Algebraic Topology Jack Romo Preliminary Group Theory Constructing Spaces Homotopy The Fundamental Group Free Groups Group Presentations 1 Introduction to Algebraic Topology Jack Romo University of York June 2019
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AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

1

Introduction to Algebraic Topology

Jack Romo

University of York

[email protected]

June 2019

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

2

Introduction

• Based on lecture notes from the Oxford course Topologyand Groups, taught by Prof. Marc Lackenby

• Assumes familiarity with basic topology, but everything weneed will be re-proven properly!

• Group theory recommended but not essential, crash courseprovided at the beginning

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

3

Contents

1 Preliminary Group Theory

2 Constructing Spaces

3 Homotopy

4 The Fundamental Group

5 Free Groups

6 Group Presentations

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

4

Preliminary Group Theory

Definition 1 (Group)

A group is a pair G = 〈S , ∗〉, where S is a set and∗ : S × S → S is a binary operation, such that

1 ∃ 1G ∈ S such that g ∗ 1G = g for g ∈ S ;

2 (g ∗ h) ∗ k = g ∗ (h ∗ k) for g , h, k ∈ S ;

3 For g ∈ S , ∃ g−1 ∈ S such that g ∗ g−1 = 1G .

We often write g ∈ G to mean g ∈ S , and treat G itself as aset. We also often contract g ∗ h to gh.

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

5

Basic Laws

Proposition 1

For any group G and g , h ∈ G ,

g ∗ 1G = g = 1G ∗ g (1)

g ∗ g−1 = 1G = g−1 ∗ g (2)

(g ∗ h)−1 = h−1 ∗ g−1 (3)

g−1, 1G are unique. (4)

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

6

Group Homomorphisms

Definition 2 (Group Homomorphisms)

For two groups G ,H, a homomorphism θ : G → H is afunction such that for all g1, g2 ∈ G ,

θ(g1 ∗ g2) = θ(g1) ∗ θ(g2)

where the latter binary operation is that of H.An isomorphism is a bijective homomorphism.

Proposition 2

For any groups G ,H and a homomorphism θ : G → H,

θ(1G ) = 1H (5)

θ(g−1) = θ(g)−1. (6)

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

6

Group Homomorphisms

Definition 2 (Group Homomorphisms)

For two groups G ,H, a homomorphism θ : G → H is afunction such that for all g1, g2 ∈ G ,

θ(g1 ∗ g2) = θ(g1) ∗ θ(g2)

where the latter binary operation is that of H.An isomorphism is a bijective homomorphism.

Proposition 2

For any groups G ,H and a homomorphism θ : G → H,

θ(1G ) = 1H (5)

θ(g−1) = θ(g)−1. (6)

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

7

Subgroups

Definition 3 (Subgroup)

Given a group G , a subgroup S of G is a subset of G that is agroup itself, with the same binary operation restricted to itselements. We write S ≤ G in this case.

Necessarily, we have 1G ∈ S and for a, b ∈ S , ab ∈ S anda−1 ∈ S . These criteria are a sufficient test for a subgroup.

Definition 4 (Normal Subgroup)

A subgroup N ≤ G is a normal subgroup, written N / G , iff forall g ∈ G , n ∈ N, we have g−1ng ∈ N.

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

7

Subgroups

Definition 3 (Subgroup)

Given a group G , a subgroup S of G is a subset of G that is agroup itself, with the same binary operation restricted to itselements. We write S ≤ G in this case.

Necessarily, we have 1G ∈ S and for a, b ∈ S , ab ∈ S anda−1 ∈ S . These criteria are a sufficient test for a subgroup.

Definition 4 (Normal Subgroup)

A subgroup N ≤ G is a normal subgroup, written N / G , iff forall g ∈ G , n ∈ N, we have g−1ng ∈ N.

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

8

Quotient Groups

Definition 5 (Left Cosets)

Given a subset S ⊆ G of a group G and g ∈ G , define the leftcoset gS as

gS = {gs | s ∈ S}

Definition 6 (Quotient Groups)

Given a normal subgroup N / G , the quotient group G/N isthe group whose elements are the left cosets gN for g ∈ G and

(g1N) ∗ (g2N) = {ab | a ∈ g1N, b ∈ g2N}

It turns out that if N is normal, (g1N) ∗ (g2N) = (g1 ∗ g2)N, arequirement for G/N to satisfy the axioms of a group.

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

8

Quotient Groups

Definition 5 (Left Cosets)

Given a subset S ⊆ G of a group G and g ∈ G , define the leftcoset gS as

gS = {gs | s ∈ S}

Definition 6 (Quotient Groups)

Given a normal subgroup N / G , the quotient group G/N isthe group whose elements are the left cosets gN for g ∈ G and

(g1N) ∗ (g2N) = {ab | a ∈ g1N, b ∈ g2N}

It turns out that if N is normal, (g1N) ∗ (g2N) = (g1 ∗ g2)N, arequirement for G/N to satisfy the axioms of a group.

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

9

Kernel and Image

Definition 7 (Kernel)

Given a homomorphism θ : G → H, the kernel ker θ ⊆ G isdefined as

ker θ = θ−1(1H).

Definition 8 (Image)

Given θ : G → H as above, the image Im θ ⊆ H is defined as

Im θ = θ(G ).

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

10

Kernel and Image

Proposition 3

For a homomorphism θ : G → H, Im θ ≤ H and ker θ / G .

Proposition 4

A homomorphism θ : G → H is injective iff ker θ = {1G}.

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

11

Generating Sets

Definition 9 (Generating Set)

A subset of a group S ⊆ G is said to be a generating set iffevery g ∈ G is a product of elements of S and their inverses.

For instance, a generating set of 〈Z,+〉 is {1}.Another generating set is {2, 3}.{2, 6} is NOT a generating set.

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

11

Generating Sets

Definition 9 (Generating Set)

A subset of a group S ⊆ G is said to be a generating set iffevery g ∈ G is a product of elements of S and their inverses.

For instance, a generating set of 〈Z,+〉 is {1}.Another generating set is {2, 3}.

{2, 6} is NOT a generating set.

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

11

Generating Sets

Definition 9 (Generating Set)

A subset of a group S ⊆ G is said to be a generating set iffevery g ∈ G is a product of elements of S and their inverses.

For instance, a generating set of 〈Z,+〉 is {1}.Another generating set is {2, 3}.{2, 6} is NOT a generating set.

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

12

Graphs

Definition 10 (Graph)

A graph Γ = 〈V ,E , δ〉 consists of a set of vertices V , a set ofedges E and δ : E → P(V ) which sends each edge to a subsetof V with 1 or 2 elements. We call δ(e) the endpoints of e.

Definition 11 (Orientation)

An oriented graph is a graph Γ together with functionsι : E → V and τ : E → V such that δ(e) = {ι(e), τ(e)} for alle ∈ E . We call ι and τ the source and target functions.

An oriented graph is a graph Γ together with an orientation.

