Show that a finite graph product of finite groups has a free subgroup of finite index, by constructing a finite-sheeted covering space of K Γ from universal covers of

usually taken to be slightly more restrictive than the one we have given here, namely, one considers only oriented graphs obtained from an unoriented graph by subdividing each edge by adding a vertex at its midpoint, then orienting the two resulting edges outward, away from the new vertex.

Exercises

1. Suppose a group G acts simplicially on a ∆ complex X , where ‘simplicially’ means that each element of G takes each simplex of X onto another simplex by a linear homeomorphism. If the action is free, show it is a covering space action.

2. Let X be a connected CW complex and G a group such that every homomorphism π₁(X)

→

G is trivial. Show that every map X

→

K(G, 1) is nullhomotopic.

low-dimensional spaces into X , namely loops I

→

X and homotopies of loops, maps I×I

→

X . The definition in terms of objects that are at most 2 dimensional manifests itself for example in the fact that when X is a CW complex, π₁(X) depends only on the 2 skeleton ofX . In view of the low-dimensional nature of the fundamental group, we should not expect it to be a very refined tool for dealing with high-dimensional spaces. Thus it cannot distinguish between spheres Sⁿ with n≥ 2. This limitation to low dimensions can be removed by considering the natural higher-dimensional analogs of π₁(X) , the homotopy groups π_n(X) , which are defined in terms of maps of the n dimensional cube Iⁿ intoX and homotopies Iⁿ×I

→

X of such maps. Not surprisingly, when X is a CW complex, π_n(X) depends only on the (n+ 1) skeleton ofX . And as one might hope, homotopy groups do indeed distinguish spheres of all dimensions sinceπ_i(Sⁿ) is 0 for i < n and Z for i = n.

However, the higher-dimensional homotopy groups have the serious drawback that they are extremely difficult to compute in general. Even for simple spaces like spheres, the calculation of π_i(Sⁿ) for i > n turns out to be a huge problem. For-tunately there is a more computable alternative to homotopy groups: the homology groups H_n(X) . Like π_n(X) , the homology group H_n(X) for a CW complex X de-pends only on the (n+ 1) skeleton. For spheres, the homology groups Hi(Sⁿ) are isomorphic to the homotopy groups π_i(Sⁿ) in the range 1 ≤ i ≤ n, but homology groups have the advantage that H_i(Sⁿ)= 0 for i > n.

The computability of homology groups does not come for free, unfortunately.

The definition of homology groups is decidedly less transparent than the definition of homotopy groups, and once one gets beyond the definition there is a certain amount of technical machinery to be set up before any real calculations and applications can be given. In the exposition below we approach the definition ofH_n(X) by two prelim-inary stages, first giving a few motivating examples nonrigorously, then constructing

a restricted model of homology theory called simplicial homology, before plunging into the general theory, known as singular homology. After the definition of singular homology has been assimilated, the real work of establishing its basic properties be-gins. This takes close to 20 pages, and there is no getting around the fact that it is a substantial effort. This takes up most of the first section of the chapter, with small digressions only for two applications to classical theorems of Brouwer: the fixed point theorem and ‘invariance of dimension.’

The second section of the chapter gives more applications, including the ho-mology definition of Euler characteristic and Brouwer’s notion of degree for maps Sⁿ

→

^Sn. However, the main thrust of this section is toward developing techniques for calculating homology groups efficiently. The maximally efficient method is known as cellular homology, whose power comes perhaps from the fact that it is ‘homology squared’ — homology defined in terms of homology. Another quite useful tool is Mayer–Vietoris sequences, the analog for homology of van Kampen’s theorem for the fundamental group.

An interesting feature of homology that begins to emerge after one has worked with it for a while is that it is the basic properties of homology that are used most often, and not the actual definition itself. This suggests that an axiomatic approach to homology might be possible. This is indeed the case, and in the third section of the chapter we list axioms which completely characterize homology groups for CW complexes. One could take the viewpoint that these rather algebraic axioms are all that really matters about homology groups, that the geometry involved in the definition of homology is secondary, needed only to show that the axiomatic theory is not vacuous.

The extent to which one adopts this viewpoint is a matter of taste, and the route taken here of postponing the axioms until the theory is well-established is just one of several possible approaches.

The chapter then concludes with three optional sections of Additional Topics. The first is rather brief, relatingH₁(X) to π₁(X) , while the other two contain a selection of classical applications of homology. These include then dimensional version of the Jordan curve theorem and the ‘invariance of domain’ theorem, both due to Brouwer, along with the Lefschetz fixed point theorem.

