Show that there are only countably many homotopy types of finite CW complexes

simplicial subcomplexS_αⁿhomeomorphic toSⁿat one end ofM(f_α) , and the homeo-morphismSⁿ≈ S_αⁿ is homotopic inW_nto the mapf_α, hence also toϕ_α. LetZ_n+1 be obtained fromZ_nby attachingD_αⁿ⁺¹×I ’s via these homotopies between the ϕ_α’s and the inclusionsS_αⁿ

>

^{Wn. ThusZn+1 containsXn+1}at one end, and at the other end we have a simplicial complex Y_n+1= Yn

S

αC(f_α) , where C(f_α) is obtained from M(f_α) by attaching a cone on the subcomplexS_αⁿ. Since Dⁿ⁺¹×I deformation retracts onto

∂Dⁿ⁺¹×I ∪ Dⁿ⁺¹×{1}, we see that Z_n+1 deformation retracts ontoZ_n∪ Y_n+1, which in turn deformation retracts onto Y_n∪ Y_n+1 = Y_n+1 by induction. Likewise, Z_n+1 deformation retracts ontoXⁿ⁺¹∪Wn which deformation retracts ontoXⁿ⁺¹∪Znand hence onto Xⁿ⁺¹∪ Xⁿ= Xⁿ⁺¹ by induction.

Let Y = S

nY_n and Z = S

nZ_n. The deformation retractions of Z_n onto Xⁿ give deformation retractions ofX∪ Z_n ontoX , and the infinite concatenation of the latter deformation retractions is a deformation retraction of Z onto X . Similarly, Z

deformation retracts onto Y . tu

Exercises

1. What is the minimum number of edges in simplicial complex structures K and L

much like the axioms for homology, except that induced homomorphisms go in the opposite direction as a result of the dualization. The basic distinction between homol-ogy and cohomolhomol-ogy is thus that cohomolhomol-ogy groups are contravariant functors while homology groups are covariant. In terms of intrinsic information, however, there is not a big difference between homology groups and cohomology groups. The homol-ogy groups of a space determine its cohomolhomol-ogy groups, and the converse holds at least when the homology groups are finitely generated.

What is a little surprising is that contravariance leads to extra structure in co-homology. This first appears in a natural product, calledcup product, which makes the cohomology groups of a space into a ring. This is an extremely useful piece of additional structure, and much of this chapter is devoted to studying cup products, which are considerably more subtle than the additive structure of cohomology.

How does contravariance lead to a product in cohomology that is not present in homology? Actually there is a natural product in homology, but it takes the somewhat different form of a mapH_i(X)×H_j(Y )

-→

^Hi+j(X×Y ) called the cross product. If both X and Y are CW complexes, this cross product in homology is induced from a map of cellular chains sending a pair (eⁱ, e^j) consisting of a cell of X and a cell of Y to the product celleⁱ×e^j inX×Y . The details of the construction are described in §3.B.

TakingX = Y , we thus have the first half of a hypothetical product H_i(X)×H_j(X)

-→

^Hi+j(X×X)

-→

^Hi+j(X)

The difficulty is in defining the second map. The natural thing would be for this to be induced by a map X×X

→

X . The multiplication map in a topological group, or more generally an H–space, is such a map, and the resultingPontryagin product can be quite useful when studying these spaces, as we show in§3.C. But for general X , the only

natural maps X×X

→

X are the projections onto one of the factors, and since these projections collapse the other factor to a point, the resulting product in homology is rather trivial.

With cohomology, however, the situation is better. One still has a cross product Hⁱ(X)×H^j(Y )

-→

^Hi+j(X×Y ) constructed in much the same way as in homology, so one can again takeX= Y and get the first half of a product

Hⁱ(X)×H^j(X)

-→

^Hi+j(X×X)

-→

^Hi+j(X)

But now by contravariance the second map would be induced by a map X

→

^{X×X ,}

and there is an obvious candidate for this map, the diagonal map∆(x) = (x, x). This turns out to work very nicely, giving a well-behaved product in cohomology, the cup product.

Another sort of extra structure in cohomology whose existence is traceable to contravariance is provided by cohomology operations. These make the cohomology groups of a space into a module over a certain rather complicated ring. Cohomology operations lie at a depth somewhat greater than the cup product structure, so we defer their study to§4.L.

