топология / Tom Dieck T. Algebraic topology (EMS, 2008)

58 Chapter 2. The Fundamental Group

element in 1.R2 X ˙1/ this loop represents in terms of the generators u; v in 2.8.7. Determine the winding number about points in each of the six complementary regions.

3. Let .Xj j j 2 J / be a family of subspaces of X such that the interiors Xjı cover X. Then each morphism in ….X/ is a composition of morphisms in the Xj . If the intersections Xi \ Xj are path connected and 2 Xi \ Xj , then 1.X; / is generated by loops in the

Xj .

4. Let i0 in (2.6.2) be an isomorphism. Then j1 is an isomorphism. This statement is a

general formal property of pushouts. If i0 is surjective, then j1 is surjective.

5. Projective plane. The real projective plane P 2 is defined as the quotient of S2 by the relation x x. Let Œx0; x1; x2 denote the equivalence class of x D .x0; x1; x2/. We can also obtain P 2 from S1 by attaching a 2-cell

'

S1 P 1

jJ

the boundary circles by a homeomorphism. The projective plane cannot be embedded into R3, as we will prove in (18.3.7). There exist models in R3 with self-intersections (technically,

@M to a point, we obtain a map q W K D M [@M M ! P 2 _ P 2. The induced map on the fundamental group is the homomorphism onto Z=2 Z=2.

2.9 Homotopy Groupoids

The homotopy category does not have good categorical properties. Therefore we consider “homotopy” as an additional structure on the category TOP of topological spaces. The category TOP will be enriched: The set of morphisms ….X; Y / between two objects carries the additional structure of a groupoid. The fundamental groupoid is the special case in which X is a point.

Recall that a category has the data: objects, morphisms, identities, and composition of morphisms. The data satisfy the axioms: composition of morphisms is associative; identities are right and left neutral with respect to composition.

Let X and Y be topological spaces. We define a category ….X; Y / and begin with the data. The objects are the continuous maps X ! Y . A morphism from

f W X ! Y to g W X ! Y is represented by a homotopy K W f ' g. Two such homotopies K and L define the same morphism if they are homotopic relative to X @I with @I D f0; 1g the boundary of I . Let us use a second symbol J D Œ0; 1 for the unit interval. This means: A map ˆ W .X I / J ! Y is a homotopy relative to X @I , if ˆ.x; 0; t/ is independent of t and ˆ.x; 1; t/ is also independent of t. Therefore ˆt W X I ! Y; .x; s/ 7!ˆ.x; s; t/ is for each t 2 J a homotopy from f to g. For this sort of relative homotopy one has, as before, the notion of a product and an inverse, now with respect to the J -variable. Hence we obtain

an equivalence relation on the set of homotopies from f to g. We now define: A morphism W f ! g in ….X; Y / is an equivalence class of homotopies relative to X @I from f to g. Composition of morphisms, denoted ~, is defined by the

product of homotopies

K W f ' g; L W g ' h; ŒL ~ ŒK D ŒK L W f ! h:

This is easily seen to be well-defined (use ˆt I ‰t ). The identity of f in ….X; Y / is represented by the constant homotopy kf W f ' f .

The verification of the category axioms is based on the fact that a reparametrization of a homotopy does not change its class.

(2.9.1) Lemma. Let ˛ W I ! I be a continuous map with ˛.0/ D 0 and ˛.1/ D 1. Then K and K ı .id ˛/ are homotopic relative to X @I .

(2.9.2) Proposition. The data for ….X; Y / satisfy the axioms of a category. The category is a groupoid.

Proof. The associativity of the composition follows, because

.K L/ M D K .L M / ı .id ˛/;

with ˛ defined by ˛.t/ D 2t for t 14 , ˛.t/ D t C 14 for 14 t 12 , ˛.t/ D 2t C 12 for 12 t 1.

Similarly, for each K W f ' g the homotopies kf K, K, and K kg differ by a parameter change. Therefore the constant homotopies represent the identities in the category.

The inverse homotopy K represents an inverse of the morphism defined by K.

60 Chapter 2. The Fundamental Group

The endomorphism set of an object in a groupoid is a group with respect to composition as group law. We thus see, from this view point, that the notion of homotopy directly leads to algebraic objects. This fact is a general and systematic approach to algebraic topology.

