Algebraic Topology Simplices Jack Romo Preliminary Group Theory Constructing Spaces Definition 13 (Simplex) Homotopy The The standard n-simplex is the set Fundamental Group Free Groups � � n � � ∆ n = � ( x 0 , . . . , x n ) ∈ R n + 1 Group � x i ≥ 0 ∀ i , x n = 1 � Presentations � i = 0 15
Algebraic Topology Simplices Jack Romo Preliminary Definition 14 (Vertices and Faces) Group Theory The vertices V (∆ n ) are all the elements of ∆ n where x i = 1 Constructing Spaces for some 0 ≤ i ≤ n . Homotopy Given a non-empty subset A ⊆ { 0 , . . . , n } , a face of ∆ n is the The Fundamental Group subset { ( x 0 , . . . , x n ) ∈ ∆ n | x i = 0 ∀ i �∈ A } Free Groups Group Presentations 16
Algebraic Topology Simplices Jack Romo Preliminary Definition 14 (Vertices and Faces) Group Theory The vertices V (∆ n ) are all the elements of ∆ n where x i = 1 Constructing Spaces for some 0 ≤ i ≤ n . Homotopy Given a non-empty subset A ⊆ { 0 , . . . , n } , a face of ∆ n is the The Fundamental Group subset { ( x 0 , . . . , x n ) ∈ ∆ n | x i = 0 ∀ i �∈ A } Free Groups Group Presentations Definition 15 (Inside) The inside of a simplex ∆ n is the set inside(∆ n ) = { ( x 0 , . . . , x n ) ∈ ∆ n | x i > 0 ∀ i } 16
Algebraic Topology Simplices Jack Romo Preliminary Group Theory Constructing Definition 16 (Affine Extension) Spaces Homotopy For f : V (∆ n ) → R m , the unique linear extension of f to R n + 1 The then restricted to ∆ n is the affine extension of f . Fundamental Group Free Groups Group Presentations 17
Algebraic Topology Simplices Jack Romo Preliminary Group Theory Constructing Definition 16 (Affine Extension) Spaces Homotopy For f : V (∆ n ) → R m , the unique linear extension of f to R n + 1 The then restricted to ∆ n is the affine extension of f . Fundamental Group Free Groups Definition 17 (Face Inclusion) Group Presentations A face inclusion of a standard m -simplex into a standard n -simplex, for m < n , is the affine extension of an injection V (∆ m ) → V (∆ n ) . 17
Algebraic Topology Abstract Simplicial Complexes Jack Romo Preliminary Group Theory Constructing Definition 18 (Abstract Simplicial Complex) Spaces Homotopy An abstract simplicial complex is a pair � V , Σ � , where V is a The Fundamental set of ’vertices’ and Σ is a set of finite subsets of V such that Group Free Groups 1 for each v ∈ V , { v } ∈ Σ ; Group 2 if σ ∈ Σ , so is every nonempty subset of σ . Presentations Say that � V , Σ � is finite if V is finite. We see the sets σ ∈ Σ as sets of vertices for ( | σ | − 1 ) -simplices. 18
Algebraic Topology Abstract Simplicial Complexes Jack Romo Preliminary Definition 19 (Topological Realization) Group Theory Constructing The topological realization | K | of an abstract simplicial Spaces complex K = � V , Σ � is the space obtained by: Homotopy The 1 For every σ ∈ Σ , taking a copy of the standard Fundamental Group ( | σ | − 1 ) -simplex called ∆ σ , whose vertices are labelled Free Groups with elements of σ ; Group Presentations 2 For every σ ⊂ τ ∈ Σ , identifying ∆ σ with a subset of ∆ τ by the face inclusion f where all v ∈ V (∆ σ ) and f ( v ) ∈ V (∆ τ ) share the same label. Note | K | is a quotient space of the disjoint union of the simplicial realizations of each σ ∈ Σ . 19
Algebraic Topology Abstract Simplicial Complexes Jack Romo Preliminary Group Theory • Note any point x ∈ | K | is within some n-simplex, and is a Constructing linear combination of the vertices Spaces Homotopy • So, if V = { w 0 , . . . , w n } , we have The Fundamental n Group � x = λ i w i Free Groups Group i = 0 Presentations for λ i ∈ [ 0 , 1 ] , � λ i = 1, with the understanding that λ i = 0 if x is not in the respective simplex • From now on, we say ’simplicial complex’ to refer either to an abstract simplicial complex or its topological realization 20
Algebraic Topology Triangulations Jack Romo Preliminary Group Theory Constructing Spaces Homotopy Definition 20 (Triangulation) The Fundamental Group A triangulation of a topological space X is a simplicial complex Free Groups K together with a homeomorphism h : | K | → X . Group Presentations Examples: I × I , the torus T 2 21
Algebraic Topology Subcomplexes and Maps Jack Romo Definition 21 (Subcomplex) Preliminary Group Theory Constructing A subcomplex of a simplicial complex � V , Σ � is a simplicial Spaces complex � V ′ , Σ ′ � such that V ′ ⊆ V , Σ ′ ⊆ Σ . Homotopy The Fundamental Group Free Groups Group Presentations 22