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

12

Graphs

Definition 10 (Graph)

A graph Γ = 〈V ,E , δ〉 consists of a set of vertices V , a set ofedges E and δ : E → P(V ) which sends each edge to a subsetof V with 1 or 2 elements. We call δ(e) the endpoints of e.

Definition 11 (Orientation)

An oriented graph is a graph Γ together with functionsι : E → V and τ : E → V such that δ(e) = {ι(e), τ(e)} for alle ∈ E . We call ι and τ the source and target functions.

An oriented graph is a graph Γ together with an orientation.

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

13

Cayley Graphs

Definition 12 (Cayley Graph)

For a group G and a generating set S ⊆ G , the Cayley graph isan oriented graph with vertex set G and edge set G × S , suchthat

ι : 〈g , s〉 7→ g

τ : 〈g , s〉 7→ gs

for all g ∈ G , s ∈ S .

Proposition 5

Any two points in a Cayley graph can be joined by a path.

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

13

Cayley Graphs

Definition 12 (Cayley Graph)

For a group G and a generating set S ⊆ G , the Cayley graph isan oriented graph with vertex set G and edge set G × S , suchthat

ι : 〈g , s〉 7→ g

τ : 〈g , s〉 7→ gs

for all g ∈ G , s ∈ S .

Proposition 5

Any two points in a Cayley graph can be joined by a path.

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

14

Constructing Spaces

• Turns out many spaces can be constructed from simpler,finite ones

• Will define some useful methods to construct spaces here,in particular simplicial complexes and cell complexes

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

15

Simplices

Definition 13 (Simplex)

The standard n-simplex is the set

∆n =

{(x0, . . . , xn) ∈ Rn+1

∣∣∣∣∣ xi ≥ 0 ∀ i ,n∑

i=0

xn = 1

}

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

16

Simplices

Definition 14 (Vertices and Faces)

The vertices V (∆n) are all the elements of ∆n where xi = 1for some 0 ≤ i ≤ n.

Given a non-empty subset A ⊆ {0, . . . , n}, a face of ∆n is thesubset

{(x0, . . . , xn) ∈ ∆n | xi = 0∀ i 6∈ A}

Definition 15 (Inside)

The inside of a simplex ∆n is the set

inside(∆n) = {(x0, . . . , xn) ∈ ∆n | xi > 0∀ i}

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

16

Simplices

Definition 14 (Vertices and Faces)

The vertices V (∆n) are all the elements of ∆n where xi = 1for some 0 ≤ i ≤ n.

Given a non-empty subset A ⊆ {0, . . . , n}, a face of ∆n is thesubset

{(x0, . . . , xn) ∈ ∆n | xi = 0∀ i 6∈ A}

Definition 15 (Inside)

The inside of a simplex ∆n is the set

inside(∆n) = {(x0, . . . , xn) ∈ ∆n | xi > 0 ∀ i}

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

17

Simplices

Definition 16 (Affine Extension)

For f : V (∆n)→ Rm, the unique linear extension of f to Rn+1

then restricted to ∆n is the affine extension of f .

Definition 17 (Face Inclusion)

A face inclusion of a standard m-simplex into a standardn-simplex, for m < n, is the affine extension of an injectionV (∆m)→ V (∆n).

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

17

Simplices

Definition 16 (Affine Extension)

For f : V (∆n)→ Rm, the unique linear extension of f to Rn+1

then restricted to ∆n is the affine extension of f .

Definition 17 (Face Inclusion)

A face inclusion of a standard m-simplex into a standardn-simplex, for m < n, is the affine extension of an injectionV (∆m)→ V (∆n).

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

18

Abstract Simplicial Complexes

Definition 18 (Abstract Simplicial Complex)

An abstract simplicial complex is a pair 〈V ,Σ〉, where V is aset of ’vertices’ and Σ is a set of finite subsets of V such that

1 for each v ∈ V , {v} ∈ Σ;

2 if σ ∈ Σ, so is every nonempty subset of σ.

Say that 〈V ,Σ〉 is finite if V is finite.We see the sets σ ∈ Σ as sets of vertices for (|σ| − 1)-simplices.

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

19

Abstract Simplicial Complexes

Definition 19 (Topological Realization)

The topological realization |K | of an abstract simplicialcomplex K = 〈V ,Σ〉 is the space obtained by:

1 For every σ ∈ Σ, taking a copy of the standard(|σ| − 1)-simplex called ∆σ, whose vertices are labelledwith elements of σ;

2 For every σ ⊂ τ ∈ Σ, identifying ∆σ with a subset of ∆τ

by the face inclusion f where all v ∈ V (∆σ) andf (v) ∈ V (∆τ ) share the same label.

Note |K | is a quotient space of the disjoint union of thesimplicial realizations of each σ ∈ Σ.

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

20

Abstract Simplicial Complexes

• Note any point x ∈ |K | is within some n-simplex, and is alinear combination of the vertices

• So, if V = {w0, . . . ,wn}, we have

x =

n∑i=0

λi wi

for λi ∈ [0, 1],∑λi = 1, with the understanding that

λi = 0 if x is not in the respective simplex

• From now on, we say ’simplicial complex’ to refer either toan abstract simplicial complex or its topological realization

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

21

Triangulations

Definition 20 (Triangulation)

A triangulation of a topological space X is a simplicial complexK together with a homeomorphism h : |K | → X .

Examples: I × I , the torus T2

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

22

Subcomplexes and Maps

Definition 21 (Subcomplex)

A subcomplex of a simplicial complex 〈V ,Σ〉 is a simplicialcomplex 〈V ′,Σ′〉 such that V ′ ⊆ V , Σ′ ⊆ Σ.

Definition 22 (Simplicial Map)

A simplicial map between abstract simplicial complexes〈V1,Σ1〉 and 〈V2,Σ2〉 is a function f : V1 → V2 such that, forall σ ∈ Σ1, f (σ) ∈ Σ2.

A simplicial map is a simplicial isomorphism if it has asimplicial inverse.

This induces a natural continuous map |f | : |K1| → |K2| byaffine extension of f . We also call this a simplicial map.

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

22

Subcomplexes and Maps

Definition 21 (Subcomplex)

A subcomplex of a simplicial complex 〈V ,Σ〉 is a simplicialcomplex 〈V ′,Σ′〉 such that V ′ ⊆ V , Σ′ ⊆ Σ.

Definition 22 (Simplicial Map)

A simplicial map between abstract simplicial complexes〈V1,Σ1〉 and 〈V2,Σ2〉 is a function f : V1 → V2 such that, forall σ ∈ Σ1, f (σ) ∈ Σ2.

A simplicial map is a simplicial isomorphism if it has asimplicial inverse.

This induces a natural continuous map |f | : |K1| → |K2| byaffine extension of f . We also call this a simplicial map.

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

23

Subdivisions

Triangulations are not unique; indeed, we may ’refine’ one in anatural way!

Definition 23 (Subdivision)

A subdivision of a simplicial complex K is a triangulation K ′,h : |K ′| → |K | of |K | such that, for any simplex σ′ in K ′, h(σ′)is entirely contained in some simplex of |K | and the restrictionof h to σ′ is affine.

