Notes of Topology Lesson 1: Topological Spaces Topological Spaces Definition of topological spaces A topological space is a pair ( X , T ) where X is a set and T is a collection of subsets of X such that:
∅ ∈ T and X ∈ T ,for every infinite collection { O α } α ∈ A ⊂ T , we have ⋃ α ∈ A O α ∈ T , for every finite collection { O i } 1 ≤ i ≤ n ⊂ T , we have ⋂ 1 ≤ i ≤ n O i ∈ T . The set T is called a topology on X . The elements of T are called the open sets.
Definition via closed sets Let ( X , T ) be a topological space. For every open set O ∈ T , its complement c O = { x ∈ X , x ∉ O } is called a closed set.
In other words, a set A ⊂ X is closed iff c A is open.
Topology of R n Open balls of R n Let x ∈ R n and r > 0 . The open ball of center x and radius r , denoted B ( x , r ) , is defined as:
B ( x , r ) = { y ∈ R n , ∥ x − y ∥ < r } Euclidean topology Let A ⊂ R n be a subset. Let x ∈ A .
We say that A is open around x if there exists ϵ > 0 such that B ( x , ϵ ) ⊂ A .
We say that A is open if for every x ∈ A , A is open around x .
We denote the set of such open set by T R n , the Euclidean topology on R n .
Topology of Subsets of R n Subspace topology Let ( X , T ) be a topological space, and Y ⊂ X a subset. We define the subspace topology on Y as the following set:
T | Y = { O ∩ Y , O ∈ T } Continuous Maps Continuous maps ( X , T ) ( Y , U )
Let f : X → Y be a map. We say that f is continuous if for every O ∈ U , the preimage f − 1 ( O ) = { x ∈ X , f ( x ) ∈ O } is in T .
Lesson 2: Homeomorphisms Homeomorphic Topological Spaces Definition of homeomorphism Let ( X , T ) and ( Y , U ) be two topological spaces, and f : X → Y a map.
We say that f is a homeomorphism if
f : X → Y is continuous,f is a bijection,f − 1 : Y → X is continuous.If there exist such a homeomorphism, we say that the two topological spaces are homeomorphic.
Homeomorphism equivalence relation Let us write X ≃ Y if the two topological spaces X and Y are homeomorphic, i.e., if there exists a homeomorphism f : X → Y .
For any X , we have X ≃ X .
Moreover, we have X ≃ Y ⟺ Y ≃ X .
We also have a third property X ≃ Y and Y ≃ X ⟹ X ≃ Z .
Connected Components Connectedness Let ( X , T ) be a topological space. We say that X is connected if for every open sets O , O ′ ∈ T such that O ∩ O ′ = ∅ , we have X = O ∪ O ′ ⟹ O = ∅ or O ′ = ∅ .
In other words, a connected topological space cannot be divided into two non-empty disjoint open sets.
Connectedness as an invariant Invariant property Two homeomorphic topological spaces admit the same number of connected components.
Dimension Invariance of domain If m ≠ n , the Euclidean spaces R m and R n are not homeomorphic.
Dimension Let ( X , T ) be a topological space, and n ≥ 0 . We say that it has dimension n if the following is true: for every x ∈ X , there exists an open set O such that x ∈ O , and a homeomorphism O → R n .
Dimension Invariant Let X , Y be two homeomorphic topological spaces. If X has dimension n , then Y also has dimension n .
Lesson 3: Homotopies Homotopy Equivalence between Maps Definition Let ( X , T ) and ( Y , U ) be two topological spaces, and f , g : X → Y two continuous maps. A homotopy between f and g is a map F : X × [ 0 , 1 ] → Y such that:
F ( ⋅ , 0 ) is equal to f ,F ( ⋅ , 1 ) is equal to g ,F : X × [ 0 , 1 ] → Y is continuous.If such a homotopy exists, we say that the maps f and g are homotopic.