The Idea of Homology

The difficulty with the higher homotopy groups π_n is that they are not directly computable from a cell structure as π₁ is. For example, the 2-sphere has no cells in dimensions greater than 2, yet itsn dimensional homotopy group π_n(S²) is nonzero for infinitely many values of n . Homology groups, by contrast, are quite directly related to cell structures, and may indeed be regarded as simply an algebraization of the first layer of geometry in cell structures: how cells of dimensionn attach to cells of dimensionn− 1.

Let us look at some examples to see what the idea is. Consider the graphX₁shown in the figure, consisting of two vertices joined by four edges.

When studying the fundamental group of X₁ we consider

a d

x y

c b loops formed by sequences of edges, starting and ending

at a fixed basepoint. For example, at the basepoint x , the loopab⁻¹ travels forward along the edgea , then backward along b , as indicated by the exponent−1. A more compli-cated loop would beac⁻¹bd⁻¹ca⁻¹. A salient feature of the

fundamental group is that it is generally nonabelian, which both enriches and compli-cates the theory. Suppose we simplify matters by abelianizing. Thus for example the two loops ab⁻¹ and b⁻¹a are to be regarded as equal if we make a commute with b⁻¹. These two loopsab⁻¹ and b⁻¹a are really the same circle, just with a different choice of starting and ending point: x for ab⁻¹ and y for b⁻¹a . The same thing happens for all loops: Rechoosing the basepoint in a loop just permutes its letters cyclically, so a byproduct of abelianizing is that we no longer have to pin all our loops down to a fixed basepoint. Thus loops becomecycles, without a chosen basepoint.

Having abelianized, let us switch to additive notation, so cycles become linear combinations of edges with integer coefficients, such as a− b + c − d. Let us call these linear combinations chains of edges. Some chains can be decomposed into cycles in several different ways, for example (a− c) + (b − d) = (a − d) + (b − c), and if we adopt an algebraic viewpoint then we do not want to distinguish between these different decompositions. Thus we broaden the meaning of the term ‘cycle’ to be simply any linear combination of edges for which at least one decomposition into cycles in the previous more geometric sense exists.

What is the condition for a chain to be a cycle in this more algebraic sense? A geometric cycle, thought of as a path traversed in time, is distinguished by the prop-erty that it enters each vertex the same number of times that it leaves the vertex. For an arbitrary chain ka+ `b + mc + nd, the net number of times this chain enters y is k+ ` + m + n since each of a, b , c , and d enters y once. Similarly, each of the four edges leaves x once, so the net number of times the chain ka+ `b + mc + nd entersx is −k − ` − m − n. Thus the condition for ka + `b + mc + nd to be a cycle is simply k+ ` + m + n = 0.

To describe this result in a way that would generalize to all graphs, letC₁ be the free abelian group with basis the edges a, b, c, d and let C₀ be the free abelian group with basis the vertices x, y . Elements of C₁ are chains of edges, or 1 dimensional chains, and elements of C₀ are linear combinations of vertices, or 0 dimensional chains. Define a homomorphism ∂ : C₁

→

^C0 by sending each basis element a, b, c, d toy− x , the vertex at the head of the edge minus the vertex at the tail. Thus we have

∂(ka+ `b + mc + nd) = (k + ` + m + n)y − (k + ` + m + n)x , and the cycles are precisely the kernel of∂ . It is a simple calculation to verify that a−b , b−c , and c −d

form a basis for this kernel. Thus every cycle in X₁ is a unique linear combination of these three most obvious cycles. By means of these three basic cycles we convey the geometric information that the graph X₁ has three visible ‘holes,’ the empty spaces between the four edges.

Let us now enlarge the preceding graph X₁ by attaching a 2 cell A along the cycle a− b , producing a 2 dimensional cell complex X₂. If

x y

c d

a A b

we think of the 2 cell A as being oriented clockwise, then we can regard its boundary as the cycle a− b . This cycle is now homotopically trivial since we can contract it to a point by sliding over A . In other words, it no longer encloses a hole in X₂. This suggests that we form a quotient of the group of cycles in the preceding example by factoring out

the subgroup generated by a− b . In this quotient the cycles a − c and b − c , for example, become equivalent, consistent with the fact that they are homotopic in X₂.

Algebraically, we can define now a pair of homomorphisms C₂

---→

^{∂2 C1}

---→

^{∂1 C0}

where C₂ is the infinite cyclic group generated by A and ∂₂(A)= a − b . The map

∂₁ is the boundary homomorphism in the previous example. The quotient group we are interested in is Ker∂₁/ Im ∂₂, the 1 dimensional cycles modulo those that are boundaries, the multiples ofa−b . This quotient group is the homology group H1(X₂) . The previous example can be fit into this scheme too by taking C₂ to be zero since there are no 2 cells in X₁, so in this case H₁(X₁)= Ker ∂1/ Im ∂₂= Ker ∂1, which as we saw was free abelian on three generators. In the present example, H₁(X₂) is free abelian on two generators, b− c and c − d, expressing the geometric fact that by filling in the 2 cellA we have reduced the number of ‘holes’ in our space from three to two.