Example: (I × I )(r) for r ∈ N. (A subdivision we will use often!)

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

24

Cell Complexes

Simplicial complexes are useful for finitary arguments but a bitawkward to use directly. Thankfully, there is an alternative!

Definition 24 (Attaching n-cells)

Let X be a space and f : Sn−1 → X be continuous. Then thespace obtained by attaching an n-cell to X along f , denotedX ∪f Dn, is the quotient of the disjoint union X tDn such thatthe equivalence classes are f −1({x}) ∪ {x} for every x ∈ X .

NB: We consider Sn−1 ⊂ Dn to be the boundary of Dn above,where Dn is the n-dimensional closed disk.

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

25

Cell Complexes

Definition 25 (Cell Complex)

A (finite) cell complex is a space X decomposed as

K 0 ⊂ K 1 ⊂ · · · ⊂ Kn = X

where

1 K 0 is a finite set of points, and

2 K i is obtained from K i−1 by attaching a finite number ofi-cells.

Any finite simplicial complex is clearly a finite cell complex; leteach n-simplex be an n-cell.

Examples: The torus, finite graphs

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

26

Homotopy

• A major topological property we can explore algebraically

• We will redefine all that we need from the ground up

• A major result: the Simplicial Approximation Theorem -from continuous functions to simplicial maps

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

27

Homotopy

Let X and Y henceforth be topologoical spaces.

Definition 26 (Homotopy)

A homotopy between two continuous maps f : X → Y ,g : X → Y is a continuous map H : X × I → Y such thatH(x , 0) = f (x) and H(x , 1) = g(x) for all x ∈ X . We would

then say f and g are homotopic, written f ' g or fH' g .

A standard homotopy is the straight-line homotopy, defined as

H(x , t) = (1− t)f (x) + tg(x)

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

27

Homotopy

Let X and Y henceforth be topologoical spaces.

Definition 26 (Homotopy)

A homotopy between two continuous maps f : X → Y ,g : X → Y is a continuous map H : X × I → Y such thatH(x , 0) = f (x) and H(x , 1) = g(x) for all x ∈ X . We would

then say f and g are homotopic, written f ' g or fH' g .

A standard homotopy is the straight-line homotopy, defined as

H(x , t) = (1− t)f (x) + tg(x)

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

28

Homotopy as an Equivalence

Lemma 27 (Gluing Lemma)

If {C1, . . . ,Cn} is a finite covering of a space X by closedsubsets and the restriction of f : X → Y to each Ci iscontinuous, then f is continuous.

Lemma 28

Homotopy is an equivalence relation on C(X ,Y ), the set ofcontinuous maps X → Y .

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

28

Homotopy as an Equivalence

Lemma 27 (Gluing Lemma)

If {C1, . . . ,Cn} is a finite covering of a space X by closedsubsets and the restriction of f : X → Y to each Ci iscontinuous, then f is continuous.

Lemma 28

Homotopy is an equivalence relation on C(X ,Y ), the set ofcontinuous maps X → Y .

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

29

Composition of Homotopies

Lemma 29

Consider the following continuous maps:

Wf→ X

g

⇒hY

k→ Z

Then g ' h implies gf ' hf and kg ' kh.

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

30

Homotopy Equivalence

Definition 30 (Homotopy Equivalence)

Two spaces X and Y are homotopy equivalent, writtenX ' Y , if and only if there exist maps

Xf�g

Y

such that gf ' idX and fg ' idY .

Lemma 31

Homotopy equivalence is an equivalence relation on thecollection of spaces.

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

30

Homotopy Equivalence

Definition 30 (Homotopy Equivalence)

Two spaces X and Y are homotopy equivalent, writtenX ' Y , if and only if there exist maps

Xf�g

Y

such that gf ' idX and fg ' idY .

Lemma 31

Homotopy equivalence is an equivalence relation on thecollection of spaces.

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

31

Contractible Spaces

Definition 32 (Contractible)

A space X is contractible if and only if it is homotopyequivalent to the one-point space.

Proposition 6

X is contractible iff idX ' cx for some x ∈ X .

Examples: Convex subspaces of Rn, Dn

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

31

Contractible Spaces

Definition 32 (Contractible)

A space X is contractible if and only if it is homotopyequivalent to the one-point space.

Proposition 6

X is contractible iff idX ' cx for some x ∈ X .

Examples: Convex subspaces of Rn, Dn

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

32

Homotopy Retraction

Definition 33 (Homotopy Retract)

When A is a subspace of a space X and i : A→ X is theinclusion map, r : X → A is called a homotopy retract if andonly if ri = idA and ir ' idX .

In the above case, clearly A ' X .

Example: Sn−1 and Rn − {0}

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

33

Homotopy Relative to a Set

Definition 34 (Relative Homotopy)

Let X and Y be spaces and A ⊂ X a subspace. Thenf , g : X → Y are homotopic relative to A if and only iff |A= g |A and there is a homotopy H : f ' g such thatH(x , t) = f (x) = g(x) for all x ∈ A, t ∈ I .

Note that homotopy relative to a set is an equivalence relationand Lemma 29 holds in this case.

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

34

The Simplicial ApproximationTheorem

Theorem 35 (Simplicial Approximation Theorem)

Let K and L be simplicial complexes, where K is finite, andf : |K | → |L| a continuous map. Then there exists a subdivisionK ′ of K and simplicial map g : K ′ → L such that |g | ' f .

Hence, if we can triangulate a space, we can just think in termsof finite simplicial maps.

We need more machinery before we can prove this...

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

35

Simplicial Stars

Definition 36 (Star)

Let K be a simplicial complex and x ∈ |K |. The star of x in|K |, denoted stK (x), is defined as

stK (x) =⋃{inside(σ) : σ a simplex of K, x ∈ σ}

Lemma 37

For any x ∈ |K |, stK (x) is open in |K |.

AlgebraicTopology

Jack Romo

PreliminaryGroup Theory

ConstructingSpaces

Homotopy

TheFundamentalGroup

Free Groups

GroupPresentations

35

Simplicial Stars

Definition 36 (Star)

Let K be a simplicial complex and x ∈ |K |. The star of x in|K |, denoted stK (x), is defined as

stK (x) =⋃{inside(σ) : σ a simplex of K, x ∈ σ}

Lemma 37

For any x ∈ |K |, stK (x) is open in |K |.

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Simplicial Stars

Proposition 7

Let K and L be simplicial complexes, and f : |K | → |L| becontinuous. Suppose there exists a function g : V (K )→ V (L)such that f (stK (v)) ⊆ stL(g(v)) for every v ∈ V (K ). Then gis a simplicial map and |g | ' f .

Proposition 8

Let K , L, f and g be as in Proposition 7. Let A be asubcomplex of K and B a subcomplex of L, such thatf (|A|) ⊆ |B|. Then g(A) ⊆ B and the homotopy H : |g | ' fsends |A| to |B| throughout, ie. H(|A|, t) ⊆ |B| for all t.

