Lecture 2: Algebraic

Gunnar Carlsson

Stanford University 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 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 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 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 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 in the plane? I Can we describe the “shape” of the obstacles?

Motivation

I How can we detect the presence of obstacles in a region in the plane?

I Can we count the number of distinct obstacles? Motivation

I How can we detect the presence of obstacles in a region in the plane?

I Can we count the number of distinct obstacles?

I Can we describe the “shape” of the obstacles? I Only access we have is the ability to throw a lasso into the region 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 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 the region and attempt to pull it tight 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 the region 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 We can’t count the number of obstacles

Motivation

I There is an obstacle Motivation

I There is an obstacle

I We can’t count the number of obstacles Motivation

Motivation

Motivation

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 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 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 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 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 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 of holes

Motivation

I Define an equivalence relation on the set of lasso tosses 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 of holes

Motivation

I Define an equivalence relation on the set of lasso tosses

I Called homotopy I E turns out to be countable

I Appears to be progress, but still can’t count number of holes

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 Appears to be progress, but still can’t count number of holes

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 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 of holes Motivation

Motivation

I Concatenation of lasso tosses

I For this situation E turns out to be a free group on two generators

I The number of generators of a finitely generated free group 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 For this situation E turns out to be a free group on two generators

I The number of generators of a finitely generated free group 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 The number of generators of a finitely generated free group 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 two generators 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 two generators

I The number of generators of a finitely generated free group is a well defined invariant of the group 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 two generators

I The number of generators of a finitely generated free group is a well defined invariant of the group

I Solution to the counting problem I Lasso tosses should be thought of as (based) maps of circle into our region X

I The homotopy equivalence relation is encoded by maps S1 × I → X , so that the base point always goes to base point

I The set of equivalence classes is denoted by π1(X , x)

I The set π1(X , x) obtains a group structure by concatenation of loops

I Called the fundamental group

Homotopy Groups

I Need to turn this into mathematics I The homotopy equivalence relation is encoded by maps S1 × I → X , so that the base point always goes to base point

I The set of equivalence classes is denoted by π1(X , x)

I The set π1(X , x) obtains a group structure by concatenation 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 circle into our region X I The set of equivalence classes is denoted by π1(X , x)

I The set π1(X , x) obtains a group structure by concatenation 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 circle into our region X

I The homotopy equivalence relation is encoded by maps S1 × I → X , so that the base point always goes to base point I The set π1(X , x) obtains a group structure by concatenation 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 circle into our region X

I The homotopy equivalence relation is encoded by maps S1 × I → X , so that the base point always goes to base point

I The set of equivalence classes is denoted by π1(X , x) 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 circle into our region X

I The homotopy equivalence relation is encoded by maps S1 × I → X , so that the base point always goes to base point

I The set of equivalence classes is denoted by π1(X , x)

I The set π1(X , x) obtains a group structure by concatenation of loops Homotopy Groups

I Need to turn this into mathematics

I Lasso tosses should be thought of as (based) maps of circle into our region X

I The homotopy equivalence relation is encoded by maps S1 × I → X , so that the base point always goes to base point

I The set of equivalence classes is denoted by π1(X , x)

I The set π1(X , x) obtains a group structure by concatenation of loops

I Called the fundamental group 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

n I Notion can be extended to maps of S for all n, to obtain πn(X , x) 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

n I Notion can be extended to maps of S for all n, to obtain πn(X , x) I The case n = 0 is special, carries no group structure 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

n I Notion can be extended to maps of S for all n, to obtain πn(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 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

n I Notion can be extended to maps of S for all n, to obtain πn(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 Would like to find an invariant which is easier to compute, but which captures some of the same information

Homotopy Groups

n I Notion can be extended to maps of S for all n, to obtain πn(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 Homotopy Groups

n I Notion can be extended to maps of S for all n, to obtain πn(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 I Computation is with matrices, rather than with potentially non-abelian groups

I There are several methods for computing locally to globally

I It has many useful formal properties, particularly when evaluated on

Homology

I Homology is a linear algebraic invariant. Fix a field k over which to compute throughout. F2 is a good choice. I There are several methods for computing homology locally to globally

I It has many useful formal properties, particularly when evaluated on manifolds

Homology

I Homology is a linear algebraic invariant. Fix a field k over which to compute throughout. F2 is a good choice. I Computation is with matrices, rather than with potentially non-abelian groups I It has many useful formal properties, particularly when evaluated on manifolds

