Ivancevic_Applied-Diff-Geom

184 Applied Differential Geometry: A Modern IntroductionFig. 3.5 The 1D submanifolds S 1 and S 2 represent the same homology class, since theirdifference is the boundary of U.and denoted by H p (M), with a lower index. 4 The p−chains C that satisfyδC = 0 are called p−cycles. Again, the H p (M) only exist for 0 ≤ p ≤ n.There is an interesting relation between cohomology and homologygroups. Note that we can construct a bilinear map from H p (M)×H p (M) →R by∫([ω], [C]) ↦→ ω, (3.41)where [ω] denotes the cohomology class of a p−form ω, and [Σ] the homologyclass of a p−cycle Σ. Using Stokes’ Theorem, it can be seen that theresult does not depend on the representatives for either ω or C∫∫ ∫ ∫ω + dα = ω + dα + ω + dαC+δUC C δU∫ ∫ ∫∫= ω + α + d(ω + dα) = ω,CδCwhere we used that by the definition of (co)homology classes, δC = 0 anddω = 0. As a result, the above map is indeed well–defined on homologyand cohomology classes. A very important Theorem by de Rham says thatthis map is nondegenerate [De Rham (1984)]. This means that if we takesome [ω] and we know the result of the map (3.41) for all [C], this uniquelydetermines [ω], and similarly if we start by picking an [C]. This in particularmeans that the vector space H p (M) is the dual vector space of H p (M).4 Historically, as can be seen from the terminology, homology came first and cohomologywas related to it in the way we will discuss below. However, since the cohomologygroups have a more natural additive structure, it is the name ‘cohomology’ which isactually used for generalizations.CUC