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Simplicial Stars

Proposition 7

Let K and L be simplicial complexes, and f : |K | → |L| becontinuous. Suppose there exists a function g : V (K )→ V (L)such that f (stK (v)) ⊆ stL(g(v)) for every v ∈ V (K ). Then gis a simplicial map and |g | ' f .

Proposition 8

Let K , L, f and g be as in Proposition 7. Let A be asubcomplex of K and B a subcomplex of L, such thatf (|A|) ⊆ |B|. Then g(A) ⊆ B and the homotopy H : |g | ' fsends |A| to |B| throughout, ie. H(|A|, t) ⊆ |B| for all t.

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37

Metrics on Simplices

We want a subdivision of K such that g exists as inProposition 7. When is this possible? When the subdivision is’sufficiently fine’...

Definition 38 (Standard Metric)

The standard metric d on a finite simplicial complex |K | withvertices {v0, v1, . . . , vn} is defined to be

d

(∑i

λi vi ,∑i

λ′i vi

)=∑i

|λi −λ′i |

This is clearly a metric on |K |.

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38

Metrics on Simplices

Definition 39 (Coarseness)

Let K ′ be a subdivision of K . The coarseness of K ′ is

sup{d(x , y) : x , y ∈ stK (v), v a vertex of K ′}

Example: (I × I )(r) has coarseness 4/r for r ∈ N.

We want to show that g exists when the coarseness of K ′ issufficiently small.

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39

Aside - Covering Theorem

We will need the following from metric spaces:

Definition 40 (Diameter)

The diameter of a subset A of a metric space is defined as

diam(A) = sup{d(x , y) : x , y ∈ A}

Theorem 41 (Lebesgue Covering Theorem)

Let X be a compact metric space and C an open covering ofX . Then there exists a δ > 0 such that every subset of X withdiameter less than δ is entirely contained in some member of C.

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39

Aside - Covering Theorem

We will need the following from metric spaces:

Definition 40 (Diameter)

The diameter of a subset A of a metric space is defined as

diam(A) = sup{d(x , y) : x , y ∈ A}

Theorem 41 (Lebesgue Covering Theorem)

Let X be a compact metric space and C an open covering ofX . Then there exists a δ > 0 such that every subset of X withdiameter less than δ is entirely contained in some member of C.

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40

Back to the Main Theorem

An alternate phrasing we will first prove here:

Theorem 42

Let K , L be simplicial complexes, K finite, and f : |K | → |L|continuous. Then there exists a δ > 0 such that for anysubdivision K ′ of K with coarseness less than δ, there exists asimplicial map g : K ′ → L with g ' f .

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We append the following to Theorem 43, which we will needlater:

Proposition 9

Let A1, . . . ,An be subcomplexes of K and B1, . . . ,Bn besubcomplexes of L such that f (Ai ) ⊆ Bi for all i . Then giventhe simplicial map g from Theorem 43, |g |(Ai ) ⊆ Bi and thehomotopy H : f ' |g | sends Ai to Bi throughout.

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42

Finer Subdivisions

The Simplicial Approximation Theorem follows from Theorem43 and the following:

Proposition 10

A finite simplicial complex K has subdivisions K (r) such thatthe coarseness of K (r) tends to 0 as r →∞.

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43

The Fundamental Group

• A powerful tool to consider homotopic propertiesalgebraically

• We will redefine this construct from the ground up

• Show a powerful conversion to a finite construction interms of simplicial complexes

• Major result: fundamental groups of Sn are trivial forn ≥ 2, and isomorphic to 〈Z,+〉 for n = 1

• A surprising proof at the end...

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44

Paths in a Space

Definition 43 (Path)

A path in a space X is a continuous map f : I → X . A loopbased at a point b ∈ X is a path where f (0) = f (1) = b.

Alternatively, a loop is a continuous map f : S1 → X .

Definition 44 (Composite Path)

Let X be a space and u, v paths in X such that u(1) = v(0).The composite path u.v is given by

u.v(t) =

{u(2t) if 0 ≤ t ≤ 1/2

v(2t − 1) if 1/2 ≤ t ≤ 1.

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Paths in a Space

Definition 43 (Path)

A path in a space X is a continuous map f : I → X . A loopbased at a point b ∈ X is a path where f (0) = f (1) = b.

Alternatively, a loop is a continuous map f : S1 → X .

Definition 44 (Composite Path)

Let X be a space and u, v paths in X such that u(1) = v(0).The composite path u.v is given by

u.v(t) =

{u(2t) if 0 ≤ t ≤ 1/2

v(2t − 1) if 1/2 ≤ t ≤ 1.

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45

The Fundamental Group

We consider spaces with some basepoint b ∈ X , written 〈X , b〉.Continuous maps f : 〈X , b〉 → 〈Y , c〉 must have f (b) = c .

Definition 45 (Fundamental Group)

The fundamental group of 〈X , b〉, denoted π1(X , b), is the setof homotopy classes relative to ∂I of loops based at b, with thepath composition operation.

We still need to show this is a group!

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The Fundamental Group

We consider spaces with some basepoint b ∈ X , written 〈X , b〉.Continuous maps f : 〈X , b〉 → 〈Y , c〉 must have f (b) = c .

Definition 45 (Fundamental Group)

The fundamental group of 〈X , b〉, denoted π1(X , b), is the setof homotopy classes relative to ∂I of loops based at b, with thepath composition operation.

We still need to show this is a group!

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46

Is π1(X , b) a Group?

Lemma 46 (Well-Definedness)

Suppose u and v are paths in X such that u(1) = v(0), andu′, v ′ are paths such that u ' u′, v ' v ′ relative to ∂I . Thenu.v ' u′.v ′ relative to ∂I .

Lemma 47 (Associativity)

Let u, v ,w be paths in X such that u(1) = v(0), v(1) = w(0).Then u.(v .w) ' (u.v).w relative to ∂I .

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Is π1(X , b) a Group?

Lemma 46 (Well-Definedness)

Suppose u and v are paths in X such that u(1) = v(0), andu′, v ′ are paths such that u ' u′, v ' v ′ relative to ∂I . Thenu.v ' u′.v ′ relative to ∂I .

Lemma 47 (Associativity)

Let u, v ,w be paths in X such that u(1) = v(0), v(1) = w(0).Then u.(v .w) ' (u.v).w relative to ∂I .

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47

Is π1(X , b) a Group?

NB: cx : I → X is the constant path at x .

Lemma 48 (Identity)

Let u be a path in X . Then cu(0).u ' u ' u.cu(1) relative to ∂I .

Lemma 49 (Inverses)

Let u be a path in X . Define u−1 to be the path such thatu−1(t) = u(1− t) for all t ∈ I . Then u.u−1 ' cu(0) andu−1.u ' cu(1) relative to ∂I .

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Is π1(X , b) a Group?

NB: cx : I → X is the constant path at x .

Lemma 48 (Identity)

Let u be a path in X . Then cu(0).u ' u ' u.cu(1) relative to ∂I .

Lemma 49 (Inverses)

Let u be a path in X . Define u−1 to be the path such thatu−1(t) = u(1− t) for all t ∈ I . Then u.u−1 ' cu(0) andu−1.u ' cu(1) relative to ∂I .