Algebraic Topology Subcomplexes and Maps Jack Romo Definition 21 (Subcomplex) Preliminary Group Theory Constructing A subcomplex of a simplicial complex � V , Σ � is a simplicial Spaces complex � V ′ , Σ ′ � such that V ′ ⊆ V , Σ ′ ⊆ Σ . Homotopy The Fundamental Definition 22 (Simplicial Map) Group Free Groups A simplicial map between abstract simplicial complexes Group � V 1 , Σ 1 � and � V 2 , Σ 2 � is a function f : V 1 → V 2 such that, for Presentations all σ ∈ Σ 1 , f ( σ ) ∈ Σ 2 . A simplicial map is a simplicial isomorphism if it has a simplicial inverse. This induces a natural continuous map | f | : | K 1 | → | K 2 | by affine extension of f . We also call this a simplicial map. 22
Algebraic Topology Subdivisions Jack Romo Preliminary Group Theory Constructing Triangulations are not unique; indeed, we may ’refine’ one in a Spaces natural way! Homotopy The Definition 23 (Subdivision) Fundamental Group A subdivision of a simplicial complex K is a triangulation K ′ , Free Groups h : | K ′ | → | K | of | K | such that, for any simplex σ ′ in K ′ , h ( σ ′ ) Group Presentations is entirely contained in some simplex of | K | and the restriction of h to σ ′ is affine. Example: ( I × I ) ( r ) for r ∈ N . (A subdivision we will use often!) 23
Algebraic Topology Cell Complexes Jack Romo Preliminary Group Theory Simplicial complexes are useful for finitary arguments but a bit Constructing Spaces awkward to use directly. Thankfully, there is an alternative! Homotopy Definition 24 (Attaching n-cells) The Fundamental Group Let X be a space and f : S n − 1 → X be continuous. Then the Free Groups space obtained by attaching an n-cell to X along f , denoted Group X ∪ f D n , is the quotient of the disjoint union X ⊔ D n such that Presentations the equivalence classes are f − 1 ( { x } ) ∪ { x } for every x ∈ X . NB: We consider S n − 1 ⊂ D n to be the boundary of D n above, where D n is the n -dimensional closed disk. 24
Algebraic Topology Cell Complexes Jack Romo Preliminary Definition 25 (Cell Complex) Group Theory Constructing A (finite) cell complex is a space X decomposed as Spaces Homotopy K 0 ⊂ K 1 ⊂ · · · ⊂ K n = X The Fundamental Group where Free Groups 1 K 0 is a finite set of points, and Group Presentations 2 K i is obtained from K i − 1 by attaching a finite number of i -cells. Any finite simplicial complex is clearly a finite cell complex; let each n -simplex be an n -cell. Examples: The torus, finite graphs 25
Algebraic Topology Homotopy Jack Romo Preliminary Group Theory Constructing Spaces Homotopy • A major topological property we can explore algebraically The Fundamental Group • We will redefine all that we need from the ground up Free Groups • A major result: the Simplicial Approximation Theorem - Group from continuous functions to simplicial maps Presentations 26
Algebraic Topology Homotopy Jack Romo Preliminary Group Theory Let X and Y henceforth be topologoical spaces. Constructing Spaces Definition 26 (Homotopy) Homotopy A homotopy between two continuous maps f : X → Y , The Fundamental g : X → Y is a continuous map H : X × I → Y such that Group H ( x , 0 ) = f ( x ) and H ( x , 1 ) = g ( x ) for all x ∈ X . We would Free Groups Group H then say f and g are homotopic , written f ≃ g or f ≃ g . Presentations 27
Algebraic Topology Homotopy Jack Romo Preliminary Group Theory Let X and Y henceforth be topologoical spaces. Constructing Spaces Definition 26 (Homotopy) Homotopy A homotopy between two continuous maps f : X → Y , The Fundamental g : X → Y is a continuous map H : X × I → Y such that Group H ( x , 0 ) = f ( x ) and H ( x , 1 ) = g ( x ) for all x ∈ X . We would Free Groups Group H then say f and g are homotopic , written f ≃ g or f ≃ g . Presentations A standard homotopy is the straight-line homotopy , defined as H ( x , t ) = ( 1 − t ) f ( x ) + tg ( x ) 27
Algebraic Topology Homotopy as an Equivalence Jack Romo Preliminary Group Theory Constructing Lemma 27 (Gluing Lemma) Spaces Homotopy If { C 1 , . . . , C n } is a finite covering of a space X by closed The Fundamental subsets and the restriction of f : X → Y to each C i is Group continuous, then f is continuous. Free Groups Group Presentations 28
Algebraic Topology Homotopy as an Equivalence Jack Romo Preliminary Group Theory Constructing Lemma 27 (Gluing Lemma) Spaces Homotopy If { C 1 , . . . , C n } is a finite covering of a space X by closed The Fundamental subsets and the restriction of f : X → Y to each C i is Group continuous, then f is continuous. Free Groups Group Presentations Lemma 28 Homotopy is an equivalence relation on C ( X , Y ) , the set of continuous maps X → Y . 28
