lecture 2: algebraic topology · 2019-05-15 · algebraic topology i formalism to \measure"...
TRANSCRIPT
Algebraic topology
I Formalism to “measure” aspects of shape
I Developed starting in late 19th century
I Powerful methods for understanding shape
I Regarded as esoteric
Algebraic topology
I Formalism to “measure” aspects of shape
I Developed starting in late 19th century
I Powerful methods for understanding shape
I Regarded as esoteric
Algebraic topology
I Formalism to “measure” aspects of shape
I Developed starting in late 19th century
I Powerful methods for understanding shape
I Regarded as esoteric
Algebraic topology
I Formalism to “measure” aspects of shape
I Developed starting in late 19th century
I Powerful methods for understanding shape
I Regarded as esoteric
Motivation
I How can we detect the presence of obstacles in a region inthe plane?
I Can we count the number of distinct obstacles?
I Can we describe the “shape” of the obstacles?
Motivation
I How can we detect the presence of obstacles in a region inthe plane?
I Can we count the number of distinct obstacles?
I Can we describe the “shape” of the obstacles?
Motivation
I How can we detect the presence of obstacles in a region inthe plane?
I Can we count the number of distinct obstacles?
I Can we describe the “shape” of the obstacles?
Motivation
I Suppose we can’t see or access the region directly
I Only access we have is the ability to throw a lasso into theregion and attempt to pull it tight
I If I pull it tight and the whole lasso does not return to me,it is “hung up” on an obstacle
Motivation
I Suppose we can’t see or access the region directly
I Only access we have is the ability to throw a lasso into theregion and attempt to pull it tight
I If I pull it tight and the whole lasso does not return to me,it is “hung up” on an obstacle
Motivation
I Suppose we can’t see or access the region directly
I Only access we have is the ability to throw a lasso into theregion and attempt to pull it tight
I If I pull it tight and the whole lasso does not return to me,it is “hung up” on an obstacle
Motivation
I In each case, we detect the presence of an obstacle
I All the experiments have identical results, so cannot count
I How can we attempt to count the number of obstacles
I Consider the set of all possible lasso tosses in the region
I The set of such tosses is uncountably infinite
Motivation
I In each case, we detect the presence of an obstacle
I All the experiments have identical results, so cannot count
I How can we attempt to count the number of obstacles
I Consider the set of all possible lasso tosses in the region
I The set of such tosses is uncountably infinite
Motivation
I In each case, we detect the presence of an obstacle
I All the experiments have identical results, so cannot count
I How can we attempt to count the number of obstacles
I Consider the set of all possible lasso tosses in the region
I The set of such tosses is uncountably infinite
Motivation
I In each case, we detect the presence of an obstacle
I All the experiments have identical results, so cannot count
I How can we attempt to count the number of obstacles
I Consider the set of all possible lasso tosses in the region
I The set of such tosses is uncountably infinite
Motivation
I In each case, we detect the presence of an obstacle
I All the experiments have identical results, so cannot count
I How can we attempt to count the number of obstacles
I Consider the set of all possible lasso tosses in the region
I The set of such tosses is uncountably infinite
Motivation
I Define an equivalence relation on the set of lasso tosses
I Called homotopy
I Consider the set E of equivalence classes of tosses
I E turns out to be countable
I Appears to be progress, but still can’t count number ofholes
Motivation
I Define an equivalence relation on the set of lasso tosses
I Called homotopy
I Consider the set E of equivalence classes of tosses
I E turns out to be countable
I Appears to be progress, but still can’t count number ofholes
Motivation
I Define an equivalence relation on the set of lasso tosses
I Called homotopy
I Consider the set E of equivalence classes of tosses
I E turns out to be countable
I Appears to be progress, but still can’t count number ofholes
Motivation
I Define an equivalence relation on the set of lasso tosses
I Called homotopy
I Consider the set E of equivalence classes of tosses
I E turns out to be countable
I Appears to be progress, but still can’t count number ofholes
Motivation
I Define an equivalence relation on the set of lasso tosses
I Called homotopy
I Consider the set E of equivalence classes of tosses
I E turns out to be countable
I Appears to be progress, but still can’t count number ofholes
Motivation
I Solution: recognize that E carries a group structure
I Concatenation of lasso tosses
I For this situation E turns out to be a free group on twogenerators
I The number of generators of a finitely generated freegroup is a well defined invariant of the group
I Solution to the counting problem
Motivation
I Solution: recognize that E carries a group structure
I Concatenation of lasso tosses
I For this situation E turns out to be a free group on twogenerators
I The number of generators of a finitely generated freegroup is a well defined invariant of the group
I Solution to the counting problem
Motivation
I Solution: recognize that E carries a group structure
I Concatenation of lasso tosses
I For this situation E turns out to be a free group on twogenerators
I The number of generators of a finitely generated freegroup is a well defined invariant of the group