Suppose we enlarge X₂ to a space X₃ by attaching a second 2 cell B along the same cyclea−b . This gives a 2 dimensional chain group C2

x y

a b c d

consisting of linear combinations ofA and B , and the bound-ary homomorphism∂₂:C₂

→

^{C1sends both}A and B to a−b . The homology group H₁(X₃)= Ker ∂₁/ Im ∂₂ is the same as forX₂, but now ∂₂ has a nontrivial kernel, the infinite cyclic group generated byA−B . We view A−B as a 2 dimensional cycle, generating the homology group H₂(X₃)= Ker ∂2 ≈ Z.

Topologically, the cycle A− B is the sphere formed by the cells A and B together with their common boundary circle. This spherical cycle detects the presence of a

‘hole’ in X₃, the missing interior of the sphere. However, since this hole is enclosed by a sphere rather than a circle, it is of a different sort from the holes detected by H₁(X₃)≈ Z×Z, which are detected by the cycles b − c and c − d.

Let us continue one more step and construct a complexX₄ fromX₃ by attaching a 3 cell C along the 2 sphere formed by A and B . This creates a chain group C₃

generated by this 3 cell C , and we define a boundary homomorphism ∂₃:C₃

→

^C2

sending C to A− B since the cycle A − B should be viewed as the boundary of C in the same way that the 1 dimensional cycle a− b is the boundary of A. Now we have a sequence of three boundary homomorphisms C₃

---→

^{∂3 C2}

---→

^{∂2 C1}

---→

^{∂1 C0 and}

the quotient H₂(X₄) = Ker ∂2/ Im ∂₃ has become trivial. Also H₃(X₄) = Ker ∂3 = 0.

The group H₁(X₄) is the same as H₁(X₃) , namelyZ×Z, so this is the only nontrivial homology group ofX₄.

It is clear what the general pattern of the examples is. For a cell complex X one has chain groups C_n(X) which are free abelian groups with basis the n cells of X , and there are boundary homomorphisms ∂_n:C_n(X)

→

^Cn−1(X) , in terms of which one defines the homology group H_n(X) = Ker ∂_n/ Im ∂_n+1. The major difficulty is how to define ∂_n in general. For n = 1 this is easy: The boundary of an oriented edge is the vertex at its head minus the vertex at its tail. The next case n= 2 is also not hard, at least for cells attached along cycles that are simply loops of edges, for then the boundary of the cell is this cycle of edges, with the appropriate signs taking orientations into account. But for larger n , matters become more complicated. Even if one restricts attention to cell complexes formed from polyhedral cells with nice attaching maps, there is still the matter of orientations to sort out.

The best solution to this problem seems to be to adopt an indirect approach.

Arbitrary polyhedra can always be subdivided into special polyhedra called simplices (the triangle and the tetrahedron are the 2 dimensional and 3 dimensional instances) so there is no loss of generality, though initially there is some loss of efficiency, in restricting attention entirely to simplices. For simplices there is no difficulty in defin-ing boundary maps or in handldefin-ing orientations. So one obtains a homology theory, called simplicial homology, for cell complexes built from simplices. Still, this is a rather restricted class of spaces, and the theory itself has a certain rigidity that makes it awkward to work with.

The way around these obstacles is to step back from the geometry of spaces decomposed into simplices and to consider instead something which at first glance seems wildly more complicated, the collection of all possible continuous maps of simplices into a given spaceX . These maps generate tremendously large chain groups C_n(X) , but the quotients H_n(X)= Ker ∂n/ Im ∂_n+1, called singular homology groups, turn out to be much smaller, at least for reasonably nice spacesX . In particular, for spaces like those in the four examples above, the singular homology groups coincide with the homology groups we computed from the cellular chains. And as we shall see later in this chapter, singular homology allows one to define these nice cellular homology groups for all cell complexes, and in particular to solve the problem of defining the boundary maps for cellular chains.

The most important homology theory in algebraic topology, and the one we shall be studying almost exclusively, is called singular homology. But before beginning the development of singular homology we will first introduce a more primitive version, called simplicial homology, to give some idea of how the technical apparatus works in a smaller-scale setting. The natural domain of definition for simplicial homology is a class of spaces we call∆ complexes, which are a modest generalization of the more classical notion of a simplicial complex.

在文檔中 Allen Hatcher (頁 105-111)