MMath&Phys Mathematics and Physics

The basic method of Algebraic Topology is to associate an algebraic object to each
topological space so that homeomorphic spaces have isomorphic algebraic objects. These
algebraic objects are then topological invariants and so can be used to distinguish topological
spaces. The fundamental group introduced in MATH31051/2 is such an algebraic object. This
course unit introduces higher homotopy groups and homology groups.
In order to provide motivation for homology theory, the first part of the course unit discusses
the classification theorem of closed surfaces using a combinatorial method of proof via
simplicial surfaces. Roughly speaking the theorem states that every closed surface can be
built up out of the sphere, the torus and the projective plane. Non-homeomorphic surfaces
are distinguished using two key topological invariants: the Euler characteristic and
orientability.
The second part then generalizes the idea of a simplicial surface to the idea of a simplicial
complex. This enables us to introduce a wide class of topological spaces called polyhedra which can be represented by a simplicial complex. This representation can then be used to define the homology groups of the polyhedron. The Euler characteristic and orientability type of a surface are determined by the homology groups. These powerful invariants have many attractive applications.
The supplementary reading introduces higher homotopy groups, categories and functions
and the axiomatic approach to homology theory.

Students are not permitted to take MATH41071 and MATH61071 for credit in an undergraduate programme and then a postgraduate programme.

The unit aims to:
develop further the concepts introduced in the Level 3 introductory course on Topology, emphasizing topics coming from Algebraic Topology. The course will introduce students to the basic concepts of homotopy and homology theory and explain the need for different algebraic invariants of topological spaces focussing on the classification theorem for closed surfaces

On successful completion of this course unit students will be able to:

• recognize when a collection of triangles in a Euclidean space forms a simplicial surface
   
• identify the underlying space of a simplicial surface by reducing a representing symbol to standard form
   
• state the basic definitions (including homotopy, homology groups, simplicial complexes) and main theorems from the course, and apply them in typical examples
   
• recognize whether two topological spaces are homotopic in simple cases
   
• calculate the homology groups of naturally occurring topological spaces
   
• recall proofs of basic properties of homology groups and apply these properties as topological invariants in examples
   
• determine the Euler characteristic and orientability of a simplicial surface from its homology groups
   
• calculate the homology groups of standard spaces using a variety of methods
   

Syllabus:
• Topological surfaces: definition and basic examples; the connected sum of two
surfaces; the classification theorem for compact surfaces; handles and cross-caps. [3]
• Simplicial surfaces: definition, triangulation of a topological surface and the statement
of the triangulation theorem for compact surfaces; representing the underlying space of
a simplicial surface by a symbol; equivalent symbols and the statement and proof of the
classification theorem for surface symbols; geometrical interpretation of the
classification theorem. [3]
• Topological invariants of surfaces: definition of the Euler characteristic and
orientability of a simplicial surface; statement of topological invariance; using these
invariants to recognize the underlying space of a simplicial surface. [2]
• Homotopy and homotopy equivalence: definition and simple examples, deformation retractions. [2]
• Simplicial complexes: definitions and examples; the underlying space,
polyhedral. [3]
• Simplicial homology groups: definitions and examples. [3]
• Simplicial approximation and topological invariance: statement of the Simplicial
Approximation Theorem and how this may be used to prove the topological invariance of
homology groups. [3]
• Applications: Brouwer Fixed Point Theorem, Borsuk-Ulam Theorem, Lefschetz Fixed
Point Theorem. [2]
 

22 hours of lectures plus 11 hours of tutorials. 
Feedback is available at the weekly tutorials as well as the mid-term coursework. 
 

Mid-semester coursework: weighting 20%,

End of semester examination: three hours weighting 80%.

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.
 

The following books contains most of the material in the course and much more.

• M.A. Armstrong, Basic Topology, Springer 1997 (classification of surfaces, simplicial complexes, homology)
• A. Hatcher, Algebraic Topology. (free download) (simplicial complexes, delta-complexes, homology, higher homotopy groups, categories and functors)
• W.S. Massey, Algebraic Topology: an Introduction, Springer 1990 (classification of surfaces)