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48

Path-Components

Definition 50 (Path-Component)

A path-component of a space X is a maximal path-connectedsubset A ⊆ X .

The path-components of X partition the space.

Proposition 11

If b, b′ ∈ X are in the same path-component, thenπ1(X , b) ∼= π1(X , b

′).

If X is path-connected, we omit b and just write π1(X ).

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Path-Components

Definition 50 (Path-Component)

A path-component of a space X is a maximal path-connectedsubset A ⊆ X .

The path-components of X partition the space.

Proposition 11

If b, b′ ∈ X are in the same path-component, thenπ1(X , b) ∼= π1(X , b

′).

If X is path-connected, we omit b and just write π1(X ).

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49

Induced Homomorphisms

Proposition 12

Let 〈X , x〉 and 〈Y , y〉 be spaces with basepoints. Then anycontinuous map f : 〈X , x〉 → 〈Y , y〉 induces a homomorphismf∗ : π1(X , x)→ π1(Y , y). Moreover:

1 (idX )∗ = idπ1(X ,x)

2 if g : 〈Y , y〉 → 〈Z , z〉 is continuous, then (gf )∗ = g∗f∗

3 if f ' f ′ relative to {x}, then f∗ = f ′∗ .

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50

Group Isomorphism

Theorem 51

Let X ,Y be path-connected spaces with X ' Y . Thenπ1(X ) ∼= π1(Y ).

Definition 52

A space is simply-connected if and only if it is path-connectedand has trivial fundamental group.

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Group Isomorphism

Theorem 51

Let X ,Y be path-connected spaces with X ' Y . Thenπ1(X ) ∼= π1(Y ).

Definition 52

A space is simply-connected if and only if it is path-connectedand has trivial fundamental group.

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A Simplicial Version

Definition 53 (Edge Path)

Let K be a simplicial complex. An edge path is a finitesequence (a0, . . . , an) of vertices of K such that for each i ,(ai−1, ai ) spans a simplex of K . (Clearly (ai , ai ) spans a0-simplex.)

An edge loop is a path with an = a0. We define edgecomposition by concatenation.

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Elementary Contraction

Definition 54 (Elementary Contraction)

Let α be an edge path. An elementary contraction of α is anedge path obtained from α by performing one of the followingmoves:

1 Replace (. . . , ai−1, ai , . . . ) with (. . . , ai , . . . ) if ai−1 = ai ;

2 Replace (. . . , ai−1, ai , ai+1, . . . ) with (. . . , ai , . . . ) ifai−1 = ai+1;

3 Replace (. . . , ai−1, ai , ai+1, . . . ) with (. . . , ai−1, ai+1, . . . )if {ai−1, ai , ai+1} spans a 2-simplex of K .

An elementary expansion β of α is an edge path such that α isan elementary contraction of β.

Note that rule 3 generalizes to any n-simplex contraction bycontracting along the 2-faces.

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Elementary Contraction

Definition 54 (Elementary Contraction)

Let α be an edge path. An elementary contraction of α is anedge path obtained from α by performing one of the followingmoves:

1 Replace (. . . , ai−1, ai , . . . ) with (. . . , ai , . . . ) if ai−1 = ai ;

2 Replace (. . . , ai−1, ai , ai+1, . . . ) with (. . . , ai , . . . ) ifai−1 = ai+1;

3 Replace (. . . , ai−1, ai , ai+1, . . . ) with (. . . , ai−1, ai+1, . . . )if {ai−1, ai , ai+1} spans a 2-simplex of K .

An elementary expansion β of α is an edge path such that α isan elementary contraction of β.

Note that rule 3 generalizes to any n-simplex contraction bycontracting along the 2-faces.

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Edge Loop Group

Definition 55 (Edge Equivalence)

Two edge paths α, β are said to be equivalent, written α ∼ β,if and only if β is the result of a finite series of elementarycontractions and expansions applied to α.

Definition 56 (Edge Loop Group)

The edge loop group E (K , b) for a given simplicial complex Kand b ∈ V (K ) is the set of equivalence classes of loops over ∼starting at b with the composition operation.

This is indeed a group, with identity (b) and inverses being thereversed path.

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Edge Loop Group

Definition 55 (Edge Equivalence)

Two edge paths α, β are said to be equivalent, written α ∼ β,if and only if β is the result of a finite series of elementarycontractions and expansions applied to α.

Definition 56 (Edge Loop Group)

The edge loop group E (K , b) for a given simplicial complex Kand b ∈ V (K ) is the set of equivalence classes of loops over ∼starting at b with the composition operation.

This is indeed a group, with identity (b) and inverses being thereversed path.

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Triangulating Fundamental Groups

Theorem 57

For a simplicial complex K and vertex b, E (K , b) ∼= π1(|K |, b).

This clearly shows that fundamental groups can be made intofinite, computable objects given a finite triangulation.

Also, it shows E (K , b) is independent of the choice oftriangulation. So, it doesn’t change with subdivisions.

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Triangulating Fundamental Groups

Theorem 57

For a simplicial complex K and vertex b, E (K , b) ∼= π1(|K |, b).

This clearly shows that fundamental groups can be made intofinite, computable objects given a finite triangulation.

Also, it shows E (K , b) is independent of the choice oftriangulation. So, it doesn’t change with subdivisions.

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Triangulating Fundamental Groups

Theorem 57

For a simplicial complex K and vertex b, E (K , b) ∼= π1(|K |, b).

This clearly shows that fundamental groups can be made intofinite, computable objects given a finite triangulation.

Also, it shows E (K , b) is independent of the choice oftriangulation. So, it doesn’t change with subdivisions.

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55

Computing π1(Sn)

Definition 58 (n-skeleton)

For a simplicial complex K and any non-negative integer n, then-skeleton of K , denoted skeln(K ), is the subcomplexconsisting of the simplicies with dimension ≤ n.

Lemma 59

For any simplicial complex K and vertex b,π1(|K |, b) ∼= π1(|skel2(K )|, b).

Theorem 60

For n ≥ 2, π1(Sn) is trivial.

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Computing π1(Sn)

Definition 58 (n-skeleton)

For a simplicial complex K and any non-negative integer n, then-skeleton of K , denoted skeln(K ), is the subcomplexconsisting of the simplicies with dimension ≤ n.

Lemma 59

For any simplicial complex K and vertex b,π1(|K |, b) ∼= π1(|skel2(K )|, b).

Theorem 60

For n ≥ 2, π1(Sn) is trivial.

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Computing π1(Sn)

Definition 58 (n-skeleton)

For a simplicial complex K and any non-negative integer n, then-skeleton of K , denoted skeln(K ), is the subcomplexconsisting of the simplicies with dimension ≤ n.

Lemma 59

For any simplicial complex K and vertex b,π1(|K |, b) ∼= π1(|skel2(K )|, b).

Theorem 60

For n ≥ 2, π1(Sn) is trivial.

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Computing π1(Sn)

Theorem 61

π1(S1) ∼= 〈Z,+〉.

