Intro to Homotopy Theory

Katharine Adamyk

August 23, 2017

Katharine Adamyk — Intro to Homotopy Theory 1/27

Outline

1 Intro to Homotopy

2 Fundamental Group

3 Homotopy Groups

4 An Open Problem

5 A Strategy

Katharine Adamyk — Intro to Homotopy Theory 2/27

Outline

1 Intro to Homotopy

2 Fundamental Group

3 Homotopy Groups

4 An Open Problem

5 A Strategy

Katharine Adamyk — Intro to Homotopy Theory 3/27

Algebraic TopologyWhat is homotopy theory?

Algebraic Topology

Topological space ⇒ algebraic structure ⇒ info about spacem

All kinds of fun stuff

Katharine Adamyk — Intro to Homotopy Theory 4/27

Algebraic TopologyWhat is homotopy theory?

Algebraic Topology

Topological space

⇒ algebraic structure ⇒ info about spacem

All kinds of fun stuff

Katharine Adamyk — Intro to Homotopy Theory 4/27

Algebraic TopologyWhat is homotopy theory?

Algebraic Topology

Topological space ⇒ algebraic structure

⇒ info about spacem

All kinds of fun stuff

Katharine Adamyk — Intro to Homotopy Theory 4/27

Algebraic TopologyWhat is homotopy theory?

Algebraic Topology

Topological space ⇒ algebraic structure ⇒ info about space

mAll kinds of fun stuff

Katharine Adamyk — Intro to Homotopy Theory 4/27

Algebraic TopologyWhat is homotopy theory?

Algebraic Topology

Topological space ⇒ algebraic structure ⇒ info about spacem

All kinds of fun stuff

Katharine Adamyk — Intro to Homotopy Theory 4/27

The World We Live InSome Preliminaries

All our maps are continuous and all our spaces are “nice”.

A based space is a topological space X with adistinguished point ∗.A based map is a map of based spaces that takesbasepoint to basepoint.

Katharine Adamyk — Intro to Homotopy Theory 5/27

The World We Live InSome Preliminaries

All our maps are continuous and all our spaces are “nice”.

A based space is a topological space X with adistinguished point ∗.A based map is a map of based spaces that takesbasepoint to basepoint.

Katharine Adamyk — Intro to Homotopy Theory 5/27

The World We Live InSome Preliminaries

All our maps are continuous and all our spaces are “nice”.

A based space is a topological space X with adistinguished point ∗.

A based map is a map of based spaces that takesbasepoint to basepoint.

Katharine Adamyk — Intro to Homotopy Theory 5/27

The World We Live InSome Preliminaries

All our maps are continuous and all our spaces are “nice”.

A based space is a topological space X with adistinguished point ∗.A based map is a map of based spaces that takesbasepoint to basepoint.

Katharine Adamyk — Intro to Homotopy Theory 5/27

What is a Homotopy?Formal Definition

The product space X × Y is {(x , y)|x ∈ X , y ∈ Y }

Definition

Let I = [0, 1]. Given two maps f , g : X → Y , a homotopy off and g is a map H : X × I → Y such that:

H(x , 0) = f (x)

H(x , 1) = g(x)

H(∗, t) = ∗ for all t ∈ I

Katharine Adamyk — Intro to Homotopy Theory 6/27

What is a Homotopy?Formal Definition

The product space X × Y is {(x , y)|x ∈ X , y ∈ Y }

Definition

Let I = [0, 1]. Given two maps f , g : X → Y , a homotopy off and g is a map H : X × I → Y such that:

H(x , 0) = f (x)

H(x , 1) = g(x)

H(∗, t) = ∗ for all t ∈ I

Katharine Adamyk — Intro to Homotopy Theory 6/27

What is a Homotopy?Formal Definition

The product space X × Y is {(x , y)|x ∈ X , y ∈ Y }

Definition

Let I = [0, 1]. Given two maps f , g : X → Y , a homotopy off and g is a map H : X × I → Y such that:

H(x , 0) = f (x)

H(x , 1) = g(x)

H(∗, t) = ∗ for all t ∈ I

Katharine Adamyk — Intro to Homotopy Theory 6/27

What is a Homotopy?Formal Definition

The product space X × Y is {(x , y)|x ∈ X , y ∈ Y }

Definition

Let I = [0, 1]. Given two maps f , g : X → Y , a homotopy off and g is a map H : X × I → Y such that:

H(x , 0) = f (x)

H(x , 1) = g(x)