I Solution to the counting problem
Motivation
I Solution: recognize that E carries a group structure
I Concatenation of lasso tosses
I For this situation E turns out to be a free group on twogenerators
I The number of generators of a finitely generated freegroup is a well defined invariant of the group
I Solution to the counting problem
Motivation
I Solution: recognize that E carries a group structure
I Concatenation of lasso tosses
I For this situation E turns out to be a free group on twogenerators
I The number of generators of a finitely generated freegroup is a well defined invariant of the group
I Solution to the counting problem
Homotopy Groups
I Need to turn this into mathematics
I Lasso tosses should be thought of as (based) maps of circleinto our region X
I The homotopy equivalence relation is encoded by mapsS1 × I → X , so that the base point always goes to basepoint
I The set of equivalence classes is denoted by π1(X , x)
I The set π1(X , x) obtains a group structure byconcatenation of loops
I Called the fundamental group
Homotopy Groups
I Need to turn this into mathematics
I Lasso tosses should be thought of as (based) maps of circleinto our region X
I The homotopy equivalence relation is encoded by mapsS1 × I → X , so that the base point always goes to basepoint
I The set of equivalence classes is denoted by π1(X , x)
I The set π1(X , x) obtains a group structure byconcatenation of loops
I Called the fundamental group
Homotopy Groups
I Need to turn this into mathematics
I Lasso tosses should be thought of as (based) maps of circleinto our region X
I The homotopy equivalence relation is encoded by mapsS1 × I → X , so that the base point always goes to basepoint
I The set of equivalence classes is denoted by π1(X , x)
I The set π1(X , x) obtains a group structure byconcatenation of loops
I Called the fundamental group
Homotopy Groups
I Need to turn this into mathematics
I Lasso tosses should be thought of as (based) maps of circleinto our region X
I The homotopy equivalence relation is encoded by mapsS1 × I → X , so that the base point always goes to basepoint
I The set of equivalence classes is denoted by π1(X , x)
I The set π1(X , x) obtains a group structure byconcatenation of loops
I Called the fundamental group
Homotopy Groups
I Need to turn this into mathematics
I Lasso tosses should be thought of as (based) maps of circleinto our region X
I The homotopy equivalence relation is encoded by mapsS1 × I → X , so that the base point always goes to basepoint
I The set of equivalence classes is denoted by π1(X , x)
I The set π1(X , x) obtains a group structure byconcatenation of loops
I Called the fundamental group
Homotopy Groups
I Need to turn this into mathematics
I Lasso tosses should be thought of as (based) maps of circleinto our region X
I The homotopy equivalence relation is encoded by mapsS1 × I → X , so that the base point always goes to basepoint
I The set of equivalence classes is denoted by π1(X , x)
I The set π1(X , x) obtains a group structure byconcatenation of loops
I Called the fundamental group
Homotopy Groups
I Notion can be extended to maps of Sn for all n, to obtainπn(X , x)
I The case n = 0 is special, carries no group structure
I π0(X , x) as a set is the set of connected components of X
I πn(X , x) is abelian for n ≥ 2
I Although natural easy to define, very difficult to compute
I Would like to find an invariant which is easier to compute,but which captures some of the same information
Homotopy Groups
I Notion can be extended to maps of Sn for all n, to obtainπn(X , x)
I The case n = 0 is special, carries no group structure
I π0(X , x) as a set is the set of connected components of X
I πn(X , x) is abelian for n ≥ 2
I Although natural easy to define, very difficult to compute
I Would like to find an invariant which is easier to compute,but which captures some of the same information
Homotopy Groups
I Notion can be extended to maps of Sn for all n, to obtainπn(X , x)
I The case n = 0 is special, carries no group structure
I π0(X , x) as a set is the set of connected components of X
I πn(X , x) is abelian for n ≥ 2
I Although natural easy to define, very difficult to compute
I Would like to find an invariant which is easier to compute,but which captures some of the same information
Homotopy Groups
I Notion can be extended to maps of Sn for all n, to obtainπn(X , x)
I The case n = 0 is special, carries no group structure
I π0(X , x) as a set is the set of connected components of X
I πn(X , x) is abelian for n ≥ 2
I Although natural easy to define, very difficult to compute
I Would like to find an invariant which is easier to compute,but which captures some of the same information
Homotopy Groups
I Notion can be extended to maps of Sn for all n, to obtainπn(X , x)
I The case n = 0 is special, carries no group structure
I π0(X , x) as a set is the set of connected components of X
I πn(X , x) is abelian for n ≥ 2
I Although natural easy to define, very difficult to compute
I Would like to find an invariant which is easier to compute,but which captures some of the same information
Homotopy Groups
I Notion can be extended to maps of Sn for all n, to obtainπn(X , x)
I The case n = 0 is special, carries no group structure
I π0(X , x) as a set is the set of connected components of X
I πn(X , x) is abelian for n ≥ 2
I Although natural easy to define, very difficult to compute
I Would like to find an invariant which is easier to compute,but which captures some of the same information
Homology
I Homology is a linear algebraic invariant. Fix a field k overwhich to compute throughout. F2 is a good choice.
