Algebraic Topology: A First Course (Graduate Texts in by William Fulton

By William Fulton

This ebook introduces the real principles of algebraic topology by means of emphasizing the relation of those principles with different parts of arithmetic. instead of deciding on one viewpoint of contemporary topology (homotropy concept, axiomatic homology, or differential topology, say) the writer concentrates on concrete difficulties in areas with a number of dimensions, introducing in basic terms as a lot algebraic equipment as useful for the issues encountered. This makes it attainable to determine a greater diversity of vital gains within the topic than is usual in introductory texts; it's also in concord with the ancient improvement of the topic. The ebook is aimed toward scholars who don't unavoidably intend on focusing on algebraic topology.

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As a particular case it yields one form of the so-called de Rham theorem, which states ˇ that the de Rham cohomology of a differentiable manifold and the Cech cohomology of the constant sheaf R are isomorphic. 11. Let F be a sheaf of abelian groups on X. A resolution of F is a collection of sheaves of abelian groups {Lk }k∈N with morphisms i : F → L0 , dk : Lk → Lk+1 such that the sequence d i d 0 → F → L0 →0 L1 →1 . . is exact. If the sheaves L• are acyclic (fine) the resolution is said to be acyclic (fine).

6) G2 i2 `BB BB BB k2 BB / G2 | | || || j2 | ~| E2 is exact. 5). 6) can be iterated, and we get a sequence of exact triangles Gr ir `BB BB BB kr BB / Gr | | || || jr | ~| Er where each group Er is the cohomology group of the differential module (Er−1 , dr−1 ), with dr−1 = jr−1 ◦ kr−1 . As we have already noticed, due to the assumption that the filtration K• has finite length , the groups Gr stabilize when r ≥ + 1, and the morphisms ir : Gr → Gr 2. THE SPECTRAL SEQUENCE OF A FILTERED COMPLEX 57 become injective.

One has a diagram of inclusions ? U @@ @@ jU ~~ ~ @@ ~ ~ @@ ~~ U A@ @ X ~> ~ ~~ ~~ ~~ jV @@ @@ @ V V Defining i(σ) = ( U ◦ σ, − V ◦ σ) and p(σ1 , σ2 ) = jU ◦ σ1 + jV ◦ σ2 , the exactness of the Mayer-Vietoris sequence is easily proved. The morphisms i and p commute with the homology operator ∂, so that one obtains a long homology exact sequence involving the homologies H• (A), H• (V ) ⊕ H• (V ) and H•U (X). 9. Prove that for any ring R the homology of the sphere S n with coefficients in R, n ≥ 2, is Hk (S n , R) = R for k = 0 and k = n 0 for 0 < k < n and k > n .

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