Algebraic Topology Composition of Homotopies Jack Romo Preliminary Group Theory Constructing Spaces Lemma 29 Homotopy The Consider the following continuous maps: Fundamental Group g Free Groups f k W → X Y → Z ⇒ Group h Presentations Then g ≃ h implies gf ≃ hf and kg ≃ kh. 29
Algebraic Topology Homotopy Equivalence Jack Romo Preliminary Definition 30 (Homotopy Equivalence) Group Theory Constructing Spaces Two spaces X and Y are homotopy equivalent , written Homotopy X ≃ Y , if and only if there exist maps The Fundamental f Group X g Y ⇄ Free Groups Group Presentations such that gf ≃ id X and fg ≃ id Y . 30
Algebraic Topology Homotopy Equivalence Jack Romo Preliminary Definition 30 (Homotopy Equivalence) Group Theory Constructing Spaces Two spaces X and Y are homotopy equivalent , written Homotopy X ≃ Y , if and only if there exist maps The Fundamental f Group X g Y ⇄ Free Groups Group Presentations such that gf ≃ id X and fg ≃ id Y . Lemma 31 Homotopy equivalence is an equivalence relation on the collection of spaces. 30
Algebraic Topology Contractible Spaces Jack Romo Preliminary Group Theory Constructing Spaces Definition 32 (Contractible) Homotopy A space X is contractible if and only if it is homotopy The Fundamental equivalent to the one-point space. Group Free Groups Group Presentations 31
Algebraic Topology Contractible Spaces Jack Romo Preliminary Group Theory Constructing Spaces Definition 32 (Contractible) Homotopy A space X is contractible if and only if it is homotopy The Fundamental equivalent to the one-point space. Group Free Groups Proposition 6 Group Presentations X is contractible iff id X ≃ c x for some x ∈ X. Examples: Convex subspaces of R n , D n 31
Algebraic Topology Homotopy Retraction Jack Romo Preliminary Group Theory Constructing Spaces Definition 33 (Homotopy Retract) Homotopy The When A is a subspace of a space X and i : A → X is the Fundamental Group inclusion map, r : X → A is called a homotopy retract if and Free Groups only if ri = id A and ir ≃ id X . Group Presentations In the above case, clearly A ≃ X . Example: S n − 1 and R n − { 0 } 32
Algebraic Topology Homotopy Relative to a Set Jack Romo Preliminary Group Theory Constructing Spaces Definition 34 (Relative Homotopy) Homotopy The Let X and Y be spaces and A ⊂ X a subspace. Then Fundamental Group f , g : X → Y are homotopic relative to A if and only if Free Groups f | A = g | A and there is a homotopy H : f ≃ g such that Group H ( x , t ) = f ( x ) = g ( x ) for all x ∈ A , t ∈ I . Presentations Note that homotopy relative to a set is an equivalence relation and Lemma 29 holds in this case. 33
Algebraic Topology The Simplicial Approximation Jack Romo Theorem Preliminary Group Theory Constructing Spaces Homotopy Theorem 35 (Simplicial Approximation Theorem) The Fundamental Let K and L be simplicial complexes, where K is finite, and Group f : | K | → | L | a continuous map. Then there exists a subdivision Free Groups K ′ of K and simplicial map g : K ′ → L such that | g | ≃ f . Group Presentations Hence, if we can triangulate a space, we can just think in terms of finite simplicial maps. We need more machinery before we can prove this... 34
Algebraic Topology Simplicial Stars Jack Romo Preliminary Group Theory Constructing Definition 36 (Star) Spaces Homotopy Let K be a simplicial complex and x ∈ | K | . The star of x in The | K | , denoted st K ( x ) , is defined as Fundamental Group � Free Groups st K ( x ) = { inside( σ ) : σ a simplex of K , x ∈ σ } Group Presentations 35
Algebraic Topology Simplicial Stars Jack Romo Preliminary Group Theory Constructing Definition 36 (Star) Spaces Homotopy Let K be a simplicial complex and x ∈ | K | . The star of x in The | K | , denoted st K ( x ) , is defined as Fundamental Group � Free Groups st K ( x ) = { inside( σ ) : σ a simplex of K , x ∈ σ } Group Presentations Lemma 37 For any x ∈ | K | , st K ( x ) is open in | K | . 35
Algebraic Topology Simplicial Stars Jack Romo Preliminary Group Theory Proposition 7 Constructing Spaces Let K and L be simplicial complexes, and f : | K | → | L | be Homotopy continuous. Suppose there exists a function g : V ( K ) → V ( L ) The such that f ( st K ( v )) ⊆ st L ( g ( v )) for every v ∈ V ( K ) . Then g Fundamental Group is a simplicial map and | g | ≃ f . Free Groups Group Presentations 36
