We study the topological notions of union and quotients of spaces; we discuss covering spaces and fundamental groups further, The notion of cell attachement is introduced and the Seifert-van Kampen Theorem is proven. Examples of surfaces illustrate the techniques.
Singular cohomology is defined by dualizing the singular chain complex for spaces. We will study its basic properties, see how it acquires a multiplicative structure and becomes a graded commutative algebra. We study an algebraic version, namely group cohomology, and compare both approaches.