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H-Spaces and Hopf Algebras JuniorTopologySeminar MustafaHajij

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Page 1: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

H-Spaces and Hopf Algebras

Junior Topology Seminar Mustafa Hajij

Page 2: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Basic Def init ions

Algebras

Page 3: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Basic Def init ions

Algebras

Page 4: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Basic Def init ions

Algebras

Page 5: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Basic Def init ions

Algebras

Page 6: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Basic Def init ions

Algebras

Page 7: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Basic Def init ions

Algebras

Page 8: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Basic Def init ions

Algebras

Page 9: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Basic Def init ions

Algebras

Page 10: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Basic Def init ions

Algebras

Page 11: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Basic Def init ions

Algebras

Page 12: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Basic Def init ions

Algebras

Page 13: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Basic Def init ions

Algebras

Page 14: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Basic Def init ions

Algebras

Page 15: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Basic Def init ions

Algebras

Page 16: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Basic Def init ions

Algebras

Page 17: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Basic Def init ions

Hopf Algebras

Page 18: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Basic Def init ions

Hopf Algebras

Page 19: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Basic Def init ions

Hopf Algebras

Page 20: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Basic Def init ions

Hopf Algebras

Page 21: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Basic Def init ions

Hopf Algebras

Page 22: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Hopf Algebras

Example The Polynomial ring R[X]

Page 23: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Hopf Algebras

Example The Polynomial ring R[X]

Page 24: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Hopf Algebras

Example The Polynomial ring R[X]

Page 25: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Hopf Algebras

Example The Polynomial ring R[X]

Page 26: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Hopf Algebras

Example The Polynomial ring R[X]

Page 27: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Hopf Algebras

Example

Page 28: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Hopf Algebras

Example

Page 29: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Hopf Algebras

Example

Page 30: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Hopf Algebras

Example

Page 31: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Hopf Algebras

Example

Page 32: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Hopf Algebras

Example

Page 33: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Hopf Algebras

Example

Page 34: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

H-Spaces

Definition of H-spaces

Page 35: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

H-Spaces

Examples of H-spaces

Page 36: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

H-Spaces

Examples of H-spaces

Page 37: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

H-Spaces

Examples of H-spaces

Page 38: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

H-Spaces

Examples of H-spaces

Page 39: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

H-Spaces

Examples of H-spaces

Page 40: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

H-Spaces

Examples of H-spaces

Page 41: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

H-Spaces

Examples of H-spaces

Page 42: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

The Cohomology algebra of an H-space is a Hopf algebra Introduction

Page 43: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

The Cohomology algebra of an H-space is a Hopf algebra Introduction

Page 44: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

The Cohomology algebra of an H-space is a Hopf algebra Introduction

Page 45: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

The Cohomology algebra of an H-space is a Hopf algebra Introduction

Page 46: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

The Cohomology algebra of an H-space is a Hopf algebra Introduction

Page 47: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

The Cohomology algebra of an H-space is a Hopf algebra Dual Hopf Algebra

Page 48: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

A Product structure on homology of H-spaces Pontryagin product

Page 49: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

A Product structure on homology of H-spaces

Pontryagin product

Page 50: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

A Product structure on homology of H-spaces

Pontryagin product

Page 51: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

A Product structure on homology of H-spaces

Pontryagin product

Page 52: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Dual Hopf Algebra

Page 53: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Dual Hopf Algebra

Page 54: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Dual Hopf Algebra

Page 55: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Dual Hopf Algebra

Page 56: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Dual Hopf Algebra

Page 57: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Dual Hopf Algebra

Page 58: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Dual Hopf Algebra

Page 59: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Dual Hopf Algebra

Page 60: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Dual Hopf Algebra

What is the relation between Cohomology and Homology in H-spaces?

They are dual Hopf algebras

Page 61: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Dual Hopf Algebra

What is the relation between Cohomology and Homology in H-spaces?

They are dual Hopf algebras

Page 62: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

Dual Hopf Algebra

Page 63: H-Spaces and Hopf Algebras - Mustafa HajijThe Cohomology algebra of an H-space is a Hopf algebra . Dual Hopf Algebra . A Product structure on homology of H-spaces . Pontryagin product

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