h-spaces and hopf algebras - mustafa hajijthe cohomology algebra of an h-space is a hopf algebra ....

H-Spaces and Hopf Algebras

Junior Topology Seminar Mustafa Hajij

Basic Def init ions

Algebras

Basic Def init ions

Algebras

Basic Def init ions

Algebras

Basic Def init ions

Algebras

Basic Def init ions

Algebras

Basic Def init ions

Algebras

Basic Def init ions

Algebras

Basic Def init ions

Algebras

Basic Def init ions

Algebras

Basic Def init ions

Algebras

Basic Def init ions

Algebras

Basic Def init ions

Algebras

Basic Def init ions

Algebras

Basic Def init ions

Algebras

Basic Def init ions

Algebras

Basic Def init ions

Hopf Algebras

Basic Def init ions

Hopf Algebras

Basic Def init ions

Hopf Algebras

Basic Def init ions

Hopf Algebras

Basic Def init ions

Hopf Algebras

Hopf Algebras

Example The Polynomial ring R[X]

Hopf Algebras

Example The Polynomial ring R[X]

Hopf Algebras

Example The Polynomial ring R[X]

Hopf Algebras

Example The Polynomial ring R[X]

Hopf Algebras

Example The Polynomial ring R[X]

Hopf Algebras

Example

Hopf Algebras

Example

Hopf Algebras

Example

Hopf Algebras

Example

Hopf Algebras

Example

Hopf Algebras

Example

Hopf Algebras

Example

H-Spaces

Definition of H-spaces

H-Spaces

Examples of H-spaces

H-Spaces

Examples of H-spaces

H-Spaces

Examples of H-spaces

H-Spaces

Examples of H-spaces

H-Spaces

Examples of H-spaces

H-Spaces

Examples of H-spaces

H-Spaces

Examples of H-spaces

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

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

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

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

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

The 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

A Product structure on homology of H-spaces

Pontryagin product

A Product structure on homology of H-spaces

Pontryagin product

Dual Hopf Algebra

Dual Hopf Algebra

Dual Hopf Algebra

Dual Hopf Algebra

Dual Hopf Algebra

Dual Hopf Algebra

Dual Hopf Algebra

Dual Hopf Algebra

Dual Hopf Algebra

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

They are dual Hopf algebras

Dual Hopf Algebra

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

They are dual Hopf algebras

Dual Hopf Algebra

T h a n k y o u