H(∗, t) = ∗ for all t ∈ I

Katharine Adamyk — Intro to Homotopy Theory 6/27

What is a Homotopy?Formal Definition

The product space X × Y is {(x , y)|x ∈ X , y ∈ Y }

Definition

Let I = [0, 1]. Given two maps f , g : X → Y , a homotopy off and g is a map H : X × I → Y such that:

H(x , 0) = f (x)

H(x , 1) = g(x)

H(∗, t) = ∗ for all t ∈ I

Katharine Adamyk — Intro to Homotopy Theory 6/27

What is a Homotopy?Formal Definition

The product space X × Y is {(x , y)|x ∈ X , y ∈ Y }

Definition

Let I = [0, 1]. Given two maps f , g : X → Y , a homotopy off and g is a map H : X × I → Y such that:

H(x , 0) = f (x)

H(x , 1) = g(x)

H(∗, t) = ∗ for all t ∈ I

Katharine Adamyk — Intro to Homotopy Theory 6/27

What is a Homotopy?Formal Definition

Katharine Adamyk — Intro to Homotopy Theory 7/27

What is a Homotopy?Intuition

Katharine Adamyk — Intro to Homotopy Theory 8/27

Homotopy Classes

If a homotopy of f and g exists, we say f ' g , “f ishomotopic to g”.

For any f : X → Y , f ' f .

If f ' g , then g ' f .

If f ' g and g ' h, then f ' h.

So, ' is an equivalence relation on maps X → Y .

Katharine Adamyk — Intro to Homotopy Theory 9/27

Homotopy Classes

If a homotopy of f and g exists, we say f ' g , “f ishomotopic to g”.

For any f : X → Y , f ' f .

If f ' g , then g ' f .

If f ' g and g ' h, then f ' h.

So, ' is an equivalence relation on maps X → Y .

Katharine Adamyk — Intro to Homotopy Theory 9/27

Homotopy Classes

If a homotopy of f and g exists, we say f ' g , “f ishomotopic to g”.

For any f : X → Y , f ' f .

If f ' g , then g ' f .

If f ' g and g ' h, then f ' h.

So, ' is an equivalence relation on maps X → Y .

Katharine Adamyk — Intro to Homotopy Theory 9/27

Homotopy Classes

If a homotopy of f and g exists, we say f ' g , “f ishomotopic to g”.

For any f : X → Y , f ' f .

If f ' g , then g ' f .

If f ' g and g ' h, then f ' h.

So, ' is an equivalence relation on maps X → Y .

Katharine Adamyk — Intro to Homotopy Theory 9/27

Homotopy Classes

If a homotopy of f and g exists, we say f ' g , “f ishomotopic to g”.

For any f : X → Y , f ' f .

If f ' g , then g ' f .

If f ' g and g ' h, then f ' h.

So, ' is an equivalence relation on maps X → Y .

Katharine Adamyk — Intro to Homotopy Theory 9/27

Homotopy Classes

If a homotopy of f and g exists, we say f ' g , “f ishomotopic to g”.

For any f : X → Y , f ' f .

If f ' g , then g ' f .

If f ' g and g ' h, then f ' h.

So, ' is an equivalence relation on maps X → Y .

Katharine Adamyk — Intro to Homotopy Theory 9/27

Outline

1 Intro to Homotopy

2 Fundamental Group

3 Homotopy Groups

4 An Open Problem

5 A Strategy

Katharine Adamyk — Intro to Homotopy Theory 10/27

Fundamental GroupPreliminaries

A loop on X is a map γ : S1 → X where S1 is the circle.

Two loops are in the same homotopy class if there existsa homotopy between them.

Katharine Adamyk — Intro to Homotopy Theory 11/27

Fundamental GroupPreliminaries

A loop on X is a map γ : S1 → X where S1 is the circle.

Two loops are in the same homotopy class if there existsa homotopy between them.

Katharine Adamyk — Intro to Homotopy Theory 11/27

Fundamental GroupPreliminaries

A loop on X is a map γ : S1 → X where S1 is the circle.

Two loops are in the same homotopy class if there existsa homotopy between them.

Katharine Adamyk — Intro to Homotopy Theory 11/27

Fundamental GroupDefinition

Definition

The fundamental group of a topological space X , π1(X ), isthe set of homotopy classes of loops on X with the operationof loop concatenation.

Katharine Adamyk — Intro to Homotopy Theory 12/27

Fundamental GroupDefinition

Definition

The fundamental group of a topological space X , π1(X ), isthe set of homotopy classes of loops on X with the operationof loop concatenation.

Katharine Adamyk — Intro to Homotopy Theory 12/27

Fundamental GroupDefinition

Definition

The fundamental group of a topological space X , π1(X ), isthe set of homotopy classes of loops on X with the operationof loop concatenation.