The extra layer of algebra in cohomology arising from the dualization in its def-inition may seem at first to be separating it further from topology, but there are many topological situations where cohomology arises quite naturally. One of these is Poincar´e duality, the topic of the third section of this chapter. Another is obstruction theory, covered in§4.3. Characteristic classes in vector bundle theory (see [Milnor &

Stasheff 1974] or [VBKT]) provide a further instance.

From the viewpoint of homotopy theory, cohomology is in some ways more basic than homology. As we shall see in §4.3, cohomology has a description in terms of homotopy classes of maps that is very similar to, and in a certain sense dual to, the definition of homotopy groups. There is an analog of this for homology, described in

§4.F, but the construction is more complicated.

The Idea of Cohomology

Let us look at a few low-dimensional examples to get an idea of how one might be led naturally to consider cohomology groups, and to see what properties of a space they might be measuring. For the sake of simplicity we consider simplicial cohomology of∆ complexes, rather than singular cohomology of more general spaces.

Taking the simplest case first, let X be a 1 dimensional ∆ complex, or in other words an oriented graph. For a fixed abelian groupG , the set of all functions from ver-tices ofX to G also forms an abelian group, which we denote by ∆⁰(X; G) . Similarly the set of all functions assigning an element ofG 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 ϕ ∈ ∆⁰(X; G) to the function δϕ ∈ ∆¹(X; G) whose value on an oriented

edge [v₀, v₁] is the difference ϕ(v₁)− ϕ(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 functionϕ could then assign to each junction its elevation above sea level, in which caseδϕ 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 ϕ mea-suring voltages at the connection points, the vertices, andδϕ measuring changes in voltage across the components of the circuit, represented by edges.

Regarding the mapδ :∆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 definition 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 groupH⁰(X; G) is easy to describe explicitly. A function ϕ∈ ∆⁰(X; G) has δϕ= 0 iff ϕ takes the same value at both ends of each edge of X . This is equivalent to saying thatϕ 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 ofG , one for each component of X .

The cohomology group H¹(X; G)= ∆¹(X; G)/ Im δ will be trivial iff the equation δϕ= ψ has a solution ϕ ∈ ∆⁰(X; G) for each ψ∈ ∆¹(X; G) . Solving this equation means deciding whether specifying the change inϕ across each edge of X determines an actual functionϕ∈ ∆⁰(X; G) . This is rather like the calculus problem of finding a function having a specified derivative, with the difference operatorδ playing the role of differentiation. As in calculus, if a solution of δϕ= 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 ofX .

The equationδϕ= ψ is always solvable if X is a tree since if we choose arbitrarily a value for ϕ at a basepoint vertex v₀, then if the change in ϕ across each edge of X is specified, this uniquely determines the value of ϕ 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 first 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 ϕ uniquely up to a constant on each component of X . But in order for the equation δϕ= ψ to hold, the value of ψ on each edge not in any of the maximal trees must equal the difference in the already-determined values ofϕ at the two ends of the edge. This condition need not be satisfied since ψ can have arbitrary values on these edges. Thus we see that the homology groupH¹(X; G) is a direct product of copies of the groupG , 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; G) which consists of a directsum of copies of G , one for each edge of X not in one of the maximal trees.

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.

Define ∆⁰(X; G) and ∆¹(X; G) as before, as functions from vertices and edges of X to the abelian groupG , and define ∆²(X; G) to be the functions from 2 simplices of X to G . A homomorphism δ :∆¹(X; G)

→

^∆2(X; G) is defined 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 δϕ([v₀, v₁]) for ϕ ∈ ∆⁰(X; G) was a signed sum of the values ofϕ 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 ϕ∈ ∆⁰(X; G) we haveδδϕ= ϕ(v1)−ϕ(v0)

+ ϕ(v2)−ϕ(v1)

− ϕ(v2)−ϕ(v0)

= 0. Extending this chain complex by 0 ’s on each end, the resulting homology groups are by definition the cohomology groupsHⁱ(X; G) .

The formula for the map δ :∆¹(X; G)

→

^∆2(X; G) can be looked at from several different viewpoints. Perhaps the simplest is the observation that δψ = 0 iff ψ satisfies 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 finding ϕ∈ ∆⁰(X; G) with ψ= δϕ, for if ψ = δϕ then δψ = 0 since δδϕ = 0 as we saw above. We can think of δψ as a local obstruction to solving ψ= δϕ since it depends only on the values ofψ within individual 2 simplices of X . If this local obstruction vanishes, then ψ defines an element of H¹(X; G) which is zero iff ψ = δϕ has an actual solution. This class in H¹(X; G) is thus the global obstruction to solving ψ= δϕ. This situation is similar to the calculus problem of determining whether a given vector field is the gradient vector field of some function. The local obstruction here is the vanishing of the curl of the vector field, and the global obstruction is the vanishing of all line integrals around closed loops in the domain of the vector field.