I Computation is with matrices, rather than with potentiallynon-abelian groups
I There are several methods for computing homology locallyto globally
I It has many useful formal properties, particularly whenevaluated on manifolds
Homology
I Homology is a linear algebraic invariant. Fix a field k overwhich to compute throughout. F2 is a good choice.
I Computation is with matrices, rather than with potentiallynon-abelian groups
I There are several methods for computing homology locallyto globally
I It has many useful formal properties, particularly whenevaluated on manifolds
Homology
I Homology is a linear algebraic invariant. Fix a field k overwhich to compute throughout. F2 is a good choice.
I Computation is with matrices, rather than with potentiallynon-abelian groups
I There are several methods for computing homology locallyto globally
I It has many useful formal properties, particularly whenevaluated on manifolds
Homology
I Homology is a linear algebraic invariant. Fix a field k overwhich to compute throughout. F2 is a good choice.
I Computation is with matrices, rather than with potentiallynon-abelian groups
I There are several methods for computing homology locallyto globally
I It has many useful formal properties, particularly whenevaluated on manifolds
Homology - Simplicial Complexes
I Given a finite set X of k + 1 points in Rn in generalposition (i.e. so that they are not contained in any affinek-dimensional hyperplane), the simplex σ(X ) spanned byX is the convex hull of X . For #(X ) = 2, this is the linesegment spanned by the two points and for #(X ) = 3, it isthe triangle spanned by X . X is called the set of verticesof the simplex.
I By a face of a simplex σ, we will mean the simplexspanned by a subset of its vertex set
I A simplicial complex X is a collection of simplices in someRn so that (a) for any simplex in X , all of its faces are alsoin X and (b) the intersection of any two simplices σ and τof X is a face of both σ and τ .
Homology - Simplicial Complexes
I Given a finite set X of k + 1 points in Rn in generalposition (i.e. so that they are not contained in any affinek-dimensional hyperplane), the simplex σ(X ) spanned byX is the convex hull of X . For #(X ) = 2, this is the linesegment spanned by the two points and for #(X ) = 3, it isthe triangle spanned by X . X is called the set of verticesof the simplex.
I By a face of a simplex σ, we will mean the simplexspanned by a subset of its vertex set
I A simplicial complex X is a collection of simplices in someRn so that (a) for any simplex in X , all of its faces are alsoin X and (b) the intersection of any two simplices σ and τof X is a face of both σ and τ .
Homology - Simplicial Complexes
I Given a finite set X of k + 1 points in Rn in generalposition (i.e. so that they are not contained in any affinek-dimensional hyperplane), the simplex σ(X ) spanned byX is the convex hull of X . For #(X ) = 2, this is the linesegment spanned by the two points and for #(X ) = 3, it isthe triangle spanned by X . X is called the set of verticesof the simplex.
I By a face of a simplex σ, we will mean the simplexspanned by a subset of its vertex set
I A simplicial complex X is a collection of simplices in someRn so that (a) for any simplex in X , all of its faces are alsoin X and (b) the intersection of any two simplices σ and τof X is a face of both σ and τ .
Homology
I Quotient of vector space V by subspace W , V /W , haselements the “cosets” v + W
I v + W = v ′ + W if and only if v − v ′ ∈W .
I (v + W ) + (v ′ + W ) = v + v ′ + W
I Too abstract!
I How to compute and obtain bases?
Homology
I Quotient of vector space V by subspace W , V /W , haselements the “cosets” v + W
I v + W = v ′ + W if and only if v − v ′ ∈W .
I (v + W ) + (v ′ + W ) = v + v ′ + W
I Too abstract!
I How to compute and obtain bases?
Homology
I Quotient of vector space V by subspace W , V /W , haselements the “cosets” v + W
I v + W = v ′ + W if and only if v − v ′ ∈W .
I (v + W ) + (v ′ + W ) = v + v ′ + W
I Too abstract!
I How to compute and obtain bases?
Homology
I Quotient of vector space V by subspace W , V /W , haselements the “cosets” v + W
I v + W = v ′ + W if and only if v − v ′ ∈W .
I (v + W ) + (v ′ + W ) = v + v ′ + W
I Too abstract!
I How to compute and obtain bases?
Homology
I Quotient of vector space V by subspace W , V /W , haselements the “cosets” v + W
I v + W = v ′ + W if and only if v − v ′ ∈W .
I (v + W ) + (v ′ + W ) = v + v ′ + W
I Too abstract!
I How to compute and obtain bases?