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The Fundamental Theorem ofAlgebra

You have seen FTA proven using Galois theory and withcomplex analysis. Here, we present a proof with algebraictopology.

Theorem 62 (Fundamental Theorem of Algebra)

For f ∈ C[X ], deg(f ) > 0⇒ 0 ∈ f (C).

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Free Groups

• We have shown existence of useful groups to topology;how do these groups look in general?

• Need a more formal concept of how to ’present’ a group

• Idea: elements are words over an alphabet S ∪ S−1, whereS is a generating set

• We will discover in doing this that the group-topologyconnection is two-way...

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59

Words over S

We assume that the set S is such that S ∩ S−1 = ∅, whereS−1 = {s−1 | s ∈ S}. These are not inverses in any givengroup, just elements of S with an added ·−1 superscript. Wealso specify that (x−1)−1 = x .

Definition 63 (Word)

For any set S , a word is a finite sequence w = s1s2 . . . sn,where sn ∈ S ∪ S−1.

Definition 64 (Concatenation)

For words w1 = s1 . . . sn,w2 = r1 . . . rn, the concatenationw1w2 = s1 . . . snr1 . . . rn.

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Words over S

We assume that the set S is such that S ∩ S−1 = ∅, whereS−1 = {s−1 | s ∈ S}. These are not inverses in any givengroup, just elements of S with an added ·−1 superscript. Wealso specify that (x−1)−1 = x .

Definition 63 (Word)

For any set S , a word is a finite sequence w = s1s2 . . . sn,where sn ∈ S ∪ S−1.

Definition 64 (Concatenation)

For words w1 = s1 . . . sn,w2 = r1 . . . rn, the concatenationw1w2 = s1 . . . snr1 . . . rn.

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Elementary Contractions

Definition 65 (Elementary Contraction/Expansion)

A word w ′ is an elementary contraction of a word w , writtenw ↘ w ′, if w = y1xx

−1y2 and w ′ = y1y2 for words y1, y2 andx , x−1 ∈ S ∪ S−1.

A word w ′ is an elementary expansion of a word w , writtenw ↗ w ′, if w ′ ↘ w .

Definition 66 (Word Equivalence)

Two words w ,w ′ are equivalent, written w ∼ w ′, if and only ifthere exists a finite sequence of words w = w0,w1, . . . ,wn = w ′

such that wi−1 ↘ wi or wi−1 ↗ wi for all i .

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Elementary Contractions

Definition 65 (Elementary Contraction/Expansion)

A word w ′ is an elementary contraction of a word w , writtenw ↘ w ′, if w = y1xx

−1y2 and w ′ = y1y2 for words y1, y2 andx , x−1 ∈ S ∪ S−1.

A word w ′ is an elementary expansion of a word w , writtenw ↗ w ′, if w ′ ↘ w .

Definition 66 (Word Equivalence)

Two words w ,w ′ are equivalent, written w ∼ w ′, if and only ifthere exists a finite sequence of words w = w0,w1, . . . ,wn = w ′

such that wi−1 ↘ wi or wi−1 ↗ wi for all i .

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Free Group

Definition 67 (Free Group)

The free group on the set S , written F (S), is the set ofequivalence classes of words in the alphabet S with theconcatenation operation.

This is clearly well-defined; w ∼ w ′, v ∼ v ′ ⇒ wv ∼ w ′v ′.Checking the axioms is routine.

Definition 68 (Free Generating Set)

If for a group G there is an isomorphism θ : F (S)→ G forsome set S , then θ(S) is known as a free generating set.

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Free Group

Definition 67 (Free Group)

The free group on the set S , written F (S), is the set ofequivalence classes of words in the alphabet S with theconcatenation operation.

This is clearly well-defined; w ∼ w ′, v ∼ v ′ ⇒ wv ∼ w ′v ′.Checking the axioms is routine.

Definition 68 (Free Generating Set)

If for a group G there is an isomorphism θ : F (S)→ G forsome set S , then θ(S) is known as a free generating set.

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62

Reduced Representatives

We would like the ’minimal’ version of a word if possible.

Definition 69 (Reduced)

A word is reduced if it permits no elementary contraction.

Lemma 70 (Sequential Independence)

Let w1,w2,w3 be words such that w1 ↘ w2 ↗ w3. Then eitherw1 = w3 or there is a word w ′2 such that w1 ↗ w ′2 ↘ w3.

Theorem 71

Any element of F (S) is equivalent to a reduced word.

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62

Reduced Representatives

We would like the ’minimal’ version of a word if possible.

Definition 69 (Reduced)

A word is reduced if it permits no elementary contraction.

Lemma 70 (Sequential Independence)

Let w1,w2,w3 be words such that w1 ↘ w2 ↗ w3. Then eitherw1 = w3 or there is a word w ′2 such that w1 ↗ w ′2 ↘ w3.

Theorem 71

Any element of F (S) is equivalent to a reduced word.

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Reduced Representatives

We would like the ’minimal’ version of a word if possible.

Definition 69 (Reduced)

A word is reduced if it permits no elementary contraction.

Lemma 70 (Sequential Independence)

Let w1,w2,w3 be words such that w1 ↘ w2 ↗ w3. Then eitherw1 = w3 or there is a word w ′2 such that w1 ↗ w ′2 ↘ w3.

Theorem 71

Any element of F (S) is equivalent to a reduced word.

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63

The Universal Property

Given a set S , there is a canonical inclusion i : S → F (S),namely the identity.

Theorem 72 (Universal Property)

Given any set S , any group G and function f : S → G , there isa unique homomorphism φ : F (S)→ G such that the followingdiagram commutes:

S G

F (S)

i

f

φ

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64

Fundamental Groups of Graphs

An immediate interesting application of free groups totopology: graphs!Any graph can be seen as a topology by considering theequivalent 1-dimensional cell complex.

Theorem 73

The fundamental group of a countable connected graph is free.

We will spend the rest of today proving this.

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64

Fundamental Groups of Graphs

An immediate interesting application of free groups totopology: graphs!Any graph can be seen as a topology by considering theequivalent 1-dimensional cell complex.

Theorem 73

The fundamental group of a countable connected graph is free.

We will spend the rest of today proving this.

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64

Fundamental Groups of Graphs

An immediate interesting application of free groups totopology: graphs!Any graph can be seen as a topology by considering theequivalent 1-dimensional cell complex.

Theorem 73

The fundamental group of a countable connected graph is free.

We will spend the rest of today proving this.

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65

Subgraphs and Edge Paths

Definition 74 (Subgraph)

Let Γ be a graph with vertex set V , edge set E , and endpointfunction δ. A subgraph of Γ is a graph with vertex set V ′ ⊆ V ,edge set E ′ ⊆ E and δ′ = δ |E ′ such that

⋃δ′(E ′) ⊆ V ′.

Clearly if Γ is oriented, the orientation can similarly beinherited.

Definition 75 (Edge Path)

An edge path in a graph Γ is a path concatenationu0.u1. . . . .un, where each ui is either a path running along asingle edge at unit speed or a constant path based at a vertex.An edge loop is an edge path where u(0) = u(1). An edge path(loop) u : I → Γ is embedded if u is injective (injective on I o .)

