PDF《Cohomology is an algebraic variant》

1 dimensional ? complex, or in other words an oriented graph. For a ?xed abelian group G, the set of all functions from ver- tices of X to G also forms an abelian group, which we denote by ?0 (X;

G). Similarly the set of all functions assigning an element of G to each edge of X forms an abelian group ?1 (X;

G). We will be interested in the homomorphism δ : ?0 (X;

G)→?1 (X;

G) sending ? ∈ ?0 (X;

G) to the function δ? ∈ ?1 (X;

G) whose value on an oriented The Idea of Cohomology

187 edge [v0, v1] is the di?erence ?(v1) ? ?(v0). For example, X might be the graph formed by a system of trails on a mountain, with vertices at the junctions between trails. The function ? could then assign to each junction its elevation above sea level, in which case δ? would measure the net change in elevation along the trail from one junction to the next. Or X might represent a simple electrical circuit with ? mea- suring voltages at the connection points, the vertices, and δ? measuring changes in voltage across the components of the circuit, represented by edges. Regarding the map δ : ?0 (X;

G)→?1 (X;

G) as a chain complex with 0'

s before and after these two terms, the homology groups of this chain complex are by de?nition the simplicial cohomology groups of X , namely H0 (X;

G) = Ker δ ? ?0 (X;

G) and H1 (X;

G) = ?1 (X;

G)/ Im δ. For simplicity we are using here the same notation as will be used for singular cohomology later in the chapter, in anticipation of the theorem that the two theories coincide for ? complexes, as we show in §3.1. The group H0 (X;

G) is easy to describe explicitly. A function ? ∈ ?0 (X;

G) has δ? =

0 i? ? takes the same value at both ends of each edge of X . This is equivalent to saying that ? is constant on each component of X . So H0 (X;

G) is the group of all functions from the set of components of X to G. This is a direct product of copies of G, one for each component of X . The cohomology group H1 (X;

G) = ?1 (X;

G)/ Im δ will be trivial i? the equation δ? = ψ has a solution ? ∈ ?0 (X;

G) for each ψ ∈ ?1 (X;

G). Solving this equation means deciding whether specifying the change in ? across each edge of X determines an actual function ? ∈ ?0 (X;

G). This is rather like the calculus problem of ?nding a function having a speci?ed derivative, with the di?erence operator δ playing the role of di?erentiation. As in calculus, if a solution of δ? = ψ exists, it will be unique up to adding an element of the kernel of δ, that is, a function that is constant on each component of X . The equation δ? = ψ is always solvable if X is a tree since if we choose arbitrarily a value for ? at a basepoint vertex v0 , then if the change in ? across each edge of X is speci?ed, this uniquely determines the value of ? at every other vertex v by induction along the unique path from v0 to v in the tree. When X is not a tree, we ?rst choose a maximal tree in each component of X . Then, since every vertex lies in one of these maximal trees, the values of ψ on the edges of the maximal trees determine ? uniquely up to a constant on each component of X . But in order for the equation δ? = ψ to hold, the value of ψ on each edge not in any of the maximal trees must equal the di?erence in the already-determined values of ? at the two ends of the edge. This condition need not be satis?ed since ψ can have arbitrary values on these edges. Thus we see that the cohomology group H1 (X;

G) is a direct product of copies of the group G, one copy for each edge of X not in one of the chosen maximal trees. This can be compared with the homology group H1(X;