The homotopy categories of Section 2.2 have a similar enriched structure. If we work, e.g., with pointed spaces and pointed homotopies, then we obtain for pointed spaces X and Y a category …0.X; Y /. The objects are pointed maps. Morphisms are represented by pointed homotopies, and the equivalence is defined by homotopies ˆ rel X @I such that each ˆt is a pointed homotopy.

The remainder of this section can be skipped on a first reading. We study the

dependence of the groupoids ….X; Y / on X and Y . The formal structure of this dependence can be codified in the notion of a 2-category. Suppose given ˛ W U ! X

and ˇ W Y ! V . Composition with ˛ and ˇ yield a functor

ˇ# D …#.ˇ/ W ….X; Y / ! ….X; V /;

which sends f to ˇf and ŒK to ŒˇK and a functor

˛# D …#.˛/ W ….X; Y / ! ….U; Y /;

which sends f to f ˛ and ŒK to ŒK.˛ id/ . They satisfy .ˇ1ˇ2/# D ˇ#1ˇ#2 and

.˛1˛2/# D ˛2#˛1#. These functors are compatible in the following sense:

(2.9.3) Proposition. Suppose K W f ' g W X ! Y and L W u ' v W Y ! Z are given. Then ŒL ˘ K D v#ŒK ~ f #ŒL D g#ŒL ~ u#ŒK . Here L ˘ K W uf ' vg W X I ! Z; .x; t/ 7!L.K.x; t/; t/.

Proof. We use the bi-homotopy L ı .K id/ W X I I ! Z. Restriction to the diagonal of I I defines L ˘ K. Along the boundary of the square we have the following situation.

with id.X/ , where .t/ D .2t; 0/ for t 2 and .t/ D .1; 2t 1/ for t 2 ,

Chapter 3

Covering Spaces

A covering space is a locally trivial map with discrete fibres. Objects of this type can be classified by algebraic data related to the fundamental group. The reduction of geometric properties to algebraic data is one of the aims of algebraic topology. The main result of this chapter has some formal similarity with Galois theory.

A concise formulation of the classification states the equivalence of two categories. We denote by COVB the category of covering spaces of B; it is the full subcategory of TOPB of spaces over B with objects the coverings of B. Under some restrictions on the topology of B this category is equivalent to the category TRAB D Œ….B/; SET of functors ….B/ ! SET and natural transformations between them. We call it the transport category. It is a natural idea that, when you move from one place to another, you carry something along with you. This transport of “information” is codified in moving along the fibres of a map (here: of a covering). We will show that the transport category is equivalent to something more familiar: group actions on sets.

The second important aspect of covering space theory is the existence of a universal covering of a space. The automorphism group of the universal covering is the fundamental group of the space – and in this manner the fundamental group appears as a symmetry group. Moreover, the whole category of covering spaces is obtainable by a simple construction (associated covering of bundle theory) from the universal covering.

In this chapter we study coverings from the view-point of the fundamental group. Another aspect belongs to bundle theory. In the chapter devoted to bundles we show for instance that isomorphism classes of n-fold coverings over a paracompact space

B correspond to homotopy classes B ! BS.n/ into a so-called classifying space

BS.n/.

3.1 Locally Trivial Maps. Covering Spaces

Let p W E ! B be continuous and U B open. We assume that p is surjective to avoid empty fibres. A trivialization of p over U is a homeomorphism

' W p 1.U / ! U F over U , i.e., a homeomorphism which satisfies pr1 ı' D p.

This condition determines the space F up to homeomorphism, since ' induces a homeomorphism of p 1.u/ with fug F . The map p is locally trivial if there

exists an open covering U of B such that p has a trivialization over each U 2 U. A locally trivial map is also called a bundle or fibre bundle, and a local trivialization a bundle chart. We say, p is trivial over U , if there exists a bundle chart over U . If

64 Chapter 3. Covering Spaces

(3.1.4) Proposition. Let q W E ! B Œ0; 1 be locally trivial with typical fibre F . Then B has an open cover U such that q is trivial over each set U Œ0; 1 , U 2 U.