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Subgraphs and Edge Paths

Definition 74 (Subgraph)

Let Γ be a graph with vertex set V , edge set E , and endpointfunction δ. A subgraph of Γ is a graph with vertex set V ′ ⊆ V ,edge set E ′ ⊆ E and δ′ = δ |E ′ such that

⋃δ′(E ′) ⊆ V ′.

Clearly if Γ is oriented, the orientation can similarly beinherited.

Definition 75 (Edge Path)

An edge path in a graph Γ is a path concatenationu0.u1. . . . .un, where each ui is either a path running along asingle edge at unit speed or a constant path based at a vertex.An edge loop is an edge path where u(0) = u(1). An edge path(loop) u : I → Γ is embedded if u is injective (injective on I o .)

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Subgraphs and Edge Paths

Definition 74 (Subgraph)

Let Γ be a graph with vertex set V , edge set E , and endpointfunction δ. A subgraph of Γ is a graph with vertex set V ′ ⊆ V ,edge set E ′ ⊆ E and δ′ = δ |E ′ such that

⋃δ′(E ′) ⊆ V ′.

Clearly if Γ is oriented, the orientation can similarly beinherited.

Definition 75 (Edge Path)

An edge path in a graph Γ is a path concatenationu0.u1. . . . .un, where each ui is either a path running along asingle edge at unit speed or a constant path based at a vertex.An edge loop is an edge path where u(0) = u(1). An edge path(loop) u : I → Γ is embedded if u is injective (injective on I o .)

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66

Trees

Definition 76 (Tree)

A tree is a connected graph with no embedded edge loops.

Lemma 77

In a tree, there is a unique embedded edge path betweendistinct vertices.

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Trees

Definition 76 (Tree)

A tree is a connected graph with no embedded edge loops.

Lemma 77

In a tree, there is a unique embedded edge path betweendistinct vertices.

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Maximal Trees

Definition 78 (Maximal Tree)

A maximal tree of a connected graph Γ is a subgraph T that isa tree, but adding any edge in EΓ \ ET creates an embeddededge loop.

Lemma 79

Let Γ be a connected graph and T be a subgraph that is atree. Then the following are equivalent:

1 VT = VΓ;

2 T is maximal.

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Maximal Trees

Definition 78 (Maximal Tree)

A maximal tree of a connected graph Γ is a subgraph T that isa tree, but adding any edge in EΓ \ ET creates an embeddededge loop.

Lemma 79

Let Γ be a connected graph and T be a subgraph that is atree. Then the following are equivalent:

1 VT = VΓ;

2 T is maximal.

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Maximal Trees

Lemma 80

Any connected countable graph Γ contains a maximal tree.

(Aside - This is only true for uncountable graphs if we acceptthe Axiom of Choice. However, we won’t ever need theuncountable case.)

With this, we can finally prove Theorem 73, namely that everycountable connected graph has free fundamental group.

Examples: n-bouquet, Cayley graph of Z2 with generating set{(0, 1), (1, 0)}

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Maximal Trees

Lemma 80

Any connected countable graph Γ contains a maximal tree.

(Aside - This is only true for uncountable graphs if we acceptthe Axiom of Choice. However, we won’t ever need theuncountable case.)

With this, we can finally prove Theorem 73, namely that everycountable connected graph has free fundamental group.

Examples: n-bouquet, Cayley graph of Z2 with generating set{(0, 1), (1, 0)}

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Maximal Trees

Lemma 80

Any connected countable graph Γ contains a maximal tree.

(Aside - This is only true for uncountable graphs if we acceptthe Axiom of Choice. However, we won’t ever need theuncountable case.)

With this, we can finally prove Theorem 73, namely that everycountable connected graph has free fundamental group.

Examples: n-bouquet, Cayley graph of Z2 with generating set{(0, 1), (1, 0)}

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68

Maximal Trees

Lemma 80

Any connected countable graph Γ contains a maximal tree.

(Aside - This is only true for uncountable graphs if we acceptthe Axiom of Choice. However, we won’t ever need theuncountable case.)

With this, we can finally prove Theorem 73, namely that everycountable connected graph has free fundamental group.

Examples: n-bouquet, Cayley graph of Z2 with generating set{(0, 1), (1, 0)}

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69

Group Presentations

• It’s time to develop a way to ’write out’ any group

• Groups can be seen as a free group where some words areidentified (eg. D2n); makes many infinite groups possibleto reason about finitely

• When are two presentations equal?

• What are the presentations of fundamental groups?

• End this lecture series on a high note - a deep connectionbetween group presentations and topological spaces

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70

Generating Normal Subgroups

Definition 81 (Normal Subgroup Generated by B)

Let B ⊆ G , where G is a group. The normal subgroupgenerated by B is the intersection of all normal subgroupscontaining B, denoted 〈B〉.

Proposition 13

The subgroup 〈B〉 consists of all expressions of the form

n∏i=1

gibεii g−1i

for n ∈ Z0, gi ∈ G , bi ∈ B and εi = ±1 for all i .

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Generating Normal Subgroups

Definition 81 (Normal Subgroup Generated by B)

Let B ⊆ G , where G is a group. The normal subgroupgenerated by B is the intersection of all normal subgroupscontaining B, denoted 〈B〉.

Proposition 13

The subgroup 〈B〉 consists of all expressions of the form

n∏i=1

gibεii g−1i

for n ∈ Z0, gi ∈ G , bi ∈ B and εi = ±1 for all i .

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Group Presentations

Definition 82 (Presentation)

Let X be a set, and R ⊆ F (X ). The group with presentation〈X | R〉 is defined as F (X )/〈R〉.

Example: Dihedral group D2n = 〈σ, τ | σn, τ2, τστσ〉

Natural question: when are two words w ,w ′ equivalent in〈X | R〉? We call this the word problem.

Proposition 14

Two words w ,w ′ ∈ F (X ) are equal in 〈X | R〉 if and only ifthey differ by a finite number of the following operations:

1 Elementary contractions or expansions;

2 Inserting an element of 〈R〉 into one of the words.

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Group Presentations

Definition 82 (Presentation)

Let X be a set, and R ⊆ F (X ). The group with presentation〈X | R〉 is defined as F (X )/〈R〉.

Example: Dihedral group D2n = 〈σ, τ | σn, τ2, τστσ〉

Natural question: when are two words w ,w ′ equivalent in〈X | R〉? We call this the word problem.

Proposition 14

Two words w ,w ′ ∈ F (X ) are equal in 〈X | R〉 if and only ifthey differ by a finite number of the following operations:

1 Elementary contractions or expansions;

2 Inserting an element of 〈R〉 into one of the words.

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Group Presentations

Definition 82 (Presentation)

Let X be a set, and R ⊆ F (X ). The group with presentation〈X | R〉 is defined as F (X )/〈R〉.

Example: Dihedral group D2n = 〈σ, τ | σn, τ2, τστσ〉

Natural question: when are two words w ,w ′ equivalent in〈X | R〉? We call this the word problem.