Katharine Adamyk — Intro to Homotopy Theory 12/27

Fundamental GroupGroup Operation

A : S1 → X B : S1 → X

A + B : S1 → X

Katharine Adamyk — Intro to Homotopy Theory 13/27

Fundamental GroupGroup Operation

A : S1 → X B : S1 → X

A + B : S1 → X

Katharine Adamyk — Intro to Homotopy Theory 13/27

Fundamental GroupIdentity

The identity element of π1(X ) is the constant loop.

A loop that is in the same homotopy class as theconstant loop is called contractible .

Contractible Not Contractible

Katharine Adamyk — Intro to Homotopy Theory 14/27

Fundamental GroupIdentity

The identity element of π1(X ) is the constant loop.

A loop that is in the same homotopy class as theconstant loop is called contractible .

Contractible Not Contractible

Katharine Adamyk — Intro to Homotopy Theory 14/27

Fundamental GroupIdentity

The identity element of π1(X ) is the constant loop.

A loop that is in the same homotopy class as theconstant loop is called contractible .

Contractible Not Contractible

Katharine Adamyk — Intro to Homotopy Theory 14/27

Fundamental GroupIdentity

The identity element of π1(X ) is the constant loop.

A loop that is in the same homotopy class as theconstant loop is called contractible .

Contractible Not Contractible

Katharine Adamyk — Intro to Homotopy Theory 14/27

Fundamental GroupIdentity

The identity element of π1(X ) is the constant loop.

A loop that is in the same homotopy class as theconstant loop is called contractible .

Contractible

Not Contractible

Katharine Adamyk — Intro to Homotopy Theory 14/27

Fundamental GroupIdentity

The identity element of π1(X ) is the constant loop.

A loop that is in the same homotopy class as theconstant loop is called contractible .

Contractible Not Contractible

Katharine Adamyk — Intro to Homotopy Theory 14/27

Fundamental GroupInverses

f ∈ [f ]

g ∈ [f ]−1

Katharine Adamyk — Intro to Homotopy Theory 15/27

Fundamental GroupInverses

f ∈ [f ]

g ∈ [f ]−1

Katharine Adamyk — Intro to Homotopy Theory 15/27

Fundamental GroupInverses

f ∈ [f ]

