lecture 2: algebraic topology · 2019-05-15 · algebraic topology i formalism to \measure"...

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Lecture 2: Algebraic Topology Gunnar Carlsson Stanford University

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Lecture 2: Algebraic Topology

Gunnar Carlsson

Stanford University

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

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

Motivation

I There is an obstacle

I We can’t count the number of obstacles

Motivation

I There is an obstacle

I We can’t count the number of obstacles

Motivation

Motivation

Motivation

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

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

Motivation

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 - Simplicial Complexes

A

B

C

D

E

F

G

Homology

A

B C

{A,B,C , {A,B}, {A,C}, {B,C}}

Homology

AB AC BC

A 1 1 0B 1 0 1C 0 1 1

Boundary Matrix

Homology

AB AC BC

A 1 1 0B 1 0 1C 0 1 1

Rank is 2

Homology

AB AC BC

A 1 1 0B 1 0 1C 0 1 1

Quotient by image is 1-D: means one connectedcomponent

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

Betti Numbers

I β0 = 1

I β1 = 2

I β2 = 0

Betti Numbers

I β0 = 1

I β1 = 2

I β2 = 0

Betti Numbers

I β0 = 1

I β1 = 2

I β2 = 0

Betti Numbers

I β0 = 1

I β1 = 2

I β2 = 0

Betti Numbers

I β0 = 1

I β1 = 0

I β2 = 1

Betti Numbers

I β0 = 1

I β1 = 0

I β2 = 1

Betti Numbers

I β0 = 1

I β1 = 0

I β2 = 1

Betti Numbers

I β0 = 1

I β1 = 0

I β2 = 1

Betti Numbers

I β0 = 1

I β1 = 2

I β2 = 1

Betti Numbers

I β0 = 1

I β1 = 2

I β2 = 1

Betti Numbers

I β0 = 1

I β1 = 2

I β2 = 1

Betti Numbers

I β0 = 1

I β1 = 2

I β2 = 1

Betti Numbers

I β0 = 1

I β1 = 4

I β2 = 1

Betti Numbers

I β0 = 1

I β1 = 4

I β2 = 1

Betti Numbers

I β0 = 1

I β1 = 4

I β2 = 1

Betti Numbers

I β0 = 1

I β1 = 4

I β2 = 1

Betti Numbers

I β0 = 1

I β1 = 2

I β2 = 1

Betti Numbers

I β0 = 1

I β1 = 2

I β2 = 1

Betti Numbers

I β0 = 1

I β1 = 2

I β2 = 1

Betti Numbers

I β0 = 1

I β1 = 2

I β2 = 1

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

Application: Brouwer Fixed Point Theorem

Brouwer fixed point theorem

Application: Brouwer Fixed Point Theorem

S1 → D2 → S1

Application: Brouwer Fixed Point Theorem

H1(S1)→ H1(D2)→ H1(S2)

Application: Brouwer Fixed Point Theorem

k → {0} → k

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.

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 (Sn) = 0 for i 6= 0, n, and Hn(Sn) ∼= H0(Sn) ∼= k .