Homology

I Homology is a linear algebraic invariant. Fix a field k over which to compute throughout. F2 is a good choice. I Computation is with matrices, rather than with potentially non-abelian groups

I There are several methods for computing homology locally to globally Homology

I Homology is a linear algebraic invariant. Fix a field k over which to compute throughout. F2 is a good choice. I Computation is with matrices, rather than with potentially non-abelian groups

I There are several methods for computing homology locally to globally

I It has many useful formal properties, particularly when evaluated on manifolds I By a face of a simplex σ, we will mean the simplex spanned by a subset of its vertex set

I A X is a collection of simplices in some n R so that (a) for any simplex in X , all of its faces are also in X and (b) the intersection of any two simplices σ and τ of X is a face of both σ and τ.

Homology - Simplicial Complexes

n I Given a finite set X of k + 1 points in R in general position (i.e. so that they are not contained in any affine k-dimensional hyperplane), the simplex σ(X ) spanned by X is the convex hull of X . For #(X ) = 2, this is the line segment spanned by the two points and for #(X ) = 3, it is the triangle spanned by X . X is called the set of vertices of the simplex. I A simplicial complex X is a collection of simplices in some n R so that (a) for any simplex in X , all of its faces are also in X and (b) the intersection of any two simplices σ and τ of X is a face of both σ and τ.

Homology - Simplicial Complexes

n I Given a finite set X of k + 1 points in R in general position (i.e. so that they are not contained in any affine k-dimensional hyperplane), the simplex σ(X ) spanned by X is the convex hull of X . For #(X ) = 2, this is the line segment spanned by the two points and for #(X ) = 3, it is the triangle spanned by X . X is called the set of vertices of the simplex.

I By a face of a simplex σ, we will mean the simplex spanned by a subset of its vertex set Homology - Simplicial Complexes

n I Given a finite set X of k + 1 points in R in general position (i.e. so that they are not contained in any affine k-dimensional hyperplane), the simplex σ(X ) spanned by X is the convex hull of X . For #(X ) = 2, this is the line segment spanned by the two points and for #(X ) = 3, it is the triangle spanned by X . X is called the set of vertices of the simplex.

I By a face of a simplex σ, we will mean the simplex spanned by a subset of its vertex set

I A simplicial complex X is a collection of simplices in some n R so that (a) for any simplex in X , all of its faces are also in X and (b) the intersection of any two simplices σ and τ of X is a face of both σ and τ. Homology - Simplicial Complexes

C

A

E

B

D

F

G

Homology

A

B C

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

AB AC BC A 1 1 0 B 1 0 1 C 0 1 1 Boundary Matrix Homology

AB AC BC A 1 1 0 B 1 0 1 C 0 1 1 Rank is 2 Homology

AB AC BC A 1 1 0 B 1 0 1 C 0 1 1 Quotient by image is 1-D: means one connected component 0 0 I v + W = v + W if and only if v − v ∈ W . 0 0 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 , has elements the “cosets” v + W 0 0 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 , has elements the “cosets” v + W 0 0 I v + W = v + W if and only if 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 , has elements the “cosets” v + W 0 0 I v + W = v + W if and only if v − v ∈ W . 0 0 I (v + W ) + (v + W ) = v + v + W I How to compute and obtain bases?

Homology

I Quotient of vector space V by subspace W , V /W , has elements the “cosets” v + W 0 0 I v + W = v + W if and only if v − v ∈ W . 0 0 I (v + W ) + (v + W ) = v + v + W

I Too abstract! Homology

I Quotient of vector space V by subspace W , V /W , has elements the “cosets” v + W 0 0 I v + W = v + W if and only if v − v ∈ W . 0 0 I (v + W ) + (v + W ) = v + v + W

I Too abstract!

