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

78 Chapter 3. Covering Spaces

3.6 The Universal Covering

We collect some of our results for the standard situation that B is path connected,

locally path connected and semi-locally simply connected space. Let us now call a covering p W E ! B a universal covering if E is simply connected.

(3.6.1) Theorem (Universal covering). Let B be as above.

(1)There exists up to isomorphism a unique universal covering p W E ! B.

(2)The action of the automorphism group Aut.p/ on E furnishes p with the structure of a left Aut.p/-principal covering.

(3)The group Aut.p/ is isomorphic to 1.B; b/. Given x 2 p 1.b/, an isomorphism x W Aut.p/ ! 1.B; b/ is obtained, if we assign to ˛ 2 Aut.p/ the class of the loop pw for a path w from x to ˛.x/.

(4)The space Eb is simply connected.

Proof. (1) Existence is shown in (3.3.3). Since B is locally path connected, the total space of each covering has the same property. Let pi W Ei ! B be simply connected coverings with base points xi 2 pi 1.b/. By (3.5.2), there exist morphisms ˛ W p1 ! p2 and ˇ W p2 ! p1 such that ˛.x1/ D x2 and ˇ.x2/ D x1. By uniqueness of liftings, ˛ˇ and ˇ˛ are the identity, i.e., ˛ and ˇ are isomorphisms. This shows uniqueness.

(2)By (3.1.8), the action of Aut.p/ on E is properly discontinuous. As in (1) one shows that Aut.p/ acts transitively on each fibre of p. The map Aut.p/nE ! B, induced by p, is therefore a homeomorphism. Since E ! Aut.p/nE is a principal covering, so is p.

(3)Since E is simply connected, there exists a unique homotopy class of paths w from x to ˛.x/. Since x and ˛.x/ are contained in the same fibre, pw is a loop.

Therefore x is well-defined. If we lift a loop u based at b to a path w beginning in x, then there exists ˛ 2 Aut.p/ such that ˛.x/ D w.1/. Hence x is surjective. Two paths starting in x have the same end point if and only if their images in B are homotopic. Hence x is injective. If v is a path from x to ˛.x/ and w a path from x to ˇ.x/, then v ˇw is a path from x to ˛ˇ.x/. Hence x is a homomorphism.

(4) is shown in (3.3.3).

(3.6.2) Theorem (Classification III). Suppose that B has a universal covering p W E ! B. Then p is a 1.B; b/-principal covering. Each connected covering of B is isomorphic to a covering of the form E=H ! B, H 1.B; b/ a subgroup. This covering has H as a characteristic subgroup. Two such coverings are isomorphic if and only if the corresponding subgroups of 1.B; b/ are conjugate.

Problems

1.The product Q11 S1 is not semi-locally simply connected.

2.Is the product of a countably infinite number of the universal covering of S1 a covering?

3.Identify in S1 the open upper and the open lower hemi-sphere to a point. The resulting space X has four points. Show 1.X/ Š Z. Does X have a universal covering?

4.The quotient map p W Rn ! Rn=Zn is a universal covering. The map q W Rn ! T n,

.xj / 7!.exp 2 ixj / is a universal covering of the n-dimensional torus T n D S1 S1. Let f W T n ! T n be a continuous automorphism, and let F W Rn ! Rn be a lifting of f q along q with F .0/ D 0. The assignments x 7!F .x/ C F .y/ and x 7!F .x C y/ are liftings of the same map with the same value for x D 0. Hence F .x C y/ D F .x/ C F .y/. From this relation one deduces that F is a linear map. Since F .Zn/ Zn, the map F is given

by a matrix A 2 GLn.Z/. Conversely, each matrix in GLn.Z/ gives us an automorphism of T n. The group of continuous automorphisms of T n is therefore isomorphic to GLn.Z/.

5.Classify the 2-fold coverings of S1 _ S1 and of S1 _ S1 _ S1. (Note that a subgroup of index 2 is normal.)

6.The k-fold (k 2 N) coverings of S1 _ S1 correspond to isomorphism classes of D

1.S1 _ S1/ D h ui h v i-sets of cardinality k. An action of on f1; : : : ; kg is determined by the action of u and v, and these actions can be arbitrary permutations of f1; : : : ; kg.

Figure 3.1. The 3-fold regular coverings of S1 _ S1.

Hence these actions correspond bijectively to the elements of Sk Sk (Sk the symmetric

Consider the case k D 3 and S3 D h A; B j A3 D 1; B2 D 1; BAB 1 D A 1g. The three conjugacy classes are represented by 1; A; B. We can normalize the first component of each orbit correspondingly. If we fix u, then the centralizer Z.u/ of u acts on the second component. We have Z.1/ D S3, Z.B/ D f1; Bg, and Z.A/ D f1; A; A2g. This yields the following representing pairs for the orbits:

80 Chapter 3. Covering Spaces

The transitive actions (which yield connected coverings) have the addition c, the normal subgroups (which yield regular coverings) have the addition n.

Draw figures for the connected coverings. For this purpose study the restrictions of the coverings to the two summands S1; note that under restriction a connected covering may become disconnected. Over each summand one has a 3-fold covering of S1; there are three of them.

