Math 751

Week 4

October 26, 2014

1 Free Groups.

Definition 1.1. Let G be a group, and let {xj}j∈J be a set of elements of G. We say that {xj}j∈Jgenerates the group G if every element of G can be written as a product of powers of the elementsof {xj}j∈J .

Let X be a set. We want to define a group generated by the elements of X such that it is “free”in some sense, that is, there are no relations between these generators.

Definition 1.2. The set of words in X is the set

W (X) = {w = xe11 . . . xenn |xi ∈ X, ei = ±1, n ∈ N} ∩ {1}

We will call every element of W (X) a word in X, and 1 ∈W (X) will be denoted the empty word.

We can endow the set W (X) with the binary operation of concatenation.Let us define an equivalence relation on W (X). To do it, we need the following notion.

Definition 1.3. Let w and w′ be words in X. We say that w is equivalent to w′ by an elementaryreduction (and denote it by w∼ew′) if one element of {w,w′} contains a subword of the form xx−1

or x−1x, and the other is obtained from it by deleting this subword.

Using this, we can define an equivalence relation on W (X) as follows.

Definition 1.4. Let w and w′ be words in X. We say that w ∼ w′ if and only if there exist a sequencew1, . . . , wk of words in X such that

w ∼e w1 ∼e . . . ∼e wk ∼e w′

∼ defined like this is obviously reflexive, symmetric and transitive, so it is an equivalence relation.

Definition 1.5. F (X) := W (X)/ ∼.

The relation ∼ is consistent with concatenation, that is, if w1, w′1, w2, w

′2 are words in X such that

w1 ∼ w′1w2 ∼ w′2

then,w1w2 ∼ w′1w′2

Thus, the binary law on W (X) given by concatenation descends to F (X).

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Theorem 1.1. F (X), endowed with the operation that comes from concatenation in W (X), is a groupcalled the free group on X.

Proposition 1.1 (Universal Mapping Property - UMP). Let

i : X −→ F (X)x 7−→ [x]

the map that sends every element of X to the equivalence class of the word it defines, and let

j : X −→ G

be a set map from X to a group G. Then, there is a unique group homomorphism

f : F (X) −→ G

such that f ◦ i = j.

Sketch of proof. We define the map f by

f([xe11 . . . xenn ]) = j(x1)e1 . . . j(xn)en ∈ G

This turns out to be well defined and a homomorphism of groups.

Example 1.1. Let X = {x}. Then,

F (X) = {xn|n ∈ Z} ∼= Z

Let G be the cyclic group of order n, that is,

G = 〈a|an = 1〉

Then,j : X −→ G

x 7−→ a

gives a homomorphismf : F (X) −→ G

which is surjective. Thus, we get that

G ∼= F (X)/ ker(f)

More generally, if G is a group generated by a set X, we can form a free group F (X ′) for any setX ′ such that |X| = |X ′|, and there is an epimorphism

f : F (X ′) −→ G

and therefore,G ∼= F (X)/ ker(f) = 〈x ∈ X|r ∈ ker(f)〉

which gives us a presentation of G by generators and relations.

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2 Free Products.

Let H and K be groups. We want to define a new group from them, which we will call the free productof H and K. To do that, we proceed as follows. First we consider the following set of words.

W (H,K) = {g1g2 . . . gn|gi ∈ H or gi ∈ K} ∪ {1}

As before, we will call 1 the empty word. The concatenation of words defines a binary law on W (H,K).Let us define an equivalence relation on W (H,K). To do it, we need the following notion.

Definition 2.1. Let w and w′ be words in X. We say that w is equivalent to w′ by an elementaryreduction (and denote it by w∼ew′) if one element of {w,w′} contains a subword of the form ab,with both a, b ∈ H or both a, b ∈ K, and the other is obtained from it by

• taking out the subword ab and substituting it by the single element of H (or K) which is theproduct a · b if a 6= b−1.

or

• taking out the subword ab if a = b−1.

Using this, we can define an equivalence relation on W (H,K) as follows.

Definition 2.2. Let w and w′ be words in W (H,K). We say that w ∼ w′ if and only if there exist asequence w1, . . . , wk of words in W (H,K) such that

w ∼e w1 ∼e . . . ∼e wk ∼e w′

∼ defined like this is obviously reflexive, symmetric and transitive, so it is an equivalence relation.