For any t ∈ [ 0 , 1 ] , the notation F ( ⋅ , t ) refers to the map
F ( ⋅ , t ) : X ⟶ Y x ⟼ F ( x , t ) Trivial maps From a homotopic point a view, a trivial map is a map that is homotopic to a constant map .
Homotopy Equivalence between Topological Spaces Definition of homotopy equivalence Let ( X , T ) and ( Y , U ) be two topological spaces. A homotopy equivalence between X and Y is a pair of continuous maps f : X → Y and g : Y → X such that:
g ∘ f : X → X is homotopic to the identity map id: X → X ,f ∘ g : Y → Y is homotopic to the identity map id: Y → Y .If such a homotopy equivalence exists, we say that X and Y are homotopy equivalent.
Let ( X , T ) be a topological space and Y ⊂ X a subset, endowed with the subspace topology T | Y .
A retraction is a continuous map r : X → X such that ∀ x ∈ X , r ( x ) ∈ Y and ∀ y ∈ Y , r ( y ) = y .
A deformation retraction is a homotopy F : X × [ 0 , 1 ] → Y between the identity map id: X → X and a retraction r : X → X .
Homotopy Equivalence Relation Link with Homeomorphic Spaces Homeomorphic implies homotopic Let X , Y be two topological spaces. If they are homeomorphic, then they are homotopic equivalent.
Invariants Number of connected components Two homotopy equivalent topological spaces admit the same number of connected components.
Lesson 4: Simplicial Complexes Combinatorial Simplicial Complexes Standard simplices 单纯形 - 维基百科,自由的百科全书
The standard simplex of dimension n is the following subset of R n + 1
Δ n = { x = ( x 1 , … , x n + 1 ) ∈ R n + 1 , x 1 , … , x n + 1 ≥ 0 a n d x 1 + ⋯ + x n + 1 = 1 } 凸包 - 维基百科,自由的百科全书
For any collection of points a 1 , … , a k ∈ R n , their convex hull is defined as:
c o n v ( { a 1 … a k } ) = { ∑ 1 ≤ i ≤ k t i a i , t 1 + ⋯ + t k = 1 , t 1 , … , t k ≥ 0 } We can say that Δ n is the convex hull of the vectors e 1 , … , e n + 1 of \R n + 1 , where e i = ( 0 , … , 1 , 0 , … , 0 )
Simplicial complexes 单纯复形 - 维基百科,自由的百科全书
Let V be a set (called the set of vertices ). A simplicial complex over V is a set K of subsets of V (called the simplices ) such that, for every σ ∈ K and every non-empty τ ⊂ σ , we have τ ∈ K .
Topology Topological realization Let K be a simplicial complex, with vertex V = { 1 , … , n } .
In R n , consider, for every i ∈ [ 1 , n ] , the vector e i = ( 0 , … , 1 , 0 , … , 0 ) .
Let | K | be the subset of R n defined as:
| K | = ⋃ σ ∈ K c o n v ( { e j , j ∈ σ } ) where conv represent the convex hull of points.
Endowed with the subspace topology, ( | K | , T | K | ) is a topological space, that we call the topological realization of K .
Triangulations Let ( X , T ) be a topological space. A triangulation of X is a simplicial complex K such that its topological realization | K | is homeomorphic to X .
Euler Characteristic Euler characteristic Let K be a simplicial complex of dimension n . Its Euler characteristic is the integer:
χ ( K ) = ∑ 0 ≤ i ≤ n ( − 1 ) i ⋅ ( n u m b e r o f s i m p l i c e s o f d i m e n s i o n i ) Let X be a topological space. Its Euler characteristic is defined as the Euler characteristic of any triangulation of it.
Euler characteristic is an invariant If X and Y are two homotopy equivalent topological spaces, then χ ( X ) = χ ( Y ) .