7.Classify the regular 4-fold coverings of S1 _ S1.

8.The Klein bottle has three 2-fold connected coverings. One of them is a torus, the other two are Klein bottles.

9.Let X be path connected and set Y D X I=X @I . Show 1.Y / Š Z. Show that Y has a simply connected universal Z-principal covering. Is Y always locally path connected?

10.Construct a transport space which is not locally path connected.

11.The space Rn with two origins is obtained from Rn C Rn by identifying x 6D0 in the first summand with the same element in the second summand. Let M be the line with two

origins. Construct a universal covering of M and determine 1.M /. What can you say about 1 of Rn with two origins for n > 1?

12.Make the fundamental groupoid ….B/ into a topological groupoid with object space B. (Hypothesis (3.6.1). Use (14.1.17).)

13.Let X be a compact Hausdorff space and H.X/ the group of homeomorphism. Then H.X/ together with the CO-topology is a topological group and H.X/ X ! X, .f; x/ 7! f .x/ a continuous group action.

14.The space C.S1; S1/ with CO-topology becomes a topological group under pointwise multiplication of maps.

is an isomorphism of topological groups. The space M 0.S1/ is isomorphic to the space V

is a homomorphism. The space M1.S1/ is homeomorphic to the space H of homeomorphism

H.S1/ of homeomorphisms of S1 is h-equivalent to O.2/.

Chapter 4

Elementary Homotopy Theory

Further analysis and applications of the homotopy notion require a certain amount of formal consideration. We deal with several related topics.

(1)The construction of auxiliary spaces from the basic “homotopy cylinder” X I : mapping cylinders, mapping cones, suspensions; and dual constructions based on the “path space” XI . These elementary constructions are related to the general problem of defining homotopy limits and homotopy colimits.

(2)Natural group structures on Hom-functors in TOP0. By category theory they arise from group and cogroup objects in this category. But we mainly work with the explicit constructions: suspension and loop space.

(3)Exact sequences involving homotopy functors based on “exact sequences” among pointed spaces (“space level”). These so-called cofibre and fibre sequences are a fundamental contribution of D. Puppe to homotopy theory [155]. The exact sequences have a three-periodic structure, and it has by now become clear that data of this type are an important structure in categories with (formal) homotopy (triangulated categories).

As an application, we use a theorem about homotopy equivalences of mapping cylinders to prove a gluing theorem for homotopy equivalences. The reader may have seen partitions of unity. In homotopy theory they are used to reduce homotopy colimits to ordinary colimits. Here we treat the simplest case: pushouts.

4.1 The Mapping Cylinder

Let f W X ! Y be a map. We construct the mapping cylinder Z D Z.f / of f via the pushout

Here it .x/ D .x; t/. Since h i0; i1 i is a closed embedding, the maps h j; J i, j and J are closed embeddings. We also have the projection q W Z.f / ! Y , .x; t/ 7!f .x/, y 7!y. The relations qj D f and qJ D id hold. We denote elements in Z.f / by

82 Chapter 4. Elementary Homotopy Theory

their representatives in X I C Y .

01

The map J q is homotopic to the identity relative to Y . The homotopy is the identity on Y and contracts I relative 1 to 1.

We thus have a decomposition of f into a closed embedding J and a homotopy equivalence q. From the pushout property we see:

Continuous maps ˇ W Z.f / ! B correspond bijectively to pairs h W X I ! B and ˛ W Y ! B such that h.x; 1/ D ˛f .x/.

In the following we consider Z.f / as a space under X C Y via the embedding (inclusion) h j; J i. We now study homotopy commutative diagrams

together with homotopies ˆ W f 0˛ ' ˇf . In the case that the diagram is commutative, the pair .˛; ˇ/ is a morphism from f to f 0 in the category of arrows in TOP. We consider the data .˛; ˇ; ˆ/ as a generalized morphism. These data induce a

map D Z.˛; ˇ; ˆ/ W Z.f / ! Z.f 0/ defined by

(

is commutative. The composition of two such morphisms between mapping

cylinders is homotopic to a morphism of the same type. Suppose we are given

f 00 W X00 ! Y 00, ˛0 W X0 ! X00, ˇ0 W Y 0 ! Y 00, and a homotopy ˆ0 W f 00˛0 ' ˇ0f 0.

These data yield a composed homotopy ˆ0 ˘ ˆ W f 00˛0˛ ' ˇ0ˇf defined by

(This is the product of the homotopies ˆ0t ˛ and ˇ0ˆt .)

(4.1.1) Lemma. There exists a homotopy

Z.˛0; ˇ0; ˆ0/ ı Z.˛; ˇ; ˆ/ ' Z.˛0˛; ˇ0ˇ; ˆ0 ˘ ˆ/

which is constant on X C Y .

Proof. Both maps coincide on Y and differ on X I by a parameter transformation

Proof. One applies a retraction X I I ! X .@I I [ I 0/ to the map

homotopy.