I How to compute and obtain bases? I Applies row operations to a matrix until in reduced row echelon form

I 1 ∗ − ∗ 0 ∗ − ∗ 0 0 0 − 0 1 ∗ − ∗ 0 0 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 row echelon form

Homology

I Recall Gaussian elimination I 1 ∗ − ∗ 0 ∗ − ∗ 0 0 0 − 0 1 ∗ − ∗ 0 0 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 row echelon form

Homology

I Recall Gaussian elimination

I Applies row operations to a matrix until in reduced row echelon form I Basis of null space of matrix is in one to one correspondence with non-pivot columns of the reduced row echelon form

Homology

I Recall Gaussian elimination

I Applies row operations to a matrix until in reduced row echelon form

I 1 ∗ − ∗ 0 ∗ − ∗ 0 0 0 − 0 1 ∗ − ∗ 0 0 0 − 0 0 0 − 0 1 Homology

I Recall Gaussian elimination

I Applies row operations to a matrix until in reduced row echelon form

I 1 ∗ − ∗ 0 ∗ − ∗ 0 0 0 − 0 1 ∗ − ∗ 0 0 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 row echelon form I Basis is in one to one correspondence with non pivot rows in reduced column echelon form, In fact, the cosets ei + W form a basis for the quotient as ei ranges over the non-pivot rows

I Coefficients in matrix give the expression of v + W as a linear combination of basis elements

Homology

I To obtain quotient, perform column operations until in reduced column echelon form I Coefficients in matrix give the expression of v + W as a linear combination of basis elements

Homology

I To obtain quotient, perform column operations until in reduced column echelon form

I Basis is in one to one correspondence with non pivot rows in reduced column echelon form, In fact, the cosets ei + W form a basis for the quotient as ei ranges over the non-pivot rows Homology

I To obtain quotient, perform column operations until in reduced column echelon form

I Basis is in one to one correspondence with non pivot rows in reduced column echelon form, In fact, the cosets ei + W form a basis for the quotient as ei ranges over the non-pivot rows

I Coefficients in matrix give the expression of v + W as a linear combination of basis elements Homology

AB AC BC A 1 1 0 B 1 0 1 C 0 1 1 Null space is 1-D, spanned by AB + AC + BC Represents the loop in the complex Homology

AB AC BC A 1 1 0 B 1 0 1 C 0 1 1

H0 is quotient by image of boundary map, H1 is the null space I H0 is quotient V0/∂(V1) and H1 is the null space of ∂ in V1

I The dimension of H0 is the number of connected components 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 dimensional features?

Homology

I More invariant form is a pair of vector spaces V0 andV1, together with linear transformation ∂ : V1 → V0 I The dimension of H0 is the number of connected components 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 dimensional features?

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 H1 is the number of loops in the complex I What if we want to understand higher dimensional features?

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 connected components of the complex, namely 1 I What if we want to understand higher dimensional features?

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 connected components of the complex, namely 1

I The dimension of H1 is the number of loops in the complex 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 connected components 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 dimensional features? I Ci (X ) is a vector space with basis the collection of i-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 and column pair have the property that the (i − 1)-simplex corresponding to the row is a fact of the i-simplex corresponding to the column

Homology - Chain Complexes

∂n ∂2 ∂1 ··· Cn(X ) −→ Cn−1(X ) −→ · · · −→ C1(X ) −→ C0(X )

I ∂i−1 ◦ ∂i = 0 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 and column pair have the property that the (i − 1)-simplex corresponding to the row is a fact of the i-simplex corresponding to the column

Homology - Chain Complexes

∂n ∂2 ∂1 ··· Cn(X ) −→ Cn−1(X ) −→ · · · −→ C1(X ) −→ C0(X )

I ∂i−1 ◦ ∂i = 0

I Ci (X ) is a vector space with basis the collection of i-simplices in X I An entry in the matrix is = 1 if and only if its row and column pair have the property that the (i − 1)-simplex corresponding to the row is a fact of the i-simplex corresponding to the column

Homology - Chain Complexes

∂n ∂2 ∂1 ··· Cn(X ) −→ Cn−1(X ) −→ · · · −→ C1(X ) −→ C0(X )

I ∂i−1 ◦ ∂i = 0

I Ci (X ) is a vector space with basis the collection of i-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 Homology - Chain Complexes

∂n ∂2 ∂1 ··· Cn(X ) −→ Cn−1(X ) −→ · · · −→ C1(X ) −→ C0(X )

I ∂i−1 ◦ ∂i = 0

I Ci (X ) is a vector space with basis the collection of i-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 and column pair have the property that the (i − 1)-simplex corresponding to the row is a fact of the i-simplex corresponding to the column I Bi is the image of ∂i+1. It is called the vector space of i-boundaries

I Bi is contained in Zi

I Zi /Bi is called the i-dimensional homology of the simplicial complex X , and is written as Hi (X )

I The dimension of Hi is called the i-th of X

Homology - Chain Complexes

∂n ∂2 ∂1 ··· Cn(X ) −→ Cn−1(X ) −→ · · · −→ C1(X ) −→ C0(X )

I Zi is the null space of ∂i . It is called the vector space of i-cycles. I Bi is contained in Zi

I Zi /Bi is called the i-dimensional homology of the simplicial complex X , and is written as Hi (X )

I The dimension of Hi is called the i-th Betti number of X

Homology - Chain Complexes

∂n ∂2 ∂1 ··· Cn(X ) −→ Cn−1(X ) −→ · · · −→ C1(X ) −→ C0(X )

I Zi is the null space of ∂i . It is called the vector space of i-cycles.

