Introduction to Algebraic Topology - jackromo.comjackromo.com › 2019 ›...
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AlgebraicTopology
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Introduction to Algebraic Topology
Jack Romo
University of York
June 2019
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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
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Contents
1 Preliminary Group Theory
2 Constructing Spaces
3 Homotopy
4 The Fundamental Group
5 Free Groups
6 Group Presentations
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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.
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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)
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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)
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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)
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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.
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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.
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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.
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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.
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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 ).
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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}.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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
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Simplices
Definition 13 (Simplex)
The standard n-simplex is the set
∆n =
{(x0, . . . , xn) ∈ Rn+1
∣∣∣∣∣ xi ≥ 0 ∀ i ,n∑
i=0
xn = 1
}
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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}
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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}
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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).
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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).
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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.
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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.
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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
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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
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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.
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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.
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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!)
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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.
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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
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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
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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)
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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)
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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 .
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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 .
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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.
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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.
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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.
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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
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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
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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}
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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.
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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...
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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
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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36
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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41
A Minor Addendum
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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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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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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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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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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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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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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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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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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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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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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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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51
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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53
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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53
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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54
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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54
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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54
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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56
Computing π1(Sn)
Theorem 61
π1(S1) ∼= 〈Z,+〉.
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57
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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58
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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61
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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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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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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Jack Romo
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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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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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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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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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71
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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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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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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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.