Proof. By (4.1.1) there exists a homotopy relative to X C Y of the composition in

question to Z.1X ; 1Y ; 1 / ı Z.˛0˛; ˇ0ˇ; ‰/. By (4.1.2) we have a further homotopy to Z.1X ; 1Y ; 1 / ı Z.1X ; 1Y ; 1/, which equals At C Bt on X C Y , and then by (4.1.1) a homotopy to Z.1X ; 1Y ; 1 ˘ 1/, which is constant on X C Y . The

homotopy 1 ˘ 1 W f ' f is homotopic relative to X @I to the constant homotopy kf of f . We thus have an induced homotopy relative X C Y to Z.1X ; 1Y ; kf / and finally a homotopy to the identity (Problems 1 and 2).

(4.1.4) Theorem. Suppose ˛ and ˇ are homotopy equivalences. Then the map Z.˛; ˇ; ˆ/ is a homotopy equivalence.

84 Chapter 4. Elementary Homotopy Theory

Proof. The morphism in (4.1.3) has a right homotopy inverse. We can apply (4.1.1) and (4.1.3) to and see that also has a left homotopy inverse. Henceis a homotopy equivalence. From ı Z.˛; ˇ; ˆ/ ' id we now conclude that Z.˛; ˇ; ˆ/ is a homotopy equivalence.

Problems

1. Suppose ˆ and ‰ are homotopic relative to X @I . Then Z.˛; ˇ; ˆ/ and Z.˛; ˇ; ‰/ are homotopic relative to X C Y .

2. In the case that f 0˛ D ˇf we have the map Z.˛; ˇ/ W Z.f / ! Z.f 0/ induced by ˛ id Cˇ. Let k be the constant homotopy. Then Z.˛; ˇ; k/ ' Z.˛; ˇ/ relative to X C Y . 3. Let Œˆ denote the morphism in ….X; Y 0/ represented by ˆ. We think of .˛; ˇ; Œˆ / as a morphism from ˛ to ˇ. The composition is defined by .˛0; ˇ0; Œˆ0 / ı .˛; ˇ; Œˆ / D

.˛0˛; ˇ0ˇ; Œˆ0 ˘ ˆ /. Show that we obtain in this manner a well-defined category. (This definition works in any 2-category.)

4.2 The Double Mapping Cylinder

Given a pair of maps f W A ! B and g W A ! C . The double mapping cylinder

f g

Z.f; g/ D Z.B A ! C / is the quotient of B C A I C C with respect to the relations f .a/ .a; 0/ and .a; 1/ g.a/ for each a 2 A. We can also define it via a pushout

The map h j0; j1 i is a closed embedding. In the case that f D id.A/, we can identify Z.id.A/; g/ D Z.g/. We can also glue Z.f / and Z.g/ along the common subspace A and obtain essentially Z.f; g/ (up to I [f0g I Š I ). A commutative diagram

fg

induces Z.ˇ; ˛; / W Z.f; g/ ! Z.f 0; g0/, the quotient of ˇ C ˛ id C . We can also generalize to an h-commutative diagram as in the previous section.

(4.2.1) Theorem. Suppose ˇ, ˛, are h-equivalences. Then Z.ˇ; ˛; / is an h-equivalence.

Œ0; 1=2 in Z.f; g/ and Z.g/ to the image of A Œ1=2; 1 C C . We view Z.f; g/ as a space under B C A C C . If we are given homotopies ˆB W f 0˛ ' ˇf , ˆC W g0˛ ' g, we obtain an induced map

‰ D Z.˛; ˇ; ˆB / [A Z.˛; ; ˆC / W Z.f; g/ ! Z.f 0; g0/

which extends ˇ C ˛ C .

(4.2.2) Theorem. Let ˛ be an h-equivalence with h-inverse ˛0 and suppose ˇ and have left h-inverses ˇ0; 0. Choose homotopies At W ˛0˛ ' id, Bt W ˇ0ˇ ' id,

Theorem (4.2.1) is now a consequence of (4.2.2). The reasoning is as for (4.1.4). In general, the ordinary pushout of a pair of maps f; g does not have good homotopy properties. One cannot expect to have a pushout in the homotopy category. A pushout is a colimit, in the terminology of category theory. In homotopy theory one replaces colimits by so-called homotopy colimits. We discuss this in the simplest

case of pushouts. Given a diagram

and a homotopy h W j f ' jCfC. We obtain an induced map ' W Z.f ; fC/ ! X

which is the quotient of h j ; h; jC i W X C X0 I C XC ! X. We define: The diagram (1) together with the homotopy h is called a homotopy pushout or homotopy cocartesian if the map ' is a homotopy equivalence. This definition is

in particular important if the diagram is commutative and h the constant homotopy. Suppose we have inclusions f˙ W X0 X˙ and j˙ W X˙ X such that X D X [ XC. In the case that the interiors X˙ı cover X, the space X is a pushout in the category TOP. In many cases it is also the homotopy pushout; the next proposition

is implied by (4.2.4) and (4.2.5).

(4.2.3) Proposition. Suppose the covering X˙ of X is numerable (defined below). Then X is the homotopy pushout of f˙ W X0 X˙.