Homology
I Recall Gaussian elimination
I Applies row operations to a matrix until in reduced rowechelon form
I 1 ∗ − ∗ 0 ∗ − ∗ 00 0 − 0 1 ∗ − ∗ 00 0 − 0 0 0 − 0 1
I Basis of null space of matrix is in one to one
correspondence with non-pivot columns of the reduced rowechelon form
Homology
I Recall Gaussian elimination
I Applies row operations to a matrix until in reduced rowechelon form
I 1 ∗ − ∗ 0 ∗ − ∗ 00 0 − 0 1 ∗ − ∗ 00 0 − 0 0 0 − 0 1
I Basis of null space of matrix is in one to one
correspondence with non-pivot columns of the reduced rowechelon form
Homology
I Recall Gaussian elimination
I Applies row operations to a matrix until in reduced rowechelon form
I 1 ∗ − ∗ 0 ∗ − ∗ 00 0 − 0 1 ∗ − ∗ 00 0 − 0 0 0 − 0 1
I Basis of null space of matrix is in one to onecorrespondence with non-pivot columns of the reduced rowechelon form
Homology
I Recall Gaussian elimination
I Applies row operations to a matrix until in reduced rowechelon form
I 1 ∗ − ∗ 0 ∗ − ∗ 00 0 − 0 1 ∗ − ∗ 00 0 − 0 0 0 − 0 1
I Basis of null space of matrix is in one to one
correspondence with non-pivot columns of the reduced rowechelon form
Homology
I To obtain quotient, perform column operations until inreduced column echelon form
I Basis is in one to one correspondence with non pivot rowsin reduced column echelon form, In fact, the cosets ei + Wform a basis for the quotient as ei ranges over thenon-pivot rows
I Coefficients in matrix give the expression of v + W as alinear combination of basis elements
Homology
I To obtain quotient, perform column operations until inreduced column echelon form
I Basis is in one to one correspondence with non pivot rowsin reduced column echelon form, In fact, the cosets ei + Wform a basis for the quotient as ei ranges over thenon-pivot rows
I Coefficients in matrix give the expression of v + W as alinear combination of basis elements
Homology
I To obtain quotient, perform column operations until inreduced column echelon form
I Basis is in one to one correspondence with non pivot rowsin reduced column echelon form, In fact, the cosets ei + Wform a basis for the quotient as ei ranges over thenon-pivot rows
I Coefficients in matrix give the expression of v + W as alinear combination of basis elements
Homology
AB AC BC
A 1 1 0B 1 0 1C 0 1 1
Null space is 1-D, spanned by AB + AC + BCRepresents the loop in the complex
Homology
AB AC BC
A 1 1 0B 1 0 1C 0 1 1
H0 is quotient by image of boundary map, H1 isthe null space
Homology
I More invariant form is a pair of vector spaces V0 andV1,together with linear transformation ∂ : V1 → V0
I H0 is quotient V0/∂(V1) and H1 is the null space of ∂ in V1
I The dimension of H0 is the number of connectedcomponents of the complex, namely 1
I The dimension of H1 is the number of loops in the complex
I What if we want to understand higher dimensionalfeatures?
Homology
I More invariant form is a pair of vector spaces V0 andV1,together with linear transformation ∂ : V1 → V0
I H0 is quotient V0/∂(V1) and H1 is the null space of ∂ in V1
I The dimension of H0 is the number of connectedcomponents of the complex, namely 1
I The dimension of H1 is the number of loops in the complex
I What if we want to understand higher dimensionalfeatures?
Homology
I More invariant form is a pair of vector spaces V0 andV1,together with linear transformation ∂ : V1 → V0
I H0 is quotient V0/∂(V1) and H1 is the null space of ∂ in V1
I The dimension of H0 is the number of connectedcomponents of the complex, namely 1
I The dimension of H1 is the number of loops in the complex
I What if we want to understand higher dimensionalfeatures?
Homology
I More invariant form is a pair of vector spaces V0 andV1,together with linear transformation ∂ : V1 → V0
I H0 is quotient V0/∂(V1) and H1 is the null space of ∂ in V1
I The dimension of H0 is the number of connectedcomponents of the complex, namely 1
I The dimension of H1 is the number of loops in the complex
I What if we want to understand higher dimensionalfeatures?
Homology
I More invariant form is a pair of vector spaces V0 andV1,together with linear transformation ∂ : V1 → V0
I H0 is quotient V0/∂(V1) and H1 is the null space of ∂ in V1
I The dimension of H0 is the number of connectedcomponents of the complex, namely 1
I The dimension of H1 is the number of loops in the complex
I What if we want to understand higher dimensionalfeatures?
