Homology Theory

This is the course 'Algebraic Topology I' for junior graduate students at AMSS and UCAS, and also for interested senior undergraduates. Basic topology and abstract algebra at undergraduate level are required. 

Exercies are provided during lectures, and students do not need to submit the solutions. There is only one final exam holding in the last class.

Chapter I: Fundamental Groups

1.1. review on homotopy

1.2. Some common spaces

1.3. path homotopy and fundamental groups

1.4. properties of fundamental groups

1.5. the fundamental group of circle

1.6. Van Kampen Theorem

1.7. calculations of fundamental groups and applications

1.8. covering spaces

Chapter II: Singular Homology Groups

2.1. singular chain complex and singular homology groups

2.2. the homotopy invariance of homology groups

2.3. relation between \pi_1 and H_1

2.4. relative homology groups and exact sequences

2.5. The Excision Theorem

2.6. The Mayer-Vietoris Theorem 

2.7. Some applications

2.8. homology of CW-complexes

Chapter III: Homology with general coefficients

3.1. homology with general coefficients

3.2. Tor and Ext

3.3. The Universal Coefficient Theorem of homology

3.4. The Kunneth Formula

3.5. summary and homology axiom viewpoint

Chapter IV: Singular Cohomology

4.1. singular cohomology with properties

4.2. cup and cap product

Chapter V: Duality Theorem

5.1. orientation

5.2. Poincare Duality