined, together with topological complexity, face areas, equivariant Morse thought, conjugation areas, polygon areas, among others. every one bankruptcy ends with workouts, with a few tricks and solutions on the finish of the booklet.
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Then, δ ∗ ([a]) = [δ(a)]. 3 Let I J → C∗ − → C2∗ → 0 0 → C1∗ − be a short exact sequence of cochain complexes. Then (a) I −1 (Z 2m ) = {b ∈ C m | δ(b) ∈ J (C1m+1 )}. (b) Let a ∈ Z 2m representing [a] ∈ H2m . Let b ∈ C m with I (b) = a. Then δ ∗ ([a]) = [J −1 (δ(b))] in H1m+1 . Proof Point (a) follows from the fact that I is surjective and from the equality δ2 ◦ I = I ◦ δ. For Point (b), choose a section S: C2m → Cm of I . 1, δ ∗ ([a]) = [J −1 (δ(S(a))]. The equality I (b) = a implies that b = S(a) + J (c) for some c ∈ C1m .
7 but the reader may find a proof as an exercise and this is easy to check for the particular triangulations given below. • up to isomorphism, the (co)homology of a simplicial complex K depends only of the homotopy type of |K |. This will be proved in Sect. 6. In particular, the Euler characteristic of two triangulations of a surface coincide. 7 that: Pt (S 2 ) = 1 + t 2 . The Projective Plane The projective plane RP 2 is the quotient of S 2 by the antipodal map. The triangulation of S 2 as a regular icosahedron being invariant under the antipodal map, it gives a triangulation of RP 2 given in Fig.
6 Exact Sequences In this section, we develop techniques to obtain long (co)homology exact sequences from short exact sequences of (co)chain complexes. The results are used in several forthcoming sections. All vector spaces in this section are over a fixed arbitrary field F. Let (C1∗ , δ1 ), (C2∗ , δ2 ) and (C ∗ , δ) be cochain complexes of vector spaces, giving rise to cohomology graded vector spaces H1∗ , H2∗ and H ∗ . 1) is an exact sequence. 1) a short exact sequence of cochain complexes. Choose a GrV-morphism S: C2∗ → C ∗ which is a section of I .