g ∈ [f ]−1

Katharine Adamyk — Intro to Homotopy Theory 15/27

Fundamental GroupInverses

Katharine Adamyk — Intro to Homotopy Theory 16/27

Fundamental GroupExamples

π1(S1) = Zπ1(T) = Z⊕ Z

π1(S) = 0

Katharine Adamyk — Intro to Homotopy Theory 17/27

Fundamental GroupExamples

π1(S1) = Z

π1(T) = Z⊕ Z

π1(S) = 0

Katharine Adamyk — Intro to Homotopy Theory 17/27

Fundamental GroupExamples

π1(S1) = Z

π1(T) = Z⊕ Z

π1(S) = 0

Katharine Adamyk — Intro to Homotopy Theory 17/27

Fundamental GroupExamples

π1(S1) = Zπ1(T) = Z⊕ Z

π1(S) = 0

Katharine Adamyk — Intro to Homotopy Theory 17/27

Fundamental GroupExamples

π1(S1) = Zπ1(T) = Z⊕ Z

π1(S) = 0

Katharine Adamyk — Intro to Homotopy Theory 17/27

Fundamental GroupExamples

π1(S1) = Zπ1(T) = Z⊕ Z

π1(S) = 0

Katharine Adamyk — Intro to Homotopy Theory 17/27

Outline

1 Intro to Homotopy

2 Fundamental Group

3 Homotopy Groups

4 An Open Problem

5 A Strategy

Katharine Adamyk — Intro to Homotopy Theory 18/27

Spheres

Definition

The n-sphere is

Sn =

{(x0, x1, . . . , xn) ∈ Rn+1 |

√x20 + x21 + . . . + x2n = 1

}

S0 S1 S2

Katharine Adamyk — Intro to Homotopy Theory 19/27

Spheres

Definition

The n-sphere is

Sn =

{(x0, x1, . . . , xn) ∈ Rn+1 |

√x20 + x21 + . . . + x2n = 1

}

S0 S1 S2

Katharine Adamyk — Intro to Homotopy Theory 19/27

Spheres

Definition

The n-sphere is

Sn =

{(x0, x1, . . . , xn) ∈ Rn+1 |

√x20 + x21 + . . . + x2n = 1

}

S0 S1 S2

Katharine Adamyk — Intro to Homotopy Theory 19/27

Spheres

Definition

The n-sphere is

Sn =

{(x0, x1, . . . , xn) ∈ Rn+1 |

√x20 + x21 + . . . + x2n = 1

}

S0

S1 S2

Katharine Adamyk — Intro to Homotopy Theory 19/27

Spheres

Definition

The n-sphere is

Sn =

{(x0, x1, . . . , xn) ∈ Rn+1 |

√x20 + x21 + . . . + x2n = 1

}

S0 S1

S2

Katharine Adamyk — Intro to Homotopy Theory 19/27

Spheres

Definition

The n-sphere is

Sn =

{(x0, x1, . . . , xn) ∈ Rn+1 |

√x20 + x21 + . . . + x2n = 1

}

S0 S1 S2

Katharine Adamyk — Intro to Homotopy Theory 19/27

Homotopy Group

Definition

The nth homotopy group of a space X , πn(X ), is the set ofhomotopy classes of based maps Sn → X . The groupoperation is defined by composing the wedge of two maps withthe pinch map Sn → Sn ∨ Sn.

Katharine Adamyk — Intro to Homotopy Theory 20/27

Outline

1 Intro to Homotopy

2 Fundamental Group

3 Homotopy Groups

4 An Open Problem

5 A Strategy

Katharine Adamyk — Intro to Homotopy Theory 21/27

The Dream

πn(Sm)

for all n,m ∈ Z≥0

Katharine Adamyk — Intro to Homotopy Theory 22/27

The Dream

πn(Sm)

for all n,m ∈ Z≥0

Katharine Adamyk — Intro to Homotopy Theory 22/27

Katharine Adamyk — Intro to Homotopy Theory 23/27

The DreamRevised

For large enough k , πn+k

(Sk)

is constant. We call thiseventually reached group πS

n

(S0), the nth stable homotopy

group of S0 or the nth stable stem.

πSn(S0)

for all n ∈ Z

Katharine Adamyk — Intro to Homotopy Theory 24/27

The DreamRevised

For large enough k , πn+k

(Sk)

is constant. We call thiseventually reached group πS

n

(S0), the nth stable homotopy

group of S0 or the nth stable stem.

πSn(S0)

for all n ∈ Z

Katharine Adamyk — Intro to Homotopy Theory 24/27

The DreamRevised

For large enough k , πn+k

(Sk)

is constant. We call thiseventually reached group πS

n

(S0), the nth stable homotopy

group of S0 or the nth stable stem.

πSn(S0)

for all n ∈ Z

Katharine Adamyk — Intro to Homotopy Theory 24/27

Outline

1 Intro to Homotopy

2 Fundamental Group

3 Homotopy Groups

4 An Open Problem

5 A Strategy

Katharine Adamyk — Intro to Homotopy Theory 25/27

LocalizationOne Prime at a Time

With the exception of πS0 (S0), the stable stems are all finite

abelian groups. So, they can be decomposed as a product:

Z/pe11 × Z/pe22 × · · · × Z/pemm

One strategy is to see what we can discover about thep-torsion part of πn(S0), then assemble that information forevery prime p.

Katharine Adamyk — Intro to Homotopy Theory 26/27

LocalizationOne Prime at a Time

With the exception of πS0 (S0), the stable stems are all finite

abelian groups.

So, they can be decomposed as a product:

Z/pe11 × Z/pe22 × · · · × Z/pemm

One strategy is to see what we can discover about thep-torsion part of πn(S0), then assemble that information forevery prime p.

Katharine Adamyk — Intro to Homotopy Theory 26/27

LocalizationOne Prime at a Time

With the exception of πS0 (S0), the stable stems are all finite

abelian groups. So, they can be decomposed as a product:

Z/pe11 × Z/pe22 × · · · × Z/pemm

One strategy is to see what we can discover about thep-torsion part of πn(S0), then assemble that information forevery prime p.

Katharine Adamyk — Intro to Homotopy Theory 26/27

LocalizationOne Prime at a Time

With the exception of πS0 (S0), the stable stems are all finite

abelian groups. So, they can be decomposed as a product:

Z/pe11 × Z/pe22 × · · · × Z/pemm

One strategy is to see what we can discover about thep-torsion part of πn(S0), then assemble that information forevery prime p.

Katharine Adamyk — Intro to Homotopy Theory 26/27

LocalizationOne Prime at a Time

With the exception of πS0 (S0), the stable stems are all finite

abelian groups. So, they can be decomposed as a product:

Z/pe11 × Z/pe22 × · · · × Z/pemm

One strategy is to see what we can discover about thep-torsion part of πn(S0), then assemble that information forevery prime p.

Katharine Adamyk — Intro to Homotopy Theory 26/27

Thanks!

Katharine Adamyk — Intro to Homotopy Theory 27/27