Proof. If q is trivial over U Œa; b and over U Œb; c , then q is trivial over U Œa; c . Two trivializations over U fbg differ by an automorphism, and this automorphism can be extended over U Œb; c . Use this extended automorphism to change the trivialization over U Œb; c , and then glue the trivializations. By compactness of I there exist 0 D t0 < t1 < < tn D 1 and an open set U such that q is trivial over U Œti ; tiC1 .

For the classification of covering spaces we need spaces with suitable local properties. A space X is called locally connected (locally path connected) if

for each x 2 X and each neighbourhood U of x there exists a connected (path connected) neighbourhood V of x which is contained in U . Both properties are inherited by open subspaces.

(3.1.5) Proposition. The components of a locally connected space are open. The path components of a locally path connected space Y are open and coincide with the components.

Proof. Let K be the component of x. Let V be a connected neighbourhood of x. Then K [ V is connected and therefore contained in K. This shows that K is open.

Let U be a component of Y and K a path component of U . Then U X K is a union of path components, hence open. In the case that U 6DK we would obtain a decomposition of U .

We see that each point in a locally path connected space has a neighbourhood basis of open path connected sets.

(3.1.6) Remark. Let B be path connected and locally path connected. Since a covering is a local homeomorphism, the total space E of a covering of B is locally path connected. Let E0 be a component of E and p0 W E0 ! B the restriction of p. Then p0 is also a covering: The sets U , over which p is trivial, can be taken as path connected, and then a sheet over U is either contained in E0 or disjoint to E0. Since B is path connected, we see by path lifting (3.2.9) that p0 is surjective. By (3.1.5), E is the topological sum of its components. Þ

A left action G E ! E, .g; x/ 7!gx of a discrete group G on E is called properly discontinuous if each x 2 E has an open neighbourhood U such that

U \ gU D ; for g 6De. A properly discontinuous action is free. For more details

about this notion see the chapter on bundle theory, in particular (14.1.12).

A left G -principal covering consists of a covering p W E ! B and a properly discontinuous action of G on E such that p.gx/ D p.x/ for .g; x/ 2 G E and such that the induced action on each fibre is transitive.

(3.1.7) Example. A left G-principal covering p W E ! B induces a homeomorphism of the orbit space E=G with B. The orbit map E ! E=G of a properly discontinuous action is a G-principal covering. Þ

A covering p W E ! B has an automorphism group Aut.p/. An automorphism

is a homeomorphism ˛ W E ! E such that p ı ˛ D p. Maps of this type are also called deck transformations of p. If p is a left G-principal covering, then each

left translation lg W E ! E, x 7!gx is an automorphism of p. We thus obtain a homomorphism l W G ! Aut.p/. Let E be connected. Then an automorphism ˛ is determined by its value at a single point x 2 E, and ˛.x/ is a point in the fibre p 1.p.x//. Since G acts transitively on each fibre, the map l is an isomorphism. Thus the connected principal coverings are the connected coverings with the largest possible automorphism group. Conversely, we can try to find principal coverings by studying the action of the automorphism group.

(3.1.8) Proposition. Let p W E ! B be a covering.

(1)If E is connected, then the action of Aut.p/ (and of each subgroup of Aut.p/) on E is properly discontinuous.

(2)Let B be locally path connected and let H be a subgroup of Aut.p/. Then the map q W E=H ! B induced by p is a covering.

Proof. (1) Let x 2 E and g 2 Aut.p/. Let U be a neighbourhood of p.x/ which is evenly covered, and let Ux be a sheet over U containing x. For y 2 Ux \ gUx we have p.y/ D p.g 1y/, since g 1 is an automorphism. Hence y D g 1y, since both elements are contained in Ux . This shows g 1 D id, hence Ux \ gUx D ; for g 6De, and we see that the action is properly discontinuous.

j 2J Uj be the decomposition into the sheets over U . An element h 2 H permutes

the sheets, since they are the path components of p 1.U /. The equivalence classes with respect to H are therefore open in the quotient topology of E=H and are mapped bijectively and continuously under q. Since p is open, so is q. Hence q is trivial over U . Since B is locally path connected, it has an open covering by such sets U .

