C3.1 Algebraic Topology - Material for the year 2019-2020

2019-2020
Lecturer(s): 
Prof. Alexander Ritter
General Prerequisites: 

A3 Rings and Modules is essential, in particular a solid understanding of groups, rings, fields, modules, homomorphisms of modules, kernels and cokernels, and classification of finitely generated abelian groups.
A5 Topology is essential, in paticular a solid understanding of topological spaces, connectedness, compactness, and classification of compact surfaces.
Some but not all material from B2.1 Representation Theory is essential, specifically students must have a solid understanding of tensor products of abelian groups.
Some but not all material from B3.5 Topology and Groups is essential, specifically students must have a solid understanding of homotopic maps, homotopy equivalence, and the fundamental group.
It is recommended, but not required, that students take C2.2 Homological Algebra concurrently.

Course Term: 
Michaelmas
Course Lecture Information: 

16 lectures.

Course Weight: 
1.00 unit(s)
Course Level: 
M

Assessment type:

Course Overview: 

Homology theory is a subject that pervades much of modern mathematics. Its basic ideas are used in nearly every branch, pure and applied. In this course, the homology groups of topological spaces are studied. These powerful invariants have many attractive applications. For example we will prove that the dimension of a vector space is a topological invariant and the fact that 'a hairy ball cannot be combed'.

Learning Outcomes: 

At the end of the course, students are expected to understand the basic algebraic and geometric ideas that underpin homology and cohomology theory. These include the cup product and Poincaré Duality for manifolds. They should be able to choose between the different homology theories and to use calculational tools such as the Mayer-Vietoris sequence to compute the homology and cohomology of simple examples, including projective spaces, surfaces, certain simplicial spaces and cell complexes. At the end of the course, students should also have developed a sense of how the ideas of homology and cohomology may be applied to problems from other branches of mathematics.

Course Synopsis: 

Chain complexes of free Abelian groups and their homology. Short exact sequences. Delta complexes and their homology. Euler characteristic.

Singular homology of topological spaces. Relative homology and the Five Lemma. Homotopy invariance and excision (details of proofs not examinable). Mayer-Vietoris Sequence. Equivalence of simplicial and singular homology.

Degree of a self-map of a sphere. Cell complexes and cellular homology. Application: the hairy ball theorem.

Cohomology of spaces and the Universal Coefficient Theorem (proof not examinable). Cup products. Künneth Theorem (without proof). Topological manifolds and orientability. The fundamental class of an orientable, closed manifold and the degree of a map between manifolds of the same dimension. Poincaré Duality (without proof).

Reading List: 
  1. A. Hatcher, Algebraic Topology (Cambridge University Press, 2001). Chapters 2 and 3.
  2. G. Bredon, Topology and Geometry (Springer, 1997). Chapters 4 and 5.
  3. J. Vick, Homology Theory, Graduate Texts in Mathematics 145 (Springer, 1973).