Given a covering space action of a group G on a path-connected, locally path- path-connected space X , then each subgroup H ⊂ G determines a composition of covering

spacesX

→

^X/H

→

X/G . Show:

(a) Every path-connected covering space between X and X/G is isomorphic to X/H for some subgroup H⊂ G.

(b) Two such covering spaces X/H₁ and X/H₂ of X/G are isomorphic iff H₁ and H₂ are conjugate subgroups of G .

(c) The covering space X/H

→

X/G is normal iff H is a normal subgroup of G , in which case the group of deck transformations of this cover isG/H .

25. Let ϕ :R²

→

^R2 be the linear transformationϕ(x, y)= (2x, y/2). This generates an action of Z on X = R²− {0}. Show this action is a covering space action and computeπ₁(X/Z). Show the orbit space X/Z is non-Hausdorff, and describe how it is a union of four subspaces homeomorphic toS¹×R, coming from the complementary components of the x axis and the y axis.

26. For a covering space p : eX

→

X with X connected, locally path-connected, and semilocally simply-connected, show:

(a) The components of eX are in one-to-one correspondence with the orbits of the action of π₁(X, x₀) on the fiber p⁻¹(x₀) .

(b) Under the Galois correspondence between connected covering spaces of X and subgroups of π₁(X, x₀) , the subgroup corresponding to the component of eX

containing a given lift xe₀ of x₀ is thestabilizer of xe₀, the subgroup consisting of elements whose action on the fiber leaves xe₀ fixed.

27. For a universal cover p : eX

→

X we have two actions of π₁(X, x₀) on the fiber p⁻¹(x₀) , namely the action given by lifting loops at x₀ and the action given by re-stricting deck transformations to the fiber. Are these two actions the same when X= S¹∨ S¹ or X= S¹×S¹? Do the actions always agree whenπ₁(X, x₀) is abelian?

28. Generalize the proof of Theorem 1.7 to show that for a covering space action of a groupG on a simply-connected space Y , π₁(Y /G) is isomorphic to G . [If Y is locally path-connected, this is a special case of part (b) of Proposition 1.40.]

29. Let Y be path-connected, locally path-connected, and simply-connected, and let G₁ and G₂ be subgroups of Homeo(Y ) defining covering space actions on Y . Show that the orbit spacesY /G₁ andY /G₂ are homeomorphic iffG₁ andG₂ are conjugate subgroups of Homeo(Y ) .

30. Draw the Cayley graph of the group Z ∗ Z₂=

a, b |||| b² .

31. Show that the normal covering spaces of S¹∨ S¹ are precisely the graphs that are Cayley graphs of groups with two generators. More generally, the normal cov-ering spaces of the wedge sum of n circles are the Cayley graphs of groups with n generators.

32. Consider covering spaces p : eX

→

^{X with e}X and X connected CW complexes, the cells of eX projecting homeomorphically onto cells of X . Restricting p to the 1 skeleton then gives a covering space eX¹

→

^X1 over the 1 skeleton of X . Show:

(a) Two such covering spaces eX₁

→

^{X and eX2}

→

X are isomorphic iff the restrictions Xe¹₁

→

^{X1 and eX21}

→

^X1 are isomorphic.

(b) eX

→

X is a normal covering space iff eX¹

→

^{X1 is normal.}

(c) The groups of deck transformations of the coverings eX

→

^{X and eX1}

→

^{X1 are}

isomorphic, via the restriction map.

33. In Example 1.44 let d be the greatest common divisor of m and n , and let m⁰ = m/d and n⁰ = n/d. Show that the graph T_m,n/K consists of m⁰ vertices labeled a , n⁰ vertices labeled b , together with d edges joining each a vertex to each b vertex. Deduce that the subgroup K ⊂ Gm,n is free on `m⁰n⁰− m⁰− n⁰+ 1 generators.

Since all groups can be realized as fundamental groups of spaces, this opens the way for using topology to study algebraic properties of groups. The topics in this section and the next give some illustrations of this principle, mainly using covering space theory.

We remind the reader that the Additional Topics which form the remainder of this chapter are not to be regarded as an essential part of the basic core of the book.

Readers who are eager to move on to new topics should feel free to skip ahead.