Homology - Chain Complexes
· · ·Cn(X )∂n−→ Cn−1(X ) −→ · · · ∂2−→ C1(X )
∂1−→ C0(X )
I ∂i−1 ◦ ∂i = 0
I Ci (X ) is a vector space with basis the collection ofi-simplices in X
I With respect to this basis, ∂i has matrix with columns(resp. rows) corresponding to the i-simplices (resp.(i − 1)-simplices) of X
I An entry in the matrix is = 1 if and only if its row andcolumn pair have the property that the (i − 1)-simplexcorresponding to the row is a fact of the i-simplexcorresponding to the column
Homology - Chain Complexes
· · ·Cn(X )∂n−→ Cn−1(X ) −→ · · · ∂2−→ C1(X )
∂1−→ C0(X )
I ∂i−1 ◦ ∂i = 0
I Ci (X ) is a vector space with basis the collection ofi-simplices in X
I With respect to this basis, ∂i has matrix with columns(resp. rows) corresponding to the i-simplices (resp.(i − 1)-simplices) of X
I An entry in the matrix is = 1 if and only if its row andcolumn pair have the property that the (i − 1)-simplexcorresponding to the row is a fact of the i-simplexcorresponding to the column
Homology - Chain Complexes
· · ·Cn(X )∂n−→ Cn−1(X ) −→ · · · ∂2−→ C1(X )
∂1−→ C0(X )
I ∂i−1 ◦ ∂i = 0
I Ci (X ) is a vector space with basis the collection ofi-simplices in X
I With respect to this basis, ∂i has matrix with columns(resp. rows) corresponding to the i-simplices (resp.(i − 1)-simplices) of X
I An entry in the matrix is = 1 if and only if its row andcolumn pair have the property that the (i − 1)-simplexcorresponding to the row is a fact of the i-simplexcorresponding to the column
Homology - Chain Complexes
· · ·Cn(X )∂n−→ Cn−1(X ) −→ · · · ∂2−→ C1(X )
∂1−→ C0(X )
I ∂i−1 ◦ ∂i = 0
I Ci (X ) is a vector space with basis the collection ofi-simplices in X
I With respect to this basis, ∂i has matrix with columns(resp. rows) corresponding to the i-simplices (resp.(i − 1)-simplices) of X
I An entry in the matrix is = 1 if and only if its row andcolumn pair have the property that the (i − 1)-simplexcorresponding to the row is a fact of the i-simplexcorresponding to the column
Homology - Chain Complexes
· · ·Cn(X )∂n−→ Cn−1(X ) −→ · · · ∂2−→ C1(X )
∂1−→ C0(X )
I Zi is the null space of ∂i . It is called the vector space ofi-cycles.
I Bi is the image of ∂i+1. It is called the vector space ofi-boundaries
I Bi is contained in Zi
I Zi/Bi is called the i-dimensional homology of the simplicialcomplex X , and is written as Hi (X )
I The dimension of Hi is called the i-th Betti number of X
Homology - Chain Complexes
· · ·Cn(X )∂n−→ Cn−1(X ) −→ · · · ∂2−→ C1(X )
∂1−→ C0(X )
I Zi is the null space of ∂i . It is called the vector space ofi-cycles.
I Bi is the image of ∂i+1. It is called the vector space ofi-boundaries
I Bi is contained in Zi
I Zi/Bi is called the i-dimensional homology of the simplicialcomplex X , and is written as Hi (X )
I The dimension of Hi is called the i-th Betti number of X
Homology - Chain Complexes
· · ·Cn(X )∂n−→ Cn−1(X ) −→ · · · ∂2−→ C1(X )
∂1−→ C0(X )
I Zi is the null space of ∂i . It is called the vector space ofi-cycles.
I Bi is the image of ∂i+1. It is called the vector space ofi-boundaries
I Bi is contained in Zi
I Zi/Bi is called the i-dimensional homology of the simplicialcomplex X , and is written as Hi (X )
I The dimension of Hi is called the i-th Betti number of X
Homology - Chain Complexes
· · ·Cn(X )∂n−→ Cn−1(X ) −→ · · · ∂2−→ C1(X )
∂1−→ C0(X )
I Zi is the null space of ∂i . It is called the vector space ofi-cycles.
I Bi is the image of ∂i+1. It is called the vector space ofi-boundaries
I Bi is contained in Zi
I Zi/Bi is called the i-dimensional homology of the simplicialcomplex X , and is written as Hi (X )
I The dimension of Hi is called the i-th Betti number of X
Homology - Chain Complexes
· · ·Cn(X )∂n−→ Cn−1(X ) −→ · · · ∂2−→ C1(X )
∂1−→ C0(X )
I Zi is the null space of ∂i . It is called the vector space ofi-cycles.