I Bi is the image of ∂i+1. It is called the vector space of i-boundaries I Zi /Bi is called the i-dimensional homology of the simplicial complex X , and is written as Hi (X )

I The dimension of Hi is called the i-th Betti number of X

Homology - Chain Complexes

∂n ∂2 ∂1 ··· Cn(X ) −→ Cn−1(X ) −→ · · · −→ C1(X ) −→ C0(X )

I Zi is the null space of ∂i . It is called the vector space of i-cycles.

I Bi is the image of ∂i+1. It is called the vector space of i-boundaries

I Bi is contained in Zi I The dimension of Hi is called the i-th Betti number of X

Homology - Chain Complexes

∂n ∂2 ∂1 ··· Cn(X ) −→ Cn−1(X ) −→ · · · −→ C1(X ) −→ C0(X )

I Zi is the null space of ∂i . It is called the vector space of i-cycles.

I Bi is the image of ∂i+1. It is called the vector space of i-boundaries

I Bi is contained in Zi

I Zi /Bi is called the i-dimensional homology of the simplicial complex X , and is written as Hi (X ) Homology - Chain Complexes

∂n ∂2 ∂1 ··· Cn(X ) −→ Cn−1(X ) −→ · · · −→ C1(X ) −→ C0(X )

I Zi is the null space of ∂i . It is called the vector space of i-cycles.

I Bi is the image of ∂i+1. It is called the vector space of i-boundaries

I Bi is contained in Zi

I Zi /Bi is called the i-dimensional homology of the simplicial complex X , and is written as Hi (X )

I The dimension of Hi is called the i-th Betti number of X Homology - Chain Complexes

012 013 023 123 01 1 1 0 0 02 1 0 1 0 03 0 1 1 0 12 1 0 0 1 13 0 1 0 1 23 0 0 1 1

∂2 for the sphere Homology - Chain Complexes

012 013 023 123 01 1 1 0 0 02 1 0 1 0 03 0 1 1 0 12 1 0 0 1 13 0 1 0 1 23 0 0 1 1

012 + 013 + 023 + 123 is a 2-cycle I Uses linear algebra to create integer invariants βi (X ) for all i ≥ 0

I βi (X ) counts number of i dimensional “holes” in X , or number of independent “cycles” or higher dimensional surfaces

Algebraic Topology

I Produces discrete “signatures” describing deformation invariant properties of a space I βi (X ) counts number of i dimensional “holes” in X , or number of independent “cycles” or higher dimensional surfaces

Algebraic Topology

I Produces discrete “signatures” describing deformation invariant properties of a space

I Uses linear algebra to create integer invariants βi (X ) for all i ≥ 0 Algebraic Topology

I Produces discrete “signatures” describing deformation invariant properties of a space

I Uses linear algebra to create integer invariants βi (X ) for all i ≥ 0

I βi (X ) counts number of i dimensional “holes” in X , or number of independent “cycles” or higher dimensional surfaces I β0 = 1