Algebraic Topology Simplicial Stars Jack Romo Preliminary Group Theory Proposition 7 Constructing Spaces Let K and L be simplicial complexes, and f : | K | → | L | be Homotopy continuous. Suppose there exists a function g : V ( K ) → V ( L ) The such that f ( st K ( v )) ⊆ st L ( g ( v )) for every v ∈ V ( K ) . Then g Fundamental Group is a simplicial map and | g | ≃ f . Free Groups Group Proposition 8 Presentations Let K, L, f and g be as in Proposition 7. Let A be a subcomplex of K and B a subcomplex of L, such that f ( | A | ) ⊆ | B | . Then g ( A ) ⊆ B and the homotopy H : | g | ≃ f sends | A | to | B | throughout, ie. H ( | A | , t ) ⊆ | B | for all t. 36
Algebraic Topology Metrics on Simplices Jack Romo Preliminary We want a subdivision of K such that g exists as in Group Theory Proposition 7. When is this possible? When the subdivision is Constructing Spaces ’sufficiently fine’... Homotopy Definition 38 (Standard Metric) The Fundamental Group The standard metric d on a finite simplicial complex | K | with Free Groups vertices { v 0 , v 1 , . . . , v n } is defined to be Group Presentations �� � � � λ ′ | λ i − λ ′ d λ i v i , i v i = i | i i i This is clearly a metric on | K | . 37
Algebraic Topology Metrics on Simplices Jack Romo Preliminary Group Theory Constructing Definition 39 (Coarseness) Spaces Homotopy Let K ′ be a subdivision of K . The coarseness of K ′ is The Fundamental Group sup { d ( x , y ) : x , y ∈ st K ( v ) , v a vertex of K ′ } Free Groups Group Presentations Example: ( I × I ) ( r ) has coarseness 4 / r for r ∈ N . We want to show that g exists when the coarseness of K ′ is sufficiently small. 38
Algebraic Topology Aside - Covering Theorem Jack Romo Preliminary We will need the following from metric spaces: Group Theory Constructing Definition 40 (Diameter) Spaces Homotopy The diameter of a subset A of a metric space is defined as The Fundamental Group diam( A ) = sup { d ( x , y ) : x , y ∈ A } Free Groups Group Presentations 39
Algebraic Topology Aside - Covering Theorem Jack Romo Preliminary We will need the following from metric spaces: Group Theory Constructing Definition 40 (Diameter) Spaces Homotopy The diameter of a subset A of a metric space is defined as The Fundamental Group diam( A ) = sup { d ( x , y ) : x , y ∈ A } Free Groups Group Presentations Theorem 41 (Lebesgue Covering Theorem) Let X be a compact metric space and C an open covering of X. Then there exists a δ > 0 such that every subset of X with diameter less than δ is entirely contained in some member of C . 39
Algebraic Topology Back to the Main Theorem Jack Romo Preliminary Group Theory Constructing Spaces An alternate phrasing we will first prove here: Homotopy The Theorem 42 Fundamental Group Let K, L be simplicial complexes, K finite, and f : | K | → | L | Free Groups continuous. Then there exists a δ > 0 such that for any Group subdivision K ′ of K with coarseness less than δ , there exists a Presentations simplicial map g : K ′ → L with g ≃ f . 40
Algebraic Topology A Minor Addendum Jack Romo Preliminary Group Theory Constructing Spaces We append the following to Theorem 43, which we will need Homotopy later: The Fundamental Proposition 9 Group Free Groups Let A 1 , . . . , A n be subcomplexes of K and B 1 , . . . , B n be Group subcomplexes of L such that f ( A i ) ⊆ B i for all i. Then given Presentations the simplicial map g from Theorem 43, | g | ( A i ) ⊆ B i and the homotopy H : f ≃ | g | sends A i to B i throughout. 41
Algebraic Topology Finer Subdivisions Jack Romo Preliminary Group Theory Constructing Spaces Homotopy The Simplicial Approximation Theorem follows from Theorem The 43 and the following: Fundamental Group Proposition 10 Free Groups A finite simplicial complex K has subdivisions K ( r ) such that Group Presentations the coarseness of K ( r ) tends to 0 as r → ∞ . 42
Algebraic Topology The Fundamental Group Jack Romo Preliminary Group Theory Constructing • A powerful tool to consider homotopic properties Spaces Homotopy algebraically The • We will redefine this construct from the ground up Fundamental Group • Show a powerful conversion to a finite construction in Free Groups terms of simplicial complexes Group Presentations • Major result: fundamental groups of S n are trivial for n ≥ 2, and isomorphic to � Z , + � for n = 1 • A surprising proof at the end... 43
Algebraic Topology Paths in a Space Jack Romo Preliminary Definition 43 (Path) Group Theory Constructing A path in a space X is a continuous map f : I → X . A loop Spaces based at a point b ∈ X is a path where f ( 0 ) = f ( 1 ) = b . Homotopy The Alternatively, a loop is a continuous map f : S 1 → X . Fundamental Group Free Groups Group Presentations 44
