Recurring questions
| Question | Page |
|---|---|
| Compute \(\pi_1\) of this space | Compute $\pi_1$ |
| Compute the homology of this space | Compute $H_*$ |
| What is this space, and what are its invariants | The standard spaces |
| Is this true for all spaces | Counterexamples |
The standard-spaces table lists \(\pi_1\) and \(H_*\) of the spaces from which wedges, products, and punctured surfaces are built.