Definition 2.3. H ∗K := W (H,K)/ ∼.

Remark 2.1. Any equivalence class contains a unique reduced word h1k1h2k2 . . . hrkr, withhi ∈ H for all i = 1, . . . , rkj ∈ K for all j = 1, . . . , rhi 6= 1 for all i = 2, . . . , rkj 6= 1 for all j = 1, . . . , r − 1

The equivalence relation ∼ is consistent with concatenation, so the binary law on W (H,K) de-scends to H ∗K, and we have the following result.

Theorem 2.1. H ∗K, endowed with the operation that comes from concatenation, is a group, calledthe free product of H and K.

Proposition 2.1 (Universal Mapping Property - UMP). Let

i : H −→ H ∗Kh 7−→ [h]

andj : K −→ H ∗K

k 7−→ [k]

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the maps that send every element of H (respectively, every element of K) to the equivalence class ofthe word they define, and let

p : H −→ G

andq : K −→ G

be any pair of group homomorphisms. Then, there is a unique group homomorphism

f : H ∗K −→ G

such that f ◦ i = p and f ◦ j = q, or equivalently, such that the following diagram is commutative.

H

H ∗K G

K

i

p

f

jq

Sketch of proof. We define the map f on the unique reduced words as

f(h1k1h2k2 . . . hrkr) = p(h1) · q(k1) · p(h2) · q(k2) · . . . · p(hr) · q(kr) ∈ G

This turns out to be well defined and a homomorphism of groups.

Corollary 2.1. H∗K is unique up to isomorphism, that is, any other group that satisfies this universalmapping property is isomorphic to H ∗K.

Remark 2.2. Free products of any number of groups can be defined.

Example 2.1. Let X = {x1, . . . , xn} be a set of n elements. We define

Fi := F (xi) ∼= Z

for all i = 1, . . . , n. Then,

F (X) ∼= F1 ∗ . . . ∗ Fn ∼= Z ∗ . . . ∗ Z =: Z∗n

Example 2.2. IfH = 〈h|rh〉

K = 〈k|rk〉

are presentations of H and K by generators and relations, then

H ∗K := 〈h, k|rh, rk〉

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Example 2.3. Z2 ∗ Z2 is a free product, but it is not a free group.

Z2 ∗ Z2 = 〈a, b|a2, b2〉 = {1, a, b, ab, ba, aba, bab, abab, . . .}

Letω : Z2 ∗ Z2 −→ Z2

x 7−→ lenght of x mod 2

ω is a homomorphism of groups, andker(ω) = 〈ab〉 ∼= Z

We define the action φ of Z2 on Z = 〈ab〉 by

φ : Z2 × Z −→ Z(a, ab) 7−→ a(ab)a−1 = ba

We have that 〈a〉 ∩ 〈ab〉 = {0}. Thus, Z2 ∗ Z2 = Z o Z2.

Remark 2.3. For a free product ∗α∈AHα, each Hα is identified with a subgroup of ∗

α∈AHα, the onewhose elements are the identity and the one letter words h with h ∈ Hα. We have that

{1} =⋂α∈A

Hα

and for all α, β ∈ A, with α 6= β(Hα\{1}) ∪ (Hβ\{1}) = ∅

For a free product of an arbitrary number of groups we also have the Universal Mapping Property,namely:

Proposition 2.2 (UMP). Let {ϕα : Hα −→ G}α∈A be a collection of group homomorphisms, and letiα : Hβ −→ ∗

α∈AHα the inclusion for all β ∈ A. Then, there exists a unique group homomorphism

ϕ : ∗α∈AHα −→ G

such that, for all α ∈ A,ϕ ◦ iα = ϕα

Sketch of proof. Let h1h2 . . . hn a word in ∗α∈AHα, with hi ∈ Hαi for all i = 1, . . . , n. We define the

map ϕ like this:ϕ(h1h2 . . . hn) = ϕα1(h1) · ϕα2(h2) · . . . · ϕαn(hn) ∈ G

This turns out to be well defined and a homomorphism of groups.

Example 2.4. Let G beG = ×

α∈AHα

the cartesian product of the Hα’s, and let ϕβ : Hβ −→ ×α∈AHα the inclusion for all β ∈ A. The UMPtells us that there exists a unique

ϕ : ∗α∈AHα −→ ×α∈AHα

that preserves every subgroup Hα.