I Bi is the image of ∂i+1. It is called the vector space ofi-boundaries
I Bi is contained in Zi
I Zi/Bi is called the i-dimensional homology of the simplicialcomplex X , and is written as Hi (X )
I The dimension of Hi is called the i-th Betti number of X
Homology - Chain Complexes
012 013 023 12301 1 1 0 002 1 0 1 003 0 1 1 012 1 0 0 113 0 1 0 123 0 0 1 1
∂2 for the sphere
Homology - Chain Complexes
012 013 023 12301 1 1 0 002 1 0 1 003 0 1 1 012 1 0 0 113 0 1 0 123 0 0 1 1
012 + 013 + 023 + 123 is a 2-cycle
Algebraic Topology
I Produces discrete “signatures” describing deformationinvariant properties of a space
I Uses linear algebra to create integer invariants βi (X ) for alli ≥ 0
I βi (X ) counts number of i dimensional “holes” in X , ornumber of independent “cycles” or higher dimensionalsurfaces
Algebraic Topology
I Produces discrete “signatures” describing deformationinvariant properties of a space
I Uses linear algebra to create integer invariants βi (X ) for alli ≥ 0
I βi (X ) counts number of i dimensional “holes” in X , ornumber of independent “cycles” or higher dimensionalsurfaces
Algebraic Topology
I Produces discrete “signatures” describing deformationinvariant properties of a space
I Uses linear algebra to create integer invariants βi (X ) for alli ≥ 0
I βi (X ) counts number of i dimensional “holes” in X , ornumber of independent “cycles” or higher dimensionalsurfaces
Relative Homology
I Exists a notion of relative homology of a pair (X ,Y)
I Build an analogous chain complex with vector spacesCn(X ,Y) ∼= Cn(X )/Cn(Y)
I Obtain relative homology groups Hn(X ,Y)
I Obey the excision property
I Hn(X0,Y0) ∼= Hn(X ,Y) when we have an inclusion
(X0,Y0) ↪→ (X ,Y)
of pairs of simplicial complexes, and when all simplices ofX not contained in Y are contained in X0
Relative Homology
I Exists a notion of relative homology of a pair (X ,Y)
I Build an analogous chain complex with vector spacesCn(X ,Y) ∼= Cn(X )/Cn(Y)
I Obtain relative homology groups Hn(X ,Y)
I Obey the excision property
I Hn(X0,Y0) ∼= Hn(X ,Y) when we have an inclusion
(X0,Y0) ↪→ (X ,Y)
of pairs of simplicial complexes, and when all simplices ofX not contained in Y are contained in X0
Relative Homology
I Exists a notion of relative homology of a pair (X ,Y)
I Build an analogous chain complex with vector spacesCn(X ,Y) ∼= Cn(X )/Cn(Y)
I Obtain relative homology groups Hn(X ,Y)
I Obey the excision property
I Hn(X0,Y0) ∼= Hn(X ,Y) when we have an inclusion
(X0,Y0) ↪→ (X ,Y)
of pairs of simplicial complexes, and when all simplices ofX not contained in Y are contained in X0
Relative Homology
I Exists a notion of relative homology of a pair (X ,Y)
I Build an analogous chain complex with vector spacesCn(X ,Y) ∼= Cn(X )/Cn(Y)
I Obtain relative homology groups Hn(X ,Y)
I Obey the excision property
I Hn(X0,Y0) ∼= Hn(X ,Y) when we have an inclusion
(X0,Y0) ↪→ (X ,Y)
of pairs of simplicial complexes, and when all simplices ofX not contained in Y are contained in X0
Relative Homology
I Exists a notion of relative homology of a pair (X ,Y)
I Build an analogous chain complex with vector spacesCn(X ,Y) ∼= Cn(X )/Cn(Y)
I Obtain relative homology groups Hn(X ,Y)
I Obey the excision property
I Hn(X0,Y0) ∼= Hn(X ,Y) when we have an inclusion
(X0,Y0) ↪→ (X ,Y)
of pairs of simplicial complexes, and when all simplices ofX not contained in Y are contained in X0
Homology of Topological Spaces
I What if our space is not given as a simplicial complex?
I Samuel Eilenberg (1944) defined Hi (X ) for a topologicalspace independent of the structure of a simplicial complex
I Essentially constructed a very infinite simplicial complexattached to any topological space in a natural way
I Called singular homology
Homology of Topological Spaces
I What if our space is not given as a simplicial complex?
I Samuel Eilenberg (1944) defined Hi (X ) for a topologicalspace independent of the structure of a simplicial complex
I Essentially constructed a very infinite simplicial complexattached to any topological space in a natural way
I Called singular homology
Homology of Topological Spaces
I What if our space is not given as a simplicial complex?
I Samuel Eilenberg (1944) defined Hi (X ) for a topologicalspace independent of the structure of a simplicial complex
I Essentially constructed a very infinite simplicial complexattached to any topological space in a natural way
I Called singular homology
Homology of Topological Spaces
I What if our space is not given as a simplicial complex?