I β1 = 2

I β2 = 0

Betti Numbers I β1 = 2

I β2 = 0

Betti Numbers

I β0 = 1 I β2 = 0

Betti Numbers

I β0 = 1

I β1 = 2 Betti Numbers

I β0 = 1

I β1 = 2

I β2 = 0 I β0 = 1

I β1 = 0

I β2 = 1

Betti Numbers I β1 = 0

I β2 = 1

Betti Numbers

I β0 = 1 I β2 = 1

Betti Numbers

I β0 = 1

I β1 = 0 Betti Numbers

I β0 = 1

I β1 = 0

I β2 = 1 I β0 = 1

I β1 = 2

I β2 = 1

Betti Numbers I β1 = 2

I β2 = 1

Betti Numbers

I β0 = 1 I β2 = 1

Betti Numbers

I β0 = 1

I β1 = 2 Betti Numbers

I β0 = 1

I β1 = 2

I β2 = 1 I β0 = 1

I β1 = 4

I β2 = 1

Betti Numbers I β1 = 4

I β2 = 1

Betti Numbers

I β0 = 1 I β2 = 1

Betti Numbers

I β0 = 1

I β1 = 4 Betti Numbers

I β0 = 1

I β1 = 4

I β2 = 1 I β0 = 1 I β1 = 2 I β2 = 1

Betti Numbers I β1 = 2 I β2 = 1

Betti Numbers

I β0 = 1 I β2 = 1

Betti Numbers

I β0 = 1 I β1 = 2 Betti Numbers

I β0 = 1 I β1 = 2 I β2 = 1 I Build an analogous chain complex with vector spaces ∼ Cn(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 of X not contained in Y are contained in X0

Relative Homology

I Exists a notion of relative homology of a pair (X , 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 of X 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 spaces ∼ Cn(X , Y) = Cn(X )/Cn(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 of X 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 spaces ∼ Cn(X , Y) = Cn(X )/Cn(Y)

I Obtain relative homology groups Hn(X , Y) ∼ 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 of X 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 spaces ∼ Cn(X , Y) = Cn(X )/Cn(Y)

I Obtain relative homology groups Hn(X , Y) I Obey the excision property Relative Homology

I Exists a notion of relative homology of a pair (X , Y)

I Build an analogous chain complex with vector spaces ∼ Cn(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 of X not contained in Y are contained in X0 I Samuel Eilenberg (1944) defined Hi (X ) for a topological space independent of the structure of a simplicial complex

I Essentially constructed a very infinite simplicial complex attached to any topological space in a natural way

I Called

Homology of Topological Spaces

I What if our space is not given as a simplicial complex? I Essentially constructed a very infinite simplicial complex attached 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 topological space independent of the structure of a simplicial complex 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 topological space independent of the structure of a simplicial complex

I Essentially constructed a very infinite simplicial complex attached to any topological space in a natural way 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 topological space independent of the structure of a simplicial complex

I Essentially constructed a very infinite simplicial complex attached to any topological space in a natural way

I Called singular homology I This feature is critical to all applications of and most computational techniques for homology

I Means that what was originally thought of purely as a counting problem is now categorified, so that information about morphisms of spaces is also encoded in the invariant

Functoriality and Categorification

I observed that given a map of simplicial complexes f : X → Y, one obtains a linear transformation Hi (f ): Hi (X ) → Hi (Y) I Means that what was originally thought of purely as a counting problem is now categorified, so that information about morphisms of spaces is also encoded in the invariant

Functoriality and Categorification

I Emmy Noether observed that given a map of simplicial complexes f : X → Y, one obtains a linear transformation Hi (f ): Hi (X ) → Hi (Y) I This feature is critical to all applications of and most computational techniques for homology Functoriality and Categorification

I Emmy Noether observed that given a map of simplicial complexes f : X → Y, one obtains a linear transformation Hi (f ): Hi (X ) → Hi (Y) I This feature is critical to all applications of and most computational techniques for homology

I Means that what was originally thought of purely as a counting problem is now categorified, so that information about morphisms of spaces is also encoded in the invariant I Analogous to the notion of homotopy of paths discussed above

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 is g : Y → X so that f ◦ g is homotopic to idY and g ◦ f is homotopic 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 a continuous map H : X × [0, 1] → Y with H(x, 0) = f (x) and H(x, 1) = g(x) 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 is g : Y → X so that f ◦ g is homotopic to idY and g ◦ f is homotopic 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 a continuous 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 discussed above I f : X → Y is a homotopy equivalence if there is g : Y → X so that f ◦ g is homotopic to idY and g ◦ f is homotopic 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 a continuous 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 discussed above

I If f and g from X to Y are homotopic, then Hi (f ) = Hi (g) I Of critical importance in computation and application

Homotopy Invariance

I Two maps f , g : X → Y are homotopic if there is a continuous 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 discussed above

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 is g : Y → X so that f ◦ g is homotopic to idY and g ◦ f is homotopic to idX . Means Hi (f ) is an isomorphism for all i. Homotopy Invariance

I Two maps f , g : X → Y are homotopic if there is a continuous 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 discussed above

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 is g : Y → X so that f ◦ g is homotopic to idY and g ◦ f is homotopic 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

