Introduction to Geometry and Topology by Werner Ballmann

Author:Werner Ballmann
Language: eng
Format: epub, pdf
ISBN: 9783034809832
Publisher: Springer Basel

is a diffeomorphism. For all k ≥ 0, therefore,

is an isomorphism. By Corollary 3.4.5 and Exercise 3.9.10.1, however,

Thus, we are in a position to recursively determine the cohomology of S m. We assume that for k = 0, m − 1, and that H k(S m−1) = {0} for all other k. Next, we consider the beginning

of the Mayer-Vietoris sequence, and recall for the following discussion that this sequence is exact. Since m ≥ 2, W 1 ∩ W 2 is connected, as are W 1, W 2 and W 1 ∪ W 2. Now i ∗ is injective on H 0(S m), so the image of i ∗ is a one-dimensional subspace in . Therefore, the image of j ∗ in is one-dimensional, and so is equal to H 0(W 1 ∩ W 2), i.e. j ∗ is surjective. It therefore follows that δ = 0 on H 0(W 1 ∩ W 2). So we obtain an exact sequence

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