A right G-principal covering p W E ! B gives rise to associated coverings. Let F be a set with left G-action. Denote by E G F the quotient space of E F under the equivalence relation .x; f / .xg 1; gf / for x 2 E; f 2 F; g 2 G. The continuous map pF W E G F ! B, .x; f / 7!p.x/ is a covering with typical fibre F .

A G-map W F1 ! F2 induces a morphism of coverings

id G W E G F1 ! E G F2; .x; f / 7!.x; .f //:

66 Chapter 3. Covering Spaces

We thus have obtained a functor “associated coverings”

A.p/ W G-SET ! COVB

from the category G-SET of left G-sets and G-equivariant maps. We call a G-prin- cipal covering p W E ! B over the path connected space B universal if the functor

A.p/ is an equivalence of categories.

3.2 Fibre Transport. Exact Sequence

The relation of a covering space to the fundamental groupoid is obtained via path

lifting. For this purpose we now introduce the notion of a fibration which will be studied later in detail. A map p W E ! B has the homotopy lifting property (HLP)

for the space X if the following holds: For each homotopy h W X I ! B and each

map a W X ! E such that pa.x/ D hi.x/, i.x/ D .x; 0/ there exists a homotopy H W X I ! E with pH D h and H i D a. We call H a lifting of h with initial condition a. The map p is called a fibration if it has the HLP for all spaces.

(3.2.1) Example. A projection p W B F ! B is a fibration. Let a.x/ D

.a1.x/; a2.x//. The condition pa D hi says a1.x/ D h.x; 0/. If we set H.x; t/ D

Proof. Let the homotopy h W X I ! B and the initial condition a be given. Since I is connected, a lifting with given initial condition is uniquely determined (see (3.1.3)). Therefore it suffices to find for each x 2 X an open neighbourhood Vx such that hjVx I admits a lifting with initial condition ajVx . By uniqueness (3.1.3), these partial liftings combine to a well-defined continuous map.

By (3.2.3) there exists for each x 2 B an open neighbourhood Vx and an n 2 N such that h maps Vx Œi=n; .i C 1/=n into a set U over which p is trivial. Since p W p 1.U / ! U is, by (3.2.1), a fibration, hjVx Œi=n; .i C 1/=n has a lifting for each initial condition. Therefore we find by induction over i a lifting of hjVx Œ0; i=n with initial condition ajVx .

(3.2.3) Lemma. Let U be an open covering of B Œ0; 1 . For each b 2 B there exists an open neighbourhood V .b/ of b in B and n D n.b/ 2 N such that for 0 i < n the set V .b/ Œi=n; i=.n C 1/ is contained in some member of U.

The fact that the lifted homotopy is uniquely determined implies that for a covering p W E ! B the diagram

p

EB

used to solve the next homotopy lifting problem with a modified initial condition. It reduces the problem to the HLP for I n.

(3.2.4) Proposition. Let p W E ! B have the HLP for the cube I n. For each commutative diagram

there exists H W I n I ! E with H i D a and pH D h.

Let p W E ! B be a map which has the HLP for a point and for I . Write Fb D p 1.b/. We associate to each path v W I ! B from b to c a map v# W 0.Fb/ !0.Fc / which only depends on Œv 2 ….B/. Let x 2 Fb. Choose a lifting V W I ! E of v with V .0/ D x. We set v#Œx D ŒV .1/ . We have to show that this assignment is well-defined. For this purpose assume given:

(1)u W I ! Fb;

(2)h W I I ! B a homotopy of paths from b to c;

(3)V0; V1 W I ! E liftings of h0, h1 with initial points u.0/; u.1/.

These data yield a map a W I @I [ 0 I ! E, defined by a.s; "/ D V".s/ and a.0; t/ D u.t/. The lifting H of h with initial condition a, according to (3.2.4), yields a path t 7!H.1; t/ in Fc from V0.1/ to V1.1/. This shows that the map v# is well-defined and depends only on the morphism Œv in the fundamental groupoid. The rule w#v# D .v w/# is easily verified from the definitions. Thus we have shown:

(3.2.5) Proposition. The assignments b 7! 0.Fb/ and Œv 7!v# yield a functor

a right action of the fundamental group on the set 0.Fb/. We write 0.F; x/ if Œx is chosen as base point of the set 0.F /. We use the action to define