Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 183Clearly, Ker d is a group under addition: if two forms ω (1) and ω (2) satisfydω (1) = dω (2) = 0, then so does ω (1) + ω (2) . Moreover, if we changeω (i) by adding some dα (i) , the result of the addition will still be in thesame cohomology class, since it differs from ω (1) + ω (2) by d(α (1) + α (2) ).Therefore, we can view this addition really as an addition of cohomologyclasses: H p (M) is itself an additive group. Also note that if ω (3) and ω (4)are in the same cohomology class (that is, their difference is of the formdα (3) ), then so are cω (3) and cω (4) for any constant factor c. In otherwords, we can multiply a cohomology class by a constant to get anothercohomology class: cohomology classes actually form a vector space.3.6.3.2 Intuition Behind HomologyAnother operator similar to the exterior derivative d is the boundary operatorδ, which maps compact submanifolds of a smooth manifold M totheir boundary. Here, δC = 0 means that a submanifold C of M has noboundary, and C = δU means that C is itself the boundary of some submanifoldU. It is intuitively clear, and not very hard to prove, that δ 2 = 0:the boundary of a compact submanifold does not have a boundary itself.That the objects on which δ acts are independent of its coordinates is alsoclear. So is the grading of the objects: the degree p is the dimension of thesubmanifold C. 3 What is less clear is that the collection of submanifoldsactually forms a vector space, but one can always define this vector spaceto consist of formal linear combinations of submanifolds, and this is preciselyhow one proceeds. The pD elements of this vector space are calledp−chains. One should think of -C as C with its orientation reversed, andof the sum of two disjoint sets, C 1 + C 2 , as their union. The equivalenceclasses constructed from δ are called homology classes.For example, in Figure 3.5, C 1 and C 2 both satisfy δC = 0, so theyare elements of Ker δ. Moreover, it is clear that neither of them separatelycan be viewed as the boundary of another submanifold, so they are not inthe trivial homology class Im δ. However, the boundary of U is C 1 − C 2 .(The minus sign in front of C 2 is a result of the fact that C 2 itself actuallyhas the wrong orientation to be considered a boundary of U.) This can bewritten as C 1 − C 2 = δU, or equivalently C 1 = C 2 + δU, showing that C 1and C 2 are in the same homology class.The cohomology groups for the δ−operator are called homology groups,3 Note that here we have an example of an operator that maps objects of degree p toobjects of degree p − 1 instead of p + 1.