Cohomology

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Cohomology

Homology is concerned with a countable set of functors \(F_i\) mapping manifolds to simplicial complexes (roughly), and a natural transformation, namely the boundary operator \(\delta_i : F_i \to F_{i-1}\).

Cohomology is concerned with a countable number of contravariant functors \(F^i\), mapping manifolds to spaces of differential forms, and a natural transformation, namely the exterior derivative \(d: F^{i-1} \to F^i\).

The duality between these comes from Stokes' theorem, which given the inner product between a simplicial complex \(\gamma\) and a form \(\omega\)

\[ \langle \gamma, \omega \rangle := \int_\gamma \omega\]

says that \(d\) and \(\delta\) are adjoint:

\[ \langle \delta\gamma, \omega \rangle = \langle \gamma, d\omega \rangle \]

A crucial consequence is that \(d^2=0\).

The cohomology groups

Dual to homology groups, we have for a given manifold \(\mathcal{M}\):

\(H^i = \ker(d^i)/im (d^{i-1})\) for \(i \in \Z\), where \(d^i\) is the exterior derivative acting on the i-forms.

Each \(H^i\) gives information about the space. \(H^0 = B^0/\{0\} = B^0\) is the vector space of functions that are constant on each connected component of \(\mathcal{M}\), since a 0-form \(df = \partial_\mu fdx^\mu\) is closed iff the partial derivatives all vanish. So it measures connectedness.

In a similar vein, \(H^1\) measures simple connectedness, which means existence of all homotopies between paths. When a space is simply connected, closed forms can be integrated along any path to define their integral, so exact and closed forms coincide and \(H^1 = \{0\}\).

Characterizing exactness

For \(S\) a compact (therefore boundaryless, i.e. \(\partial S = 0\)) manifold, and \(\phi : S \to \mathcal{M}\), and \(\omega = df\), we have:

\[ \int_S \phi^*\omega = \int_S \phi^*df = \int_S d(\phi^*f) = \int_{\partial S} \phi^*f = 0 \]

where we have invoked the naturality of \(d\).