Lesson 5: Homological Algebra Reminder of Algebra Groups We recall that a group ( G , + ) is a set G endowed with an operation
G × G ⟶ G ( g , h ) ⟼ g + h such that:
(associativity) ∀ a , b , c ∈ G , ( a + b ) + c = a + ( b + c ) , (identity) ∃ 0 ∈ G , ∀ a ∈ G , a + 0 = 0 + a = a , (inverse) ∀ a ∈ G , ∃ b ∈ G , a + b = b + a = 0 . Quotient group 商群 - 维基百科,自由的百科全书
A subgroup of ( G , + ) is a subset H ⊂ G such that
∀ a , b ∈ H , a + b ∈ H If H is a subgroup of G , the operation + : G × G → G restricts to an operation + : H × H → H , making H a group on its own.
Suppose that ∗ ∗ G is commutative**, and that ∗ ∗ H is a subgroup of G **. We define the following equivalence relation on G : ∀ a , b ∈ G ,
a ∼ b ⟺ a − b ∈ H Denoted by G / H the quotient set of G under this relation. For any a ∈ G , one shows that the equivalence class of a is equal to a + H = { a + h , h ∈ H }
Let a 0 , a 1 , … , a n be a choice of representants of equivalence classes of the relation ∼ .
The quotient set can be written as G / H = { 0 + H , a 1 + H , … , a n + H } .
One defines a group structure ⊕ on G / H as follows: for any i , j ∈ [ 0 , n ] ,
( a i + H ) ⊕ ( a j + H ) = ( a i + a j ) + H The group ( G / H , ⊕ ) is called the quotient group.
The group Z / 2 Z The subgroup 2 Z ⊂ Z consists of all even numbers.
The relation a ∼ b ⟺ a − b ∈ 2 Z admits two equivalence classes: 2 Z = { 2 n , n ∈ Z } and 1 + 2 Z = { 1 + 2 n , n ∈ Z }
The quotient group can be seen as the group Z / 2 Z = { 0 , 1 } with the operation
0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0 For any n ≥ 1 , the product group ( ( Z / 2 Z ) n , + ) is the group whose underlying set is
( Z / 2 Z ) n = { ( ϵ 1 , … , ϵ n ) , ϵ 1 , … , ϵ n ∈ Z / 2 Z } and whose operation is defined as
( ϵ 1 , … , ϵ n ) + ( ϵ 1 ′ , … , ϵ n ′ ) = ( ϵ 1 + ϵ 1 ′ , … , ϵ n + ϵ n ′ ) Note that the set ( Z / 2 Z ) n has 2 n elements.
Vector spaces Let ( F , + , × ) be a field. We recall that a vector space over F is a group ( V , + ) endowed with an operation:
F × V ⟶ V ( λ , v ) ⟼ λ ⋅ v such that
(compatibility of multiplication) ∀ λ , μ ∈ F , ∀ v ∈ V , λ ⋅ ( μ ⋅ v ) = ( λ × μ ) ⋅ v , (identity) ∀ v ∈ V , 1 ⋅ v = v where 1 denotes the unit of F , (scalar distributivity) ∀ λ , υ ∈ F , ∀ v ∈ V , ( λ + υ ) ⋅ v = λ ⋅ v + υ ⋅ v , (vector distributivity) ∀ μ ∈ F , ∀ v , w ∈ V , λ ⋅ ( u + v ) = λ ⋅ v + υ ⋅ v . Let { v 1 , … , v n } be a collection of elements of V . We say that it is free if
∀ λ 1 , … , λ n ∈ F , ∑ 1 ≤ i ≤ n λ i v i = 0 ⟹ λ 1 = ⋯ = λ n = 0 We say that it is spans V if
∀ v ∈ V , ∃ λ 1 , … , λ n ∈ F , ∑ 1 ≤ i ≤ n λ i v i = v If the collection { v 1 , … , v n } is free and spans V , we say that it is a basis.
A linear subspace of ( V , + , ⋅ ) is a subset W ⊂ V such that
∀ u , v ∈ W , u + v ∈ W a n d ∀ v ∈ W , ∀ λ ∈ F , λ v ∈ W Just as for groups, we can define an equivalence relation ∼ on V , and a quotient vector space V / W .