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3 Seifert-Van Kampen Theorem.

Let X be a topological space, x0 ∈ X and let Aα ⊂ X be open path connected subsets for all α ∈ Jsuch that

x0 ∈⋂α∈J

Aα

andX =

⋃α∈J

Aα

The inclusionjα : Aα −→ X

induces(jα)∗ : π1(Aα, x0) −→ π1(X,x0)

for all α ∈ J , so, by the UMP, there exists a unique

ϕ : ∗α∈Jπ1(Aα, x0) −→ π1(X,x0)

For every α, β ∈ J , with α 6= β, we denote by iαβ the following inclusion

iαβ : Aα ∩Aβ −→ Aα

We have this commutative diagram

Aα

Aα ∩Aβ X

Aβ

iαβ

iβα

jα

jβ

which induces the following commutative diagram

π1(Aα, x0)

π1(Aα ∩Aβ, x0) π1(X,x0)

π1(Aβ, x0)

(iαβ)∗

(iβα)∗

(jα)∗

(jβ)∗

Thus, by how we have defined ϕ, we know that

(iαβ)∗(ξ)((iβα)∗(ξ))−1 ∈ ker(ϕ)

for all ξ ∈ π1(Aα ∩Aβ, x0), and for all α, β ∈ J .Now that we have introduced all this notation, we can state Seifert-Van Kampen Theorem.

Theorem 3.1 (Seifert-Van Kampen). If X =⋃α∈J

Aα is a topological space, where Aα is a path-

connected open set such that x0 ∈ Aα for all α ∈ J , then

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1) If Aα ∩Aβ is path-connected for all α, β ∈ J , then

ϕ : ∗α∈Jπ1(Aα, x0) −→ π1(X,x0)

is surjective.

2) If Aα ∩Aβ ∩Aγ is path connected for all α, β, γ ∈ J , then

ker(ϕ) = N〈(iαβ)∗(ξ)((iβα)∗(ξ))−1|ξ ∈ π1(Aα ∩Aβ, x0);α, β ∈ J〉

Proof. 1) Let f : I −→ X be a loop at x0 ∈ X.

By continuity of f and compactness of I, there exists a partition 0 = s0 < s1 < . . . < sm = 1such that

f([si−1, si]) ⊂ Aαifor some αi ∈ J .

We denote by Ai the set Aαi . Let fi := f |[si−1,si]. We have that

f = f1 ∗ f2 ∗ . . . ∗ fm

with fi a path in Ai.

Figure 1: In this example, m = 2. f1 is the path in A1 := Aα that goes from x0 to f(s1) and f2 is the path in A2 := Aβthat goes from f(s1) to x0.

Ai∩Ai+1 is path-connected, and {x0, f(si)} ⊂ Ai∩Ai+1. Thus, there exists gi a path in Ai∩Ai+1

that goes from x0 to f(si) for all i = 1, . . . ,m− 1.

Then,f ∼ (f1 ∗ g1) ∗ (g1 ∗ f2 ∗ g2) ∗ . . . ∗ (gm−1 ∗ fm)

Each of these paths in brackets is a loop at x0, so

[f ] = [f1 ∗ g1][g1 ∗ f2 ∗ g2] . . . [gm−1 ∗ fm]

where f1 ∗ g1 is contained in A1

gi ∗ fi+1 ∗ gi+1 is contained in Ai+1 for all i = 1, . . . ,m− 1gm−1 ∗ fm is contained in Am

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Thus, if we see the classes of these loops as letters in ∗α∈Jπ1(Aα, x0), we get that

[f ] = ϕ([f1 ∗ g1][g1 ∗ f2 ∗ g2] . . . [gm−1 ∗ fm])

Therefore, ϕ is surjective.

2) Clearly, (iαβ)∗(ξ)((iβα)∗(ξ))−1 ∈ ker(ϕ) for all α, β ∈ J and for all ξ ∈ π1(Aα ∩Aβ, x0), so

N〈(iαβ)∗(ξ)((iβα)∗(ξ))−1|ξ ∈ π1(Aα ∩Aβ, x0);α, β ∈ J〉 ⊂ ker(ϕ)

The general proof of the other containment is in Hatcher’s book, and Munkres has the case inwhich |J | = 2, which is easier.