I Samuel Eilenberg (1944) defined Hi (X ) for a topologicalspace independent of the structure of a simplicial complex
I Essentially constructed a very infinite simplicial complexattached to any topological space in a natural way
I Called singular homology
Functoriality and Categorification
I Emmy Noether observed that given a map of simplicialcomplexes f : X → Y, one obtains a linear transformationHi (f ) : Hi (X )→ Hi (Y)
I This feature is critical to all applications of and mostcomputational techniques for homology
I Means that what was originally thought of purely as acounting problem is now categorified, so that informationabout morphisms of spaces is also encoded in the invariant
Functoriality and Categorification
I Emmy Noether observed that given a map of simplicialcomplexes f : X → Y, one obtains a linear transformationHi (f ) : Hi (X )→ Hi (Y)
I This feature is critical to all applications of and mostcomputational techniques for homology
I Means that what was originally thought of purely as acounting problem is now categorified, so that informationabout morphisms of spaces is also encoded in the invariant
Functoriality and Categorification
I Emmy Noether observed that given a map of simplicialcomplexes f : X → Y, one obtains a linear transformationHi (f ) : Hi (X )→ Hi (Y)
I This feature is critical to all applications of and mostcomputational techniques for homology
I Means that what was originally thought of purely as acounting problem is now categorified, so that informationabout morphisms of spaces is also encoded in the invariant
Homotopy Invariance
I Two maps f , g : X → Y are homotopic if there is acontinuous map H : X × [0, 1]→ Y with H(x , 0) = f (x)and H(x , 1) = g(x)
I Analogous to the notion of homotopy of paths discussedabove
I If f and g from X to Y are homotopic, then Hi (f ) = Hi (g)
I f : X → Y is a homotopy equivalence if there isg : Y → X so that f ◦ g is homotopic to idY and g ◦ f ishomotopic to idX . Means Hi (f ) is an isomorphism for all i .
I Of critical importance in computation and application
Homotopy Invariance
I Two maps f , g : X → Y are homotopic if there is acontinuous map H : X × [0, 1]→ Y with H(x , 0) = f (x)and H(x , 1) = g(x)
I Analogous to the notion of homotopy of paths discussedabove
I If f and g from X to Y are homotopic, then Hi (f ) = Hi (g)
I f : X → Y is a homotopy equivalence if there isg : Y → X so that f ◦ g is homotopic to idY and g ◦ f ishomotopic to idX . Means Hi (f ) is an isomorphism for all i .
I Of critical importance in computation and application
Homotopy Invariance
I Two maps f , g : X → Y are homotopic if there is acontinuous map H : X × [0, 1]→ Y with H(x , 0) = f (x)and H(x , 1) = g(x)
I Analogous to the notion of homotopy of paths discussedabove
I If f and g from X to Y are homotopic, then Hi (f ) = Hi (g)
I f : X → Y is a homotopy equivalence if there isg : Y → X so that f ◦ g is homotopic to idY and g ◦ f ishomotopic to idX . Means Hi (f ) is an isomorphism for all i .
I Of critical importance in computation and application
Homotopy Invariance
I Two maps f , g : X → Y are homotopic if there is acontinuous map H : X × [0, 1]→ Y with H(x , 0) = f (x)and H(x , 1) = g(x)
I Analogous to the notion of homotopy of paths discussedabove
I If f and g from X to Y are homotopic, then Hi (f ) = Hi (g)
I f : X → Y is a homotopy equivalence if there isg : Y → X so that f ◦ g is homotopic to idY and g ◦ f ishomotopic to idX . Means Hi (f ) is an isomorphism for all i .
I Of critical importance in computation and application
Homotopy Invariance
I Two maps f , g : X → Y are homotopic if there is acontinuous map H : X × [0, 1]→ Y with H(x , 0) = f (x)and H(x , 1) = g(x)
I Analogous to the notion of homotopy of paths discussedabove
I If f and g from X to Y are homotopic, then Hi (f ) = Hi (g)
I f : X → Y is a homotopy equivalence if there isg : Y → X so that f ◦ g is homotopic to idY and g ◦ f ishomotopic to idX . Means Hi (f ) is an isomorphism for all i .
