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Department of Mathematical Sciences

# MATH4161 Algebraic Topology IV

The basic method of algebraic topology is to assign an algebraic system (say, a group or a ring) to each topological space in such a way that homeomorphic spaces have isomorphic systems. Geometrical problems about spaces can then be solved by "pushing" them into algebra and doing computations there. This idea can be illustrated by the theory of fundamental group which is familiar from the Topology III course.

In this course we meet other, more sophisticated theories such as singular homology and cohomology. The course begins with an introduction to cell complexes, a convenient class of topological spaces which contain many important examples and which are amenable to homology calculations. Properties of singular homology are studied and used to prove important theorems such as the Brouwer fixed point theorem and the Jordan curve theorem.

In the second term cohomology is introduced, which appears formally very similar to homology, but adds useful structure to the theory. Applied to manifolds, the full power of these theories lies in their interplay, which culminates in the duality theorems of Poincaré and Alexander. As an application we show that non-orientable surfaces such as the real projective plane and the Klein bottle cannot be embedded in 3-dimensional Euclidean space.

## Outline of Course

Aim: The module will provide a deeper knowledge in the field of topology (a balanced introduction having been provided in Topology III).

### Term 1

• Homotopy properties of cell complexes: Cell complexes, main constructions (mapping cones and cylinders, products), examples.
• Elements of homological algebra: Chain complexes, homology, chain homotopy, exact sequences, Euler characteristic.
• Homology theory of topological spaces: Singular homology of topological spaces, homotopy invariance, Mayer-Vietoris sequences, relation between homology and the fundamental group, geometric interpretation of homology classes, homology groups of cell complexes.
• Applications: Brouwer fixed point theorem, Jordan curve theorem, invariance of domain.

### Term 2

• Cohomology theory of topological spaces: Singular cohomology of topological spaces, statement of the universal coefficient theorem, cup products and ring structure, Künneth formula.
• (Co)homology of manifolds: Fundamental classes, orientations in terms of homology, intersection numbers, Poincaré duality.
• Applications: Alexander duality, non-embeddability of non-orientable closed surface in Euclidean 3-space.

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.