PDF《Cohomology is an algebraic variant》

G) which consists of a direct sum of copies of G, one for each edge of X not in one of the maximal trees.

188 Chapter

3 Cohomology Note that the relation between H1 (X;

G) and H1(X;

G) is the same as the relation between H0 (X;

G) and H0(X;

G), with H0 (X;

G) being a direct product of copies of G and H0(X;

G) a direct sum, with one copy for each component of X in either case. Now let us move up a dimension, taking X to be a

2 dimensional ? complex. De?ne ?0 (X;

G) and ?1 (X;

G) as before, as functions from vertices and edges of X to the abelian group G, and de?ne ?2 (X;

G) to be the functions from

2 simplices of X to G. A homomorphism δ : ?1 (X;

G)→?2 (X;

G) is de?ned by δψ([v0, v1, v2]) = ψ([v0, v1]) + ψ([v1, v2]) ? ψ([v0, v2]), a signed sum of the values of ψ on the three edges in the boundary of [v0, v1, v2], just as δ?([v0, v1]) for ? ∈ ?0 (X;

G) was a signed sum of the values of ? on the boundary of [v0, v1]. The two homomorphisms ?0 (X;

G) δ →?1 (X;

G) δ →?2 (X;

G) form a chain complex since for ? ∈ ?0 (X;

G) we have δδ? = ?(v1)??(v0) + ?(v2)??(v1) ? ?(v2)??(v0) = 0. Extending this chain complex by 0'

s on each end, the resulting homology groups are by de?nition the cohomology groups Hi (X;

G). The formula for the map δ : ?1 (X;

G)→?2 (X;

G) can be looked at from several di?erent viewpoints. Perhaps the simplest is the observation that δψ =

0 i? ψ satis?es the additivity property ψ([v0, v2]) = ψ([v0, v1]) + ψ([v1, v2]), where we think of the edge [v0, v2] as the sum of the edges [v0, v1] and [v1, v2]. Thus δψ measures the deviation of ψ from being additive. From another point of view, δψ can be regarded as an obstruction to ?nding ? ∈ ?0 (X;

G) with ψ = δ?, for if ψ = δ? then δψ =

0 since δδ? =

0 as we saw above. We can think of δψ as a local obstruction to solving ψ = δ? since it depends only on the values of ψ within individual

2 simplices of X . If this local obstruction vanishes, then ψ de?nes an element of H1 (X;

G) which is zero i? ψ = δ? has an actual solution. This class in H1 (X;

G) is thus the global obstruction to solving ψ = δ?. This situation is similar to the calculus problem of determining whether a given vector ?eld is the gradient vector ?eld of some function. The local obstruction here is the vanishing of the curl of the vector ?eld, and the global obstruction is the vanishing of all line integrals around closed loops in the domain of the vector ?eld. The condition δψ =

0 has an interpretation of a more geometric nature when X is a surface and the group G is Z or Z2 . Consider ?rst the simpler case G = Z2 . The condition δψ =

0 means that the number of times that ψ takes the value

1 on the edges of each

2 simplex is even, either

0 or 2. This means we can associate to ψ a collection Cψ of disjoint curves in X crossing the

1 skeleton transversely, such that the number o........