I Of critical importance in computation and application
Computing Homology
I There are direct linear algebraic methods for computingthe homology of simplicial complexes
I Often computationally intensive, not applicable for singularhomology
I Requires the development of indirect methods
I One such method is the method of exact sequences
Computing Homology
I There are direct linear algebraic methods for computingthe homology of simplicial complexes
I Often computationally intensive, not applicable for singularhomology
I Requires the development of indirect methods
I One such method is the method of exact sequences
Computing Homology
I There are direct linear algebraic methods for computingthe homology of simplicial complexes
I Often computationally intensive, not applicable for singularhomology
I Requires the development of indirect methods
I One such method is the method of exact sequences
Computing Homology
I There are direct linear algebraic methods for computingthe homology of simplicial complexes
I Often computationally intensive, not applicable for singularhomology
I Requires the development of indirect methods
I One such method is the method of exact sequences
Exact Sequences
I A sequence of linear transformations of vector spaces
Uf→ V
g→W is exact if (a) g ◦ f is identically zero and(b) the kernel of g is equal to the image of f
I A longer sequence
Vnfn→ Vn−1
fn−1→ · · · f2→ V1f1→ V0
is exact if and only if its length three subsequences are allexact
Exact Sequences
I A sequence of linear transformations of vector spaces
Uf→ V
g→W is exact if (a) g ◦ f is identically zero and(b) the kernel of g is equal to the image of f
I A longer sequence
Vnfn→ Vn−1
fn−1→ · · · f2→ V1f1→ V0
is exact if and only if its length three subsequences are allexact
Exact Sequences
I For a five term exact sequence
V4f4→ V3
f3→ V2f2→ V1
f1→ V0
on has that the dimension of V2 is equal to the sum of thedimensions of V3/f4(V4) and Ker(f1)
I Means that one can compute V2 in terms of V4,V3,V1,and V0 and the transformations relating them.
I 0→ Vf→W → 0 exact means f is isomorphism.
Exact Sequences
I For a five term exact sequence
V4f4→ V3
f3→ V2f2→ V1
f1→ V0
on has that the dimension of V2 is equal to the sum of thedimensions of V3/f4(V4) and Ker(f1)
I Means that one can compute V2 in terms of V4,V3,V1,and V0 and the transformations relating them.
I 0→ Vf→W → 0 exact means f is isomorphism.
Exact Sequences
I For a five term exact sequence
V4f4→ V3
f3→ V2f2→ V1
f1→ V0
on has that the dimension of V2 is equal to the sum of thedimensions of V3/f4(V4) and Ker(f1)
I Means that one can compute V2 in terms of V4,V3,V1,and V0 and the transformations relating them.
I 0→ Vf→W → 0 exact means f is isomorphism.
Exact Sequence of a Pair
· · ·Hi+1(X ,Y )→ Hi (Y )→ Hi (X )→ Hi (X ,Y )→ Hi−1(Y )→ · · ·
Means one can compute the homology of Hi (X ) in terms ofHi (Y ) and Hi (X ,Y )
Exact Sequence of a Pair
· · ·Hi+1(X ,Y )→ Hi (Y )→ Hi (X )→ Hi (X ,Y )→ Hi−1(Y )→ · · ·
Means one can compute the homology of Hi (X ) in terms ofHi (Y ) and Hi (X ,Y )
Mayer-Vietoris Sequence
When X = U ∪ V , we have an exact sequence
· · ·Hi+1(X )→ Hi (U ∩ V )→ Hi (U)⊕ Hi (V )→
→ Hi (X )→ Hi−1(U ∩ V )→ · · ·
Means we can compute the homology of the union of two setsin terms of the homology of the two sets and their intersection.Can be viewed as categorification of inclusion/exclusionprinciple.
Mayer-Vietoris Sequence
When X = U ∪ V , we have an exact sequence
· · ·Hi+1(X )→ Hi (U ∩ V )→ Hi (U)⊕ Hi (V )→
→ Hi (X )→ Hi−1(U ∩ V )→ · · ·
Means we can compute the homology of the union of two setsin terms of the homology of the two sets and their intersection.Can be viewed as categorification of inclusion/exclusionprinciple.
Mayer-Vietoris Sequence
Suppose we take Sn = Dn+ ∪ Dn
−, with Dn+ ∩ Dn
−∼= Sn−1, get
exact sequence
Hi (Dn+)⊕Hi (D
n+)→ Hi (S
n)→ Hi−1(Sn−1)→ Hi−1(Dn+)⊕Hi−1(Dn
+)
Hi (Dn±) are zero for i > 0, since Dn
± are contractible, i.e. mapDn± → ∗ is an equivalence
Mayer-Vietoris Sequence
Suppose we take Sn = Dn+ ∪ Dn
−, with Dn+ ∩ Dn
−∼= Sn−1, get
exact sequence
Hi (Dn+)⊕Hi (D
n+)→ Hi (S
n)→ Hi−1(Sn−1)→ Hi−1(Dn+)⊕Hi−1(Dn
+)
Hi (Dn±) are zero for i > 0, since Dn
± are contractible, i.e. mapDn± → ∗ is an equivalence
Mayer-Vietoris Sequence
Suppose we take Sn = Dn+ ∪ Dn
−, with Dn+ ∩ Dn
−∼= Sn−1, get
exact sequence
Hi (Dn+)⊕Hi (D
n+)→ Hi (S
n)→ Hi−1(Sn−1)→ Hi−1(Dn+)⊕Hi−1(Dn
+)
Follows that Hi (Sn) ∼= Hi−1(Sn−1) for i > 2. Computation of
Hi (Sn) follows.