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 first 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 C_ψ of disjoint curves inX crossing the

1 skeleton transversely, such that the number of intersections of C_ψ with each edge is equal to the value of ψ on that edge. If ψ= δϕ for some ϕ, then the curves of C_ψ divide X into two regions X₀ and X₁ where the subscript indicates the value ofϕ on all vertices in the region.

When G = Z we can refine this construction by building Cψ from a number of arcs in each 2 simplex, each arc having a transverse orientation, the orientation which agrees or disagrees with the orientation

of each edge according to the sign of the value of ψ on the edge, as in the figure at the right. The resulting collection C_ψ of disjoint curves in X can be thought of as something like level curves for a functionϕ with δϕ= ψ, if such a func-tion exists. The value of ϕ changes by 1 each time a curve of C_ψ is crossed.

For example, if X is a disk then we will

v 5

2 3

0 v₁

v₂

v 1

3 2

-0 v₁

v₂

4 3 4

2

1 1

0

1 0

1

0 0

show thatH¹(X;Z) = 0, so δψ = 0 im-plies ψ= δϕ for some ϕ, hence every

transverse curve system C_ψ forms the level curves of a function ϕ . On the other hand, if X is an annulus then this need no longer be true, as illustrated in the example shown in the figure at the left, where the equation ψ= δϕ obviously has no solution even though δψ = 0. By identifying the inner and outer boundary circles of this annulus we obtain a similar example on the torus. Even with G = Z₂ the equation ψ− δϕ has no solution since the curve C_ψ does not separate X into two regions X₀ andX₁.

1 1

?

?

0 0

The key to relating cohomology groups to homology groups is the observation that a function fromi simplices of X to G is equivalent to a homomorphism from the simplicial chain group∆i(X) to G . This is because∆i(X) is free abelian with basis the i simplices of X , and a homomorphism with domain a free abelian group is uniquely determined by its values on basis elements, which can be assigned arbitrarily. Thus we have an identification of∆ⁱ(X; G) with the group Hom(∆i(X), G) of homomorphisms

∆i(X)

→

G , which is called the dual group of∆i(X) . There is also a simple relationship of duality between the homomorphism δ :∆ⁱ(X; G)

→

^∆i+1(X; G) and the boundary homomorphism∂ :∆_i+1(X)

→

^∆i(X) . The general formula for δ is

δϕ([v₀,··· , v_i+1])=X

j

(−1)^jϕ([v₀,··· , bv_j,··· , v_i+1])

and the latter sum is justϕ(∂[v₀,··· , v_i+1]) . Thus we have δϕ= ϕ∂ . In other words, δ sends each ϕ∈ Hom(∆i(X), G) to the composition∆_i+1(X)

---→

^{∂ ∆i(X)}

---→

^{ϕ G , which}

in the language of linear algebra means that δ is the dual map of ∂ .

Thus we have the algebraic problem of understanding the relationship between the homology groups of a chain complex and the homology groups of the dual complex obtained by applying the functorC

,

Hom(C, G) . This is the first topic of the chapter.

Homology groups H_n(X) are the result of a two-stage process: First one forms a chain complex ···

---→

^Cn

---→

^{∂ C}n−1

---→

··· of singular, simplicial, or cellular chains, then one takes the homology groups of this chain complex, Ker∂/ Im ∂ . To obtain the cohomology groups Hⁿ(X; G) we interpolate an intermediate step, replacing the chain groups C_n by the dual groups Hom(C_n, G) and the boundary maps ∂ by their dual maps δ , before forming the cohomology groups Ker δ/ Im δ . The plan for this section is first to sort out the algebra of this dualization process and show that the cohomology groups are determined algebraically by the homology groups, though in a somewhat subtle way. Then after this algebraic excursion we will define the cohomology groups of spaces and show that these satisfy basic properties very much like those for homology. The payoff for all this formal work will begin to be apparent in subsequent sections.

在文檔中 Allen Hatcher (頁 193-199)