By definition, agraph is a 1 dimensional CW complex, in other words, a space X obtained from a discrete set X⁰ by attaching a collection of 1 cells e_α. Thus X is obtained from the disjoint union of X⁰ with closed intervals I_α by identifying the two endpoints of each I_α with points of X⁰. The points of X⁰ are thevertices and the 1 cells theedges of X . Note that with this definition an edge does not include its endpoints, so an edge is an open subset of X . The two endpoints of an edge can be the same vertex, so the closure e_α of an edgee_α is homeomorphic either to I or S¹.

SinceX has the quotient topology from the disjoint union X⁰`

αI_α, a subset ofX is open (or closed) iff it intersects the closuree_αof each edgee_αin an open (or closed) set ine_α. One says thatX has the weak topology with respect to the subspaces e_α. In this topology a sequence of points in the interiors of distinct edges forms a closed subset, hence never converges. This is true in particular if the edges containing the sequence all have a common vertex and one tries to choose the sequence so that it gets ‘closer and closer’ to the vertex. Thus if there is a vertex that is the endpoint of infinitely many edges, then the weak topology cannot be a metric topology. An exercise at the end of this section is to show the converse, that the weak topology is a metric topology if each vertex is an endpoint of only finitely many edges.

A basis for the topology of X consists of the open intervals in the edges together with the path-connected neighborhoods of the vertices. A neighborhood of the latter sort about a vertex v is the union of connected open neighborhoods U_α of v in e_α for alle_α containingv . In particular, we see that X is locally path-connected. Hence a graph is connected iff it is path-connected.

If X has only finitely many vertices and edges, then X is compact, being the continuous image of the compact spaceX⁰`

αI_α. The converse is also true, and more generally, a compact subset C of a graph X can meet only finitely many vertices and edges of X . To see this, let the subspace D⊂ C consist of the vertices in C together with one point in each edge that C meets. Then D is a closed subset of X since it

meets each e_α in a closed set. For the same reason, any subset of D is closed, so D has the discrete topology. But D is compact, being a closed subset of the compact spaceC , so D must be finite. By the definition of D this means that C can meet only finitely many vertices and edges.

A subgraph of a graph X is a subspace Y ⊂ X that is a union of vertices and edges of X , such that e_α ⊂ Y implies eα ⊂ Y . The latter condition just says that Y is a closed subspace of X . A tree is a contractible graph. By a tree in a graph X we mean a subgraph that is a tree. We call a tree in X maximal if it contains all the vertices of X . This is equivalent to the more obvious meaning of maximality, as we will see below.

P

roposition 1A.1. Every connected graph contains a maximal tree, and in fact any tree in the graph is contained in a maximal tree.

P

roof: Let X be a connected graph. We will describe a construction that embeds an arbitrary subgraph X₀ ⊂ X as a deformation retract of a subgraph Y ⊂ X that contains all the vertices of X . By choosing X₀ to be any subtree of X , for example a single vertex, this will prove the proposition.

As a preliminary step, we construct a sequence of subgraphsX₀⊂ X1⊂ X2⊂ ···, lettingX_i+1be obtained fromX_i by adjoining the closurese_α of all edgese_α⊂ X −X_i having at least one endpoint inX_i. The unionS

iX_iis open inX since a neighborhood of a point in X_i is contained inX_i+1. Furthermore, S

iX_i is closed since it is a union of closed edges andX has the weak topology. So X=S

iX_i since X is connected.

Now to construct Y we begin by setting Y₀ = X0. Then inductively, assuming that Y_i⊂ X_i has been constructed so as to contain all the vertices of X_i, let Y_i+1 be obtained fromY_iby adjoining one edge connecting each vertex ofX_i+1−X_i toY_i, and let Y =S

iY_i. It is evident that Y_i+1 deformation retracts to Y_i, and we may obtain a deformation retraction of Y to Y₀= X0 by performing the deformation retraction of Y_i+1 to Y_i during the time interval [1/2ⁱ⁺¹, 1/2ⁱ] . Thus a point x∈ Y_i+1− Yi is stationary until this interval, when it moves into Y_i and thereafter continues mov-ing until it reaches Y₀. The resulting homotopy h_t:Y

→

Y is continuous since it is continuous on the closure of each edge andY has the weak topology. tu

Given a maximal tree T ⊂ X and a base vertex x₀ ∈ T , then each edge e_α of X− T determines a loop f_α in X that goes first from x₀ to one endpoint of e_α by a path in T , then across e_α, then back to x₀ by a path in T . Strictly speaking, we should first orient the edge e_α in order to specify which direction to cross it. Note that the homotopy class of f_α is independent of the choice of the paths inT since T is simply-connected.