Isomorphism & Isomorphic We have dim V / W = dim V − dim W
Let ( V , + , ⋅ ) and ( W , + , ⋅ ) be two vector spaces. A linear map is a map f : V → W such that
∀ u , v ∈ V , f ( u + v ) = f ( u ) + f ( v ) a n d ∀ v ∈ V , ∀ λ ∈ F , f ( λ v ) = λ ⋅ f ( v ) If f is a bijection, it is called an isomorphism, and we say that V and W are isomorphic.
Z / 2 Z -vector spaces Let ( V , + ) be a commutative group.
It can be given a Z / 2 Z -vector space structure iff ∀ v ∈ V , v + v = 0 .
Chains, cycles, and boundaries Skeleton Let K be a simplicial complex. For any n ≥ 0 , define the n -skeleton of K :
K n = { σ ∈ K , dim ( σ ) ≤ n } Also, define
K ( n ) = { σ ∈ K , dim ( σ ) = n } Chains Let n ≥ 0 . The n -chains of K is the set C n ( K ) whose elements are the formal sums
∑ σ ∈ K ( n ) ϵ σ ⋅ σ w h e r e ∀ σ ∈ K ( n ) , ϵ σ ∈ Z / 2 Z Boundary operator Let n ≥ 1 , and σ = [ x 0 , … , x n ] ∈ K ( n ) a simplex of dimension n . We define its boundary as the following element of C n − 1 ( K ) :
∂ n σ = ∑ τ ⊂ σ , | τ | = | σ | − 1 τ We can extend the operator ∂ n as a linear map ∂ n : C n ( K ) → C n − 1 ( K ) as follows: for any element of C n ( K ) ,
∂ n ∑ σ ∈ K ( n ) ϵ σ ⋅ σ = ∑ σ ∈ K ( n ) ϵ σ ⋅ ∂ n σ For any n ≥ 1 , for any c ∈ C n ( K ) , we have ∂ n − 1 ∘ ∂ n ( c ) = 0 .
Cycles and boundaries Let n ≥ 0 . We have a triplet of vector spaces
C n + 1 ( K ) → C n ( K ) → C n − 1 ( K ) We can consider their kernel and image.
We define:
Kernel (set theory) - Wikipedia
The n -cycles: Z n ( K ) = K e r ( ∂ n ) , The n -boundaries: B n ( K ) = I m ( ∂ n + 1 ) . We say that two chains c , c ′ ∈ C n ( K ) are homologous if there exists b ∈ B n ( K ) such that c = c ′ + b . Two chains are homologous if they are equal up to a boundary.
Homology Groups Homology groups n t h homology group of K :
H n ( K ) = Z n ( K ) / B n ( K ) dim H n ( K ) = dim B n ( K ) − dim Z n ( K ) .
Let K be a simplicial complex and n ≥ 0 . Its n t h Betti number is the integer β n ( K ) = dim H n ( K ) .
Homology Groups of Topological Spaces Invariant property The homology groups of a topological space are the homology groups of any triangulation of it. We define its Betti numbers similarly.
If X and Y are two homotopy equivalent topological spaces, then for any n ≥ 0 we have isomorphic homology groups H n ( X ) ≃ H n ( Y ) . As a consequence, β n ( X ) = β n ( Y ) .
Incremental Algorithm Incremental Algorithm Ordering the simplicial complex Positivity of simplices Let i ∈ [ 1 , n ] , and d = dim ( σ i ) . Recall that K i = K i + 1 ∪ { σ i } .
The simplex σ i is positive if there exists a cycle c ∈ Z d ( K i ) that contains σ i .
In other words, there exists c = ∑ σ ∈ K ( n ) i ϵ σ ⋅ σ ∈ C n ( K i ) such that ϵ σ i = 1 and ∂ n ( c ) = 0 . Otherwise, σ i is negative.
Input: an increasing sequence of simplicial complexes K 1 ⊂ ⋯ ⊂ K n = K
Output: the Betti numbers β 0 ( K ) , … , β d ( K )