Proposition 14

Two words w ,w ′ ∈ F (X ) are equal in 〈X | R〉 if and only ifthey differ by a finite number of the following operations:

1 Elementary contractions or expansions;

2 Inserting an element of 〈R〉 into one of the words.

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Group Presentations

Definition 82 (Presentation)

Let X be a set, and R ⊆ F (X ). The group with presentation〈X | R〉 is defined as F (X )/〈R〉.

Example: Dihedral group D2n = 〈σ, τ | σn, τ2, τστσ〉

Natural question: when are two words w ,w ′ equivalent in〈X | R〉? We call this the word problem.

Proposition 14

Two words w ,w ′ ∈ F (X ) are equal in 〈X | R〉 if and only ifthey differ by a finite number of the following operations:

1 Elementary contractions or expansions;

2 Inserting an element of 〈R〉 into one of the words.

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72

Group Presentations

Definition 83 (Finite Presentation)

A presentation 〈X | R〉 is finite if and only if X and R arefinite. Likewise, a group is finitely presented if it has a finitepresentation.

Aside: There is a rewriting system such that any two finitepresentations present the same group iff they can be rewrittento each other in this system. Called Tietze transformations.

Proposition 15

Let 〈X | R〉, H be groups. Let f : X → H induce ahomomorphism φ : F (X )→ H. This descends to ahomomorphism 〈X | R〉 → H if and only if φ(R) = {1H}, ie.R ⊆ ker(φ).

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Group Presentations

Definition 83 (Finite Presentation)

A presentation 〈X | R〉 is finite if and only if X and R arefinite. Likewise, a group is finitely presented if it has a finitepresentation.

Aside: There is a rewriting system such that any two finitepresentations present the same group iff they can be rewrittento each other in this system. Called Tietze transformations.

Proposition 15

Let 〈X | R〉, H be groups. Let f : X → H induce ahomomorphism φ : F (X )→ H. This descends to ahomomorphism 〈X | R〉 → H if and only if φ(R) = {1H}, ie.R ⊆ ker(φ).

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Group Presentations

Definition 83 (Finite Presentation)

A presentation 〈X | R〉 is finite if and only if X and R arefinite. Likewise, a group is finitely presented if it has a finitepresentation.

Aside: There is a rewriting system such that any two finitepresentations present the same group iff they can be rewrittento each other in this system. Called Tietze transformations.

Proposition 15

Let 〈X | R〉, H be groups. Let f : X → H induce ahomomorphism φ : F (X )→ H. This descends to ahomomorphism 〈X | R〉 → H if and only if φ(R) = {1H}, ie.R ⊆ ker(φ).

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73

Push-outs

Definition 84 (Push-out)

Let G0,G1,G2 be groups and φ1 : G0 → G1, φ2 : G0 → G2 behomomorphisms. Let 〈X1 | R1〉 and 〈X2 | R2〉 be presentationsof G1,G2 respectively where X1 ∩ X2 = ∅.The push-out G1 ∗G0 G2 of

G1φ1← G0

φ2→ G2

is the group

〈X1 ∪ X2 | R1 ∪ R2 ∪ {φ1(g) = φ2(g) | g ∈ G0}〉

Push-outs are independent of the G1,G2 presentations (proofomitted.)

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Push-outs

Proposition 16 (Universal Property)

Given a pushout G1 ∗G0 G2 of

G1φ1← G0

φ2→ G2

and a group H with morphisms βi : Gi → H such that thediagram

G0φ1 //

φ2

��

G2

β2

��G1

β1 // H

commutes, then there exists a unique homomorphismφ : G1 ∗G0 G2 → H such that the above diagram together withG1 → G1 ∗G0 G2 ← G2 commutes.

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Push-outs

Definition 85 (Free Product)

When G0 in our definition of a push-out is trivial, the push-outis called the free product of G1 and G2.

Definition 86 (Amalgamated Free Product)

When φ1 : G0 → G1 and φ2 : G0 → G2 are injective, we say thepush-out is the amalgamated free product of G1 and G2 alongG0.

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Push-outs of Fundamental Groups

Theorem 87 (Seifert - van Kampen Theorem)

Let K be a space which is a union of path-connected open setsK1,K2, where K1 ∩K2 is path-connected. Then for b ∈ K1 ∩K2

and ix : K1 ∩ K2 → Kx the inclusion maps, we have

π1(K , b) ∼= π1(K1, b) ∗π1(K1∩K2,b) π1(K2, b)

Moreover, the homomorphisms π1(Ki , b)→ π1(K , b) are themaps induced by inclusion.

This gives us a way to build presentations of π1(K , b) fromsmaller parts.

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Push-outs of Fundamental Groups

Theorem 87 (Seifert - van Kampen Theorem)

Let K be a space which is a union of path-connected open setsK1,K2, where K1 ∩K2 is path-connected. Then for b ∈ K1 ∩K2

and ix : K1 ∩ K2 → Kx the inclusion maps, we have

π1(K , b) ∼= π1(K1, b) ∗π1(K1∩K2,b) π1(K2, b)

Moreover, the homomorphisms π1(Ki , b)→ π1(K , b) are themaps induced by inclusion.

This gives us a way to build presentations of π1(K , b) fromsmaller parts.

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77

Topological Application

Recall that conjugacy classes in π1(K , b) correspond tohomotopy classes of baseless loops in K .

Theorem 88

Let K be a connected cell complex, and let li : S1 → K 1 be theattaching maps of its 2-cells, where 1 ≤ i ≤ n. Let b be abasepoint in K 0. Let [li ] be the conjugacy class of the loop li inπ1(K

1, b). Then

π1(K , b) ∼= π1(K1, b)/〈[l1] ∪ · · · ∪ [ln]〉.

Example: T2.

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Topological Application

Recall that conjugacy classes in π1(K , b) correspond tohomotopy classes of baseless loops in K .

Theorem 88

Let K be a connected cell complex, and let li : S1 → K 1 be theattaching maps of its 2-cells, where 1 ≤ i ≤ n. Let b be abasepoint in K 0. Let [li ] be the conjugacy class of the loop li inπ1(K

1, b). Then

π1(K , b) ∼= π1(K1, b)/〈[l1] ∪ · · · ∪ [ln]〉.

Example: T2.

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Topological Application

Recall that conjugacy classes in π1(K , b) correspond tohomotopy classes of baseless loops in K .

Theorem 88

Let K be a connected cell complex, and let li : S1 → K 1 be theattaching maps of its 2-cells, where 1 ≤ i ≤ n. Let b be abasepoint in K 0. Let [li ] be the conjugacy class of the loop li inπ1(K

1, b). Then

π1(K , b) ∼= π1(K1, b)/〈[l1] ∪ · · · ∪ [ln]〉.

Example: T2.

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From Presentations to Spaces

The major result of this course:

Theorem 89

The following are equivalent for a group G:

1 G is finitely presented;

2 G is the fundamental group of a finite connected cellcomplex;

3 G is the fundamental group of a finite connected simplicialcomplex.