4 The Classification of Surfaces.

Definition 4.1. An n-dimensional manifold with no boundary is a topological space X such thatevery x ∈ X has a neighborhood Ux homeomorphic to Rn.

Definition 4.2. A surface is a 2-dimensional manifold with no boundary.

For us, a surface is going to be a compact manifold with no boundary.Let P be a polygonal region in the plane, with vertices p0, p1, . . . , pm−1 and edges with oriented

labels like the one we have in the picture.

Going through the vertices starting in p0 in counter-clockwise order gives us a labeling scheme. Inthe example we just drew, the labeling scheme is

a1a2a1a−13 a2a

−13

From P and the labeling scheme, we get an identification space as follows:

• The points in the interior of P are identified only to themselves.

• Two edges of the same label are identified by an orientation preserving linear homeomorphism.

Example 4.1 (The torus T 2). We start with the following polygonal region,

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with labeling scheme aba−1b−1. First, we glue the a labels together to get

Now, we glue the b labels together to get the torus T 2.

Example 4.2 (The Sphere S2). From the polygonal region

with labeling scheme aa−1 we get the Sphere S2 gluing the a labels together.

Example 4.3 (The Projective Plane RP 2). From the polygonal region

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with labeling scheme aa we get the projective plane RP 2 gluing the a labels together.

Example 4.4 (The Klein Bottle K). We start with the following polygonal region

with labeling scheme aba−1b. First, we glue the a labels together to get a cilinder just like we did inExample 4.1, but this time, the b labels don’t glue together so nicely, that is, the surface we get bygluing them together can’t be embedded in R3. The resulting surface is called the Klein Bottle.

Proposition 4.1. The identification space X obtained from a polygonal region P like the ones we arediscussing is Hausdorff and compact.

Proof. Let π : P −→ X the projection, where X has the quotient topology. π is continuous by thedefinition of the quotient topology. P is compact, so X = π(P ) is compact, since continuous mapstake compact sets to compact sets.

Let us see that π is a closed map. Let C be a closed set in X. We have that π(C) is closed if andonly if X\π(C) is open, which happens if and only if π−1(X\π(C)) is an open set in P . We have that

π−1(X\π(C)) = P\π−1(π(C))

The only nontrivial identifications occur in the edges of P , which are closed in P , and thus theintersection of C with any edge is again a closed set. Therefore π−1(π(C)) is just the union of C anda finite number of other closed sets. Thus, P\π−1(π(C)) is open, and π is closed.

A quotient map f : Y −→ Z from a compact Hausdorff space Y is closed if and only if Z isHausdorff, so applying this result to π we get that X is Hausdorff.

Definition 4.3. Let M,N be surfaces. We define the connected sum of M and N , and denote itby M#N , as follows:

M#N = (M\D1) t (N\D2)/(∂D1 ∼ ∂D2)

where D1 is a disk in M and D2 is a disk in N .

Example 4.5 (T 2#T 2). Let T1 and T2 be two tori. In the picture, we can see T1 minus a disk andT2 minus a disk.

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Now, identifying the boundaries of these two disks, we get the following surface

Lemma 4.1. If L1 and L2 are labeling schemes for M and N , then their concatenation L1L2 is alabeling scheme for M#N .

Example 4.6. T 2#T 2 has a labeling scheme a1b1a−11 b−11 a2b2a

−12 b−12 .

Let T 21 be the following torus, and D1 a disk inside it.

Let T 22 be the following torus, and D2 a disk inside it.

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These polygonal regions represent T 21 \D1 and T 2

2 \D2 respectively.

To get the connected sum of the two tori, we need to glue ∂D1 with ∂D2, and we get

and this polygonal region has the labeling scheme a1b1a−11 b−11 a2b2a

−12 b−12 , that is, the concatenation

of the labeling schemes of T 21 and T 2

2 .

Definition 4.4. We introduce some new notation.

Tn :=

n times︷ ︸︸ ︷T 2# . . .#T 2

Pn :=

n times︷ ︸︸ ︷RP 2# . . .#RP 2

Our goal is going to be to prove the following result.

Theorem 4.1. Any surface is homeomorphic to S2, Tn or Pn for some n ∈ N.

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