1 2 2 H1(S ) → H1(D ) → H1(S ) Application: Brouwer Fixed Point Theorem

k → {0} → k I Often computationally intensive, not applicable for singular homology

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 computing the homology of simplicial complexes 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 computing the homology of simplicial complexes

I Often computationally intensive, not applicable for singular homology I One such method is the method of exact sequences

Computing Homology

I There are direct linear algebraic methods for computing the homology of simplicial complexes

I Often computationally intensive, not applicable for singular homology

I Requires the development of indirect methods Computing Homology

I There are direct linear algebraic methods for computing the homology of simplicial complexes

I Often computationally intensive, not applicable for singular homology

I Requires the development of indirect methods

I One such method is the method of exact sequences I A longer sequence

fn fn−1 f2 f1 Vn → Vn−1 → · · · → V1 → V0

is exact if and only if its length three subsequences are all exact

Exact Sequences

I A sequence of linear transformations of vector spaces g U →f V → W is exact if (a) g ◦ f is identically zero and (b) the kernel of g is equal to the image of f Exact Sequences

I A sequence of linear transformations of vector spaces g U →f V → 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

fn fn−1 f2 f1 Vn → Vn−1 → · · · → V1 → V0

is exact if and only if its length three subsequences are all exact I Means that one can compute V2 in terms of V4, V3, V1, and V0 and the transformations relating them. f I 0 → V → W → 0 exact means f is isomorphism.

Exact Sequences

I For a five term exact sequence

f4 f3 f2 f1 V4 → V3 → V2 → V1 → V0

on has that the dimension of V2 is equal to the sum of the dimensions of V3/f4(V4) and Ker(f1) f I 0 → V → W → 0 exact means f is isomorphism.

Exact Sequences

I For a five term exact sequence

f4 f3 f2 f1 V4 → V3 → V2 → V1 → V0

on has that the dimension of V2 is equal to the sum of the dimensions 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. Exact Sequences

I For a five term exact sequence

f4 f3 f2 f1 V4 → V3 → V2 → V1 → V0

on has that the dimension of V2 is equal to the sum of the dimensions 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. f I 0 → V → W → 0 exact means f is isomorphism. Means one can compute the homology of Hi (X ) in terms of Hi (Y ) and Hi (X , Y )

Exact Sequence of a Pair

··· Hi+1(X , Y ) → Hi (Y ) → Hi (X ) → Hi (X , Y ) → Hi−1(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 of Hi (Y ) and Hi (X , Y ) Means we can compute the homology of the union of two sets in terms of the homology of the two sets and their intersection. Can be viewed as categorification of inclusion/exclusion principle.

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 ) → · · · 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 sets in terms of the homology of the two sets and their intersection. Can be viewed as categorification of inclusion/exclusion principle. n n Hi (D±) are zero for i > 0, since D± are contractible, i.e. map n D± → ∗ is an equivalence

Mayer-Vietoris Sequence

n n n n n ∼ n−1 Suppose we take S = D+ ∪ D−, with D+ ∩ D− = S , get exact sequence

n n n n−1 n n Hi (D+)⊕Hi (D+) → Hi (S ) → Hi−1(S ) → Hi−1(D+)⊕Hi−1(D+) Mayer-Vietoris Sequence

n n n n n ∼ n−1 Suppose we take S = D+ ∪ D−, with D+ ∩ D− = S , get exact sequence

n n n n−1 n n Hi (D+)⊕Hi (D+) → Hi (S ) → Hi−1(S ) → Hi−1(D+)⊕Hi−1(D+)

n n Hi (D±) are zero for i > 0, since D± are contractible, i.e. map n D± → ∗ is an equivalence Mayer-Vietoris Sequence

n n n n n ∼ n−1 Suppose we take S = D+ ∪ D−, with D+ ∩ D− = S , get exact sequence

n n n n−1 n n Hi (D+)⊕Hi (D+) → Hi (S ) → Hi−1(S ) → Hi−1(D+)⊕Hi−1(D+)

n ∼ n−1 Follows that Hi (S ) = Hi−1(S ) for i > 2. Computation of n Hi (S ) follows. Mayer-Vietoris Sequence

n n n n n ∼ n−1 Suppose we take S = D+ ∪ D−, with D+ ∩ D− = S , get exact sequence

n n n n−1 n n Hi (D+)⊕Hi (D+) → Hi (S ) → Hi−1(S ) → Hi−1(D+)⊕Hi−1(D+)

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