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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 .