P

roposition 1A.2. For a connected graph X with maximal tree T , π₁(X) is a free group with basis the classes [f_α] corresponding to the edges e_α of X− T .

In particular this implies that a maximal tree is maximal in the sense of not being contained in any larger tree, since adjoining any edge to a maximal tree produces a graph with nontrivial fundamental group. Another consequence is that a graph is a tree iff it is simply-connected.

P

roof: The quotient map X

→

X/T is a homotopy equivalence by Proposition 0.17.

The quotient X/T is a graph with only one vertex, hence is a wedge sum of circles, whose fundamental group we showed in Example 1.21 to be free with basis the loops given by the edges of X/T , which are the images of the loops f_α in X . tu

Here is a very useful fact about graphs:

L

emma 1A.3. Every covering space of a graph is also a graph, with vertices and edges the lifts of the vertices and edges in the base graph.

P

roof: Let p : eX

→

X be the covering space. For the vertices of eX we take the discrete set eX⁰ = p⁻¹(X⁰) . Writing X as a quotient space of X⁰`

αI_α as in the definition of a graph and applying the path lifting property to the resulting maps I_α

→

^{X , we}

get a unique lift I_α

→

X passing through each point in p^{e −1}(x) , for x ∈ e_α. These lifts define the edges of a graph structure on eX . The resulting topology on eX is the same as its original topology since both topologies have the same basic open sets, the covering projection eX

→

X being a local homeomorphism. tu We can now apply what we have proved about graphs and their fundamental groups to prove a basic fact of group theory:

T

heorem 1A.4. Every subgroup of a free group is free.

P

roof: Given a free group F , choose a graph X with π₁(X)≈ F , for example a wedge of circles corresponding to a basis for F . For each subgroup G of F there is by Proposition 1.36 a covering space p : eX

→

^{X with p}∗ π₁( eX)

= G, hence π₁( eX)≈ G since p_∗ is injective by Proposition 1.31. Since eX is a graph by the preceding lemma, the groupG≈ π1( eX) is free by Proposition 1A.2. tu

The structure of trees can be elucidated by looking more closely at the construc-tions in the proof of Proposition 1A.1. IfX is a tree and v₀ is any vertex ofX , then the construction of a maximal tree Y ⊂ X starting with Y0 = {v0}

yields an increasing sequence of subtreesY_n⊂ X whose union is all ofX since a tree has only one maximal subtree, namely itself.

We can think of the vertices in Y_n− Y_n−1 as being at ‘height’ n , with the edges ofY_n− Y_n−1 connecting these vertices to vertices of height n− 1. In this way we get a ‘height function’ h : X

→

^R

assigning to each vertex its height, and monotone on edges.

For each vertex v of X there is exactly one edge leading downward from v , so by following these downward edges we obtain a path from v to the base vertex v₀. This is an example of anedgepath, which is a composition of finitely many paths each consisting of a single edge traversed monotonically. For any edgepath joiningv to v₀ other than the downward edgepath, the height function would not be monotone and hence would have local maxima, occurring when the edgepath backtracked, retracing some edge it had just crossed. Thus in a tree there is a unique nonbacktracking edgepath joining any two points. All the vertices and edges along this edgepath are distinct.

A tree can contain no subgraph homeomorphic to a circle, since two vertices in such a subgraph could be joined by more than one nonbacktracking edgepath.

Conversely, if a connected graph X contains no circle subgraph, then it must be a tree. For ifT is a maximal tree in X that is not equal to X , then the union of an edge ofX−T with the nonbacktracking edgepath in T joining the endpoints of this edge is a circle subgraph ofX . So if there are no circle subgraphs of X , we must have X= T , a tree.

For an arbitrary connected graphX and a pair of vertices v₀ andv₁ inX there is a unique nonbacktracking edgepath in each homotopy class of paths from v₀ tov₁. This can be seen by lifting to the universal cover eX , which is a tree since it is simply-connected. Choosing a liftve₀ ofv₀, a homotopy class of paths from v₀ tov₁ lifts to a homotopy class of paths starting at ve₀ and ending at a unique lift ve₁ of v₁. Then the unique nonbacktracking edgepath in eX from ve₀ to ve₁ projects to the desired nonbacktracking edgepath in X .

Exercises

在文檔中 Allen Hatcher (頁 90-95)