Algebraic Topology Paths in a Space Jack Romo Preliminary Definition 43 (Path) Group Theory Constructing A path in a space X is a continuous map f : I → X . A loop Spaces based at a point b ∈ X is a path where f ( 0 ) = f ( 1 ) = b . Homotopy The Alternatively, a loop is a continuous map f : S 1 → X . Fundamental Group Free Groups Definition 44 (Composite Path) Group Presentations 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 ( 2 t ) if 0 ≤ t ≤ 1 / 2 u . v ( t ) = v ( 2 t − 1 ) if 1 / 2 ≤ t ≤ 1 . 44
Algebraic Topology The Fundamental Group Jack Romo Preliminary Group Theory Constructing We consider spaces with some basepoint b ∈ X , written � X , b � . Spaces Homotopy Continuous maps f : � X , b � → � Y , c � must have f ( b ) = c . The Fundamental Definition 45 (Fundamental Group) Group Free Groups The fundamental group of � X , b � , denoted π 1 ( X , b ) , is the set Group of homotopy classes relative to ∂ I of loops based at b , with the Presentations path composition operation. 45
Algebraic Topology The Fundamental Group Jack Romo Preliminary Group Theory Constructing We consider spaces with some basepoint b ∈ X , written � X , b � . Spaces Homotopy Continuous maps f : � X , b � → � Y , c � must have f ( b ) = c . The Fundamental Definition 45 (Fundamental Group) Group Free Groups The fundamental group of � X , b � , denoted π 1 ( X , b ) , is the set Group of homotopy classes relative to ∂ I of loops based at b , with the Presentations path composition operation. We still need to show this is a group! 45
Algebraic Topology Is π 1 ( X , b ) a Group? Jack Romo Preliminary Group Theory Constructing Lemma 46 (Well-Definedness) Spaces Homotopy Suppose u and v are paths in X such that u ( 1 ) = v ( 0 ) , and The u ′ , v ′ are paths such that u ≃ u ′ , v ≃ v ′ relative to ∂ I. Then Fundamental Group u . v ≃ u ′ . v ′ relative to ∂ I. Free Groups Group Presentations 46
Algebraic Topology Is π 1 ( X , b ) a Group? Jack Romo Preliminary Group Theory Constructing Lemma 46 (Well-Definedness) Spaces Homotopy Suppose u and v are paths in X such that u ( 1 ) = v ( 0 ) , and The u ′ , v ′ are paths such that u ≃ u ′ , v ≃ v ′ relative to ∂ I. Then Fundamental Group u . v ≃ u ′ . v ′ relative to ∂ I. Free Groups Group Presentations 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. 46
Algebraic Topology Is π 1 ( X , b ) a Group? Jack Romo Preliminary Group Theory NB: c x : I → X is the constant path at x . Constructing Spaces Lemma 48 (Identity) Homotopy The Let u be a path in X. Then c u ( 0 ) . u ≃ u ≃ u . c u ( 1 ) relative to ∂ I. Fundamental Group Free Groups Group Presentations 47
Algebraic Topology Is π 1 ( X , b ) a Group? Jack Romo Preliminary Group Theory NB: c x : I → X is the constant path at x . Constructing Spaces Lemma 48 (Identity) Homotopy The Let u be a path in X. Then c u ( 0 ) . u ≃ u ≃ u . c u ( 1 ) relative to ∂ I. Fundamental Group Free Groups Lemma 49 (Inverses) Group Presentations Let u be a path in X. Define u − 1 to be the path such that u − 1 ( t ) = u ( 1 − t ) for all t ∈ I. Then u . u − 1 ≃ c u ( 0 ) and u − 1 . u ≃ c u ( 1 ) relative to ∂ I. 47
Algebraic Topology Path-Components Jack Romo Preliminary Group Theory Definition 50 (Path-Component) Constructing Spaces A path-component of a space X is a maximal path-connected Homotopy The subset A ⊆ X . Fundamental Group The path-components of X partition the space. Free Groups Group Presentations 48
Algebraic Topology Path-Components Jack Romo Preliminary Group Theory Definition 50 (Path-Component) Constructing Spaces A path-component of a space X is a maximal path-connected Homotopy The subset A ⊆ X . Fundamental Group The path-components of X partition the space. Free Groups Group Proposition 11 Presentations If b , b ′ ∈ X are in the same path-component, then π 1 ( X , b ) ∼ = π 1 ( X , b ′ ) . If X is path-connected, we omit b and just write π 1 ( X ) . 48
Algebraic Topology Induced Homomorphisms Jack Romo Preliminary Group Theory Constructing Proposition 12 Spaces Homotopy Let � X , x � and � Y , y � be spaces with basepoints. Then any The Fundamental continuous map f : � X , x � → � Y , y � induces a homomorphism Group f ∗ : π 1 ( X , x ) → π 1 ( Y , y ) . Moreover: Free Groups Group 1 ( id X ) ∗ = id π 1 ( X , x ) Presentations 2 if g : � Y , y � → � Z , z � is continuous, then ( gf ) ∗ = g ∗ f ∗ 3 if f ≃ f ′ relative to { x } , then f ∗ = f ′ ∗ . 49
