View
16
Download
0
Category
Preview:
Citation preview
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
Recommended