# Course Identification

## Lecturers and Teaching Assistants

## Course Schedule and Location

## Field of Study, Course Type and Credit Points

## Comments

## Prerequisites

## Restrictions

## Language of Instruction

## Attendance and participation

## Grade Type

## Grade Breakdown (in %)

## Evaluation Type

**Take-home exam**

## Scheduled date 1

## Estimated Weekly Independent Workload (in hours)

## Syllabus

Syllabus (preliminary!)

-- On history of algebraic topology.

-- Basic objects and categories: manifolds, simplicial complexes, cell complexes. -- Basic examples. -- Topological manifolds versus smooth manifolds. -- Every smooth manifold homotopic to a cell complex: Morse theory.

-- Classification of 2-dimensional manifolds.

-- Examples of 3-dimensional and 4-dimensional manifolds.

== Homology and cohomology ==

-- Simplicial and singular homology and cohomology. Betti numbers.

-- Axioms for the homology and cohomology theories.

-- Homological algebra - 1: resolutions, homotopic complexes, quasi-isomorphisms.

-- De Rham cohomology of smooth manifolds.

-- Sheaf (Cech) cohomology of the constant sheaf.

-- Homological algebra - 2: Derived functor of a left exact functor. Long exact sequence associated to a short exact sequence.

-- Isomomorphisms: idea of the proof.

-- (If time permits): Cohomology ring. -- Poincare duality for manifolds. -- Pairing in the middle cohomology of manifolds.

= Homotopy =

-- Fundamental group. -- Examples of computations. Covering spaces, and topological Galois theory. Example: solving polynomials of degree 5.

-- (If time permits): Higher homotopy groups. Eilenberg-Maclane spaces K(G,n). -- Postnikov tower.

= If we get lucky; or: some time in the future =

Fundamental class of a submanifold in de Rham cohomology. -- Thom isomorphism, and applications. Also: Vector bundles. -- Characteristic classes via obstruction theory, and via maps to classifying spaces. -- K-groups. -- Cohomology of some classifying spaces. -- Relation to the cohomology of Lie groups.

## Learning Outcomes

Basic understanding of manifolds, homology and cohomology, fundamental groups; ability to work with examples, and do some basic computations. -- This section will be expanded upon speaking with students, after I learn more about their knowledge level. For example, I may add characteristic classes to this section.