Algebraic Topology Group Isomorphism Jack Romo Preliminary Group Theory Constructing Spaces Theorem 51 Homotopy Let X , Y be path-connected spaces with X ≃ Y . Then The Fundamental π 1 ( X ) ∼ = π 1 ( Y ) . Group Free Groups Group Presentations 50
Algebraic Topology Group Isomorphism Jack Romo Preliminary Group Theory Constructing Spaces Theorem 51 Homotopy Let X , Y be path-connected spaces with X ≃ Y . Then The Fundamental π 1 ( X ) ∼ = π 1 ( Y ) . Group Free Groups Definition 52 Group Presentations A space is simply-connected if and only if it is path-connected and has trivial fundamental group. 50
Algebraic Topology A Simplicial Version Jack Romo Preliminary Group Theory Constructing Spaces Definition 53 (Edge Path) Homotopy Let K be a simplicial complex. An edge path is a finite The Fundamental sequence ( a 0 , . . . , a n ) of vertices of K such that for each i , Group ( a i − 1 , a i ) spans a simplex of K . (Clearly ( a i , a i ) spans a Free Groups Group 0-simplex.) Presentations An edge loop is a path with a n = a 0 . We define edge composition by concatenation. 51
Algebraic Topology Elementary Contraction Jack Romo Definition 54 (Elementary Contraction) Preliminary Group Theory Let α be an edge path. An elementary contraction of α is an Constructing edge path obtained from α by performing one of the following Spaces Homotopy moves: The Fundamental 1 Replace ( . . . , a i − 1 , a i , . . . ) with ( . . . , a i , . . . ) if a i − 1 = a i ; Group 2 Replace ( . . . , a i − 1 , a i , a i + 1 , . . . ) with ( . . . , a i , . . . ) if Free Groups a i − 1 = a i + 1 ; Group Presentations 3 Replace ( . . . , a i − 1 , a i , a i + 1 , . . . ) with ( . . . , a i − 1 , a i + 1 , . . . ) if { a i − 1 , a i , a i + 1 } spans a 2-simplex of K . An elementary expansion β of α is an edge path such that α is an elementary contraction of β . 52
Algebraic Topology Elementary Contraction Jack Romo Definition 54 (Elementary Contraction) Preliminary Group Theory Let α be an edge path. An elementary contraction of α is an Constructing edge path obtained from α by performing one of the following Spaces Homotopy moves: The Fundamental 1 Replace ( . . . , a i − 1 , a i , . . . ) with ( . . . , a i , . . . ) if a i − 1 = a i ; Group 2 Replace ( . . . , a i − 1 , a i , a i + 1 , . . . ) with ( . . . , a i , . . . ) if Free Groups a i − 1 = a i + 1 ; Group Presentations 3 Replace ( . . . , a i − 1 , a i , a i + 1 , . . . ) with ( . . . , a i − 1 , a i + 1 , . . . ) if { a i − 1 , a i , a i + 1 } spans a 2-simplex of K . An elementary expansion β of α is an edge path such that α is an elementary contraction of β . Note that rule 3 generalizes to any n -simplex contraction by contracting along the 2-faces. 52
Algebraic Topology Edge Loop Group Jack Romo Preliminary Group Theory Definition 55 (Edge Equivalence) Constructing Spaces Two edge paths α, β are said to be equivalent, written α ∼ β , Homotopy if and only if β is the result of a finite series of elementary The contractions and expansions applied to α . Fundamental Group Free Groups Group Presentations 53
Algebraic Topology Edge Loop Group Jack Romo Preliminary Group Theory Definition 55 (Edge Equivalence) Constructing Spaces Two edge paths α, β are said to be equivalent, written α ∼ β , Homotopy if and only if β is the result of a finite series of elementary The contractions and expansions applied to α . Fundamental Group Free Groups Definition 56 (Edge Loop Group) Group Presentations The edge loop group E ( K , b ) for a given simplicial complex K and 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 the reversed path. 53
Algebraic Topology Triangulating Fundamental Groups Jack Romo Preliminary Group Theory Constructing Spaces Theorem 57 Homotopy For a simplicial complex K and vertex b, E ( K , b ) ∼ The = π 1 ( | K | , b ) . Fundamental Group Free Groups Group Presentations 54
Algebraic Topology Triangulating Fundamental Groups Jack Romo Preliminary Group Theory Constructing Spaces Theorem 57 Homotopy For a simplicial complex K and vertex b, E ( K , b ) ∼ The = π 1 ( | K | , b ) . Fundamental Group This clearly shows that fundamental groups can be made into Free Groups finite, computable objects given a finite triangulation. Group Presentations 54
