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String Theory and Particle Physics by Angel M. Uranga, Luis E. Ibáñez

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Appendix B

Some topological tools

B.1 Forms and cycles: cohomology and homology

B.1.1 Differential manifolds, tangent and cotangent space

A differential manifold X of dimension d is defined as a space with local patches U(α) parametrized by local coordinates , glued together with smooth transition functions f(αβ) on the overlaps , schematically as

image

(B.1)

From now on we drop the patch labels, and the adjective “differential,” and simply ...

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