Algebraic Topology Triangulating Fundamental Groups Jack Romo Preliminary Group Theory Constructing Spaces Theorem 57 Homotopy For a simplicial complex K and vertex b, E ( K , b ) ∼ The = π 1 ( | K | , b ) . Fundamental Group This clearly shows that fundamental groups can be made into Free Groups finite, computable objects given a finite triangulation. Group Presentations Also, it shows E ( K , b ) is independent of the choice of triangulation. So, it doesn’t change with subdivisions. 54
Algebraic Topology Computing π 1 ( S n ) Jack Romo Preliminary Group Theory Definition 58 ( n -skeleton) Constructing Spaces For a simplicial complex K and any non-negative integer n , the Homotopy n-skeleton of K , denoted skel n ( K ) , is the subcomplex The consisting of the simplicies with dimension ≤ n . Fundamental Group Free Groups Group Presentations 55
Algebraic Topology Computing π 1 ( S n ) Jack Romo Preliminary Group Theory Definition 58 ( n -skeleton) Constructing Spaces For a simplicial complex K and any non-negative integer n , the Homotopy n-skeleton of K , denoted skel n ( K ) , is the subcomplex The consisting of the simplicies with dimension ≤ n . Fundamental Group Free Groups Lemma 59 Group Presentations For any simplicial complex K and vertex b, π 1 ( | K | , b ) ∼ = π 1 ( | skel 2 ( K ) | , b ) . 55
Algebraic Topology Computing π 1 ( S n ) Jack Romo Preliminary Group Theory Definition 58 ( n -skeleton) Constructing Spaces For a simplicial complex K and any non-negative integer n , the Homotopy n-skeleton of K , denoted skel n ( K ) , is the subcomplex The consisting of the simplicies with dimension ≤ n . Fundamental Group Free Groups Lemma 59 Group Presentations For any simplicial complex K and vertex b, π 1 ( | K | , b ) ∼ = π 1 ( | skel 2 ( K ) | , b ) . Theorem 60 For n ≥ 2 , π 1 ( S n ) is trivial. 55
Algebraic Topology Computing π 1 ( S n ) Jack Romo Preliminary Group Theory Constructing Spaces Homotopy The Fundamental Theorem 61 Group π 1 ( S 1 ) ∼ = � Z , + � . Free Groups Group Presentations 56
Algebraic Topology The Fundamental Theorem of Jack Romo Algebra Preliminary Group Theory Constructing Spaces Homotopy You have seen FTA proven using Galois theory and with The Fundamental complex analysis. Here, we present a proof with algebraic Group topology. Free Groups Group Theorem 62 (Fundamental Theorem of Algebra) Presentations For f ∈ C [ X ] , deg ( f ) > 0 ⇒ 0 ∈ f ( C ) . 57
Algebraic Topology Free Groups Jack Romo Preliminary Group Theory Constructing Spaces • We have shown existence of useful groups to topology; Homotopy how do these groups look in general? The Fundamental • Need a more formal concept of how to ’present’ a group Group • Idea: elements are words over an alphabet S ∪ S − 1 , where Free Groups Group S is a generating set Presentations • We will discover in doing this that the group-topology connection is two-way... 58
Algebraic Topology Words over S Jack Romo We assume that the set S is such that S ∩ S − 1 = ∅ , where Preliminary Group Theory S − 1 = { s − 1 | s ∈ S } . These are not inverses in any given Constructing group, just elements of S with an added · − 1 superscript. We Spaces Homotopy also specify that ( x − 1 ) − 1 = x . The Fundamental Definition 63 (Word) Group Free Groups For any set S , a word is a finite sequence w = s 1 s 2 . . . s n , Group where s n ∈ S ∪ S − 1 . Presentations 59
Algebraic Topology Words over S Jack Romo We assume that the set S is such that S ∩ S − 1 = ∅ , where Preliminary Group Theory S − 1 = { s − 1 | s ∈ S } . These are not inverses in any given Constructing group, just elements of S with an added · − 1 superscript. We Spaces Homotopy also specify that ( x − 1 ) − 1 = x . The Fundamental Definition 63 (Word) Group Free Groups For any set S , a word is a finite sequence w = s 1 s 2 . . . s n , Group where s n ∈ S ∪ S − 1 . Presentations Definition 64 (Concatenation) For words w 1 = s 1 . . . s n , w 2 = r 1 . . . r n , the concatenation w 1 w 2 = s 1 . . . s n r 1 . . . r n . 59
Algebraic Topology Elementary Contractions Jack Romo Preliminary Definition 65 (Elementary Contraction/Expansion) Group Theory Constructing A word w ′ is an elementary contraction of a word w , written Spaces w ց w ′ , if w = y 1 xx − 1 y 2 and w ′ = y 1 y 2 for words y 1 , y 2 and Homotopy x , x − 1 ∈ S ∪ S − 1 . The Fundamental Group A word w ′ is an elementary expansion of a word w , written Free Groups w ր w ′ , if w ′ ց w . Group Presentations 60
Algebraic Topology Elementary Contractions Jack Romo Preliminary Definition 65 (Elementary Contraction/Expansion) Group Theory Constructing A word w ′ is an elementary contraction of a word w , written Spaces w ց w ′ , if w = y 1 xx − 1 y 2 and w ′ = y 1 y 2 for words y 1 , y 2 and Homotopy x , x − 1 ∈ S ∪ S − 1 . The Fundamental Group A word w ′ is an elementary expansion of a word w , written Free Groups w ր w ′ , if w ′ ց w . Group Presentations Definition 66 (Word Equivalence) Two words w , w ′ are equivalent , written w ∼ w ′ , if and only if there exists a finite sequence of words w = w 0 , w 1 , . . . , w n = w ′ such that w i − 1 ց w i or w i − 1 ր w i for all i . 60
Algebraic Topology Free Group Jack Romo Preliminary Group Theory Definition 67 (Free Group) Constructing Spaces The free group on the set S , written F ( S ) , is the set of Homotopy equivalence classes of words in the alphabet S with the The Fundamental concatenation operation. Group This is clearly well-defined; w ∼ w ′ , v ∼ v ′ ⇒ wv ∼ w ′ v ′ . Free Groups Group Checking the axioms is routine. Presentations 61
Algebraic Topology Free Group Jack Romo Preliminary Group Theory Definition 67 (Free Group) Constructing Spaces The free group on the set S , written F ( S ) , is the set of Homotopy equivalence classes of words in the alphabet S with the The Fundamental concatenation operation. Group This is clearly well-defined; w ∼ w ′ , v ∼ v ′ ⇒ wv ∼ w ′ v ′ . Free Groups Group Checking the axioms is routine. Presentations Definition 68 (Free Generating Set) If for a group G there is an isomorphism θ : F ( S ) → G for some set S , then θ ( S ) is known as a free generating set . 61
Algebraic Topology Reduced Representatives Jack Romo Preliminary We would like the ’minimal’ version of a word if possible. Group Theory Constructing Definition 69 (Reduced) Spaces Homotopy A word is reduced if it permits no elementary contraction. The Fundamental Group Free Groups Group Presentations 62
Algebraic Topology Reduced Representatives Jack Romo Preliminary We would like the ’minimal’ version of a word if possible. Group Theory Constructing Definition 69 (Reduced) Spaces Homotopy A word is reduced if it permits no elementary contraction. The Fundamental Group Lemma 70 (Sequential Independence) Free Groups Group Let w 1 , w 2 , w 3 be words such that w 1 ց w 2 ր w 3 . Then either Presentations w 1 = w 3 or there is a word w ′ 2 such that w 1 ր w ′ 2 ց w 3 . 62
Algebraic Topology Reduced Representatives Jack Romo Preliminary We would like the ’minimal’ version of a word if possible. Group Theory Constructing Definition 69 (Reduced) Spaces Homotopy A word is reduced if it permits no elementary contraction. The Fundamental Group Lemma 70 (Sequential Independence) Free Groups Group Let w 1 , w 2 , w 3 be words such that w 1 ց w 2 ր w 3 . Then either Presentations w 1 = w 3 or there is a word w ′ 2 such that w 1 ր w ′ 2 ց w 3 . Theorem 71 Any element of F ( S ) is equivalent to a reduced word. 62
Algebraic Topology The Universal Property Jack Romo Preliminary Given a set S , there is a canonical inclusion i : S → F ( S ) , Group Theory namely the identity. Constructing Spaces Theorem 72 (Universal Property) Homotopy The Given any set S, any group G and function f : S → G, there is Fundamental Group a unique homomorphism φ : F ( S ) → G such that the following Free Groups diagram commutes: Group Presentations f S G i φ F ( S ) 63
Algebraic Topology Fundamental Groups of Graphs Jack Romo Preliminary Group Theory Constructing Spaces An immediate interesting application of free groups to Homotopy topology: graphs! The Any graph can be seen as a topology by considering the Fundamental Group equivalent 1-dimensional cell complex. Free Groups Group Presentations 64
Algebraic Topology Fundamental Groups of Graphs Jack Romo Preliminary Group Theory Constructing Spaces An immediate interesting application of free groups to Homotopy topology: graphs! The Any graph can be seen as a topology by considering the Fundamental Group equivalent 1-dimensional cell complex. Free Groups Theorem 73 Group Presentations The fundamental group of a countable connected graph is free. 64
Algebraic Topology Fundamental Groups of Graphs Jack Romo Preliminary Group Theory Constructing Spaces An immediate interesting application of free groups to Homotopy topology: graphs! The Any graph can be seen as a topology by considering the Fundamental Group equivalent 1-dimensional cell complex. Free Groups Theorem 73 Group Presentations The fundamental group of a countable connected graph is free. We will spend the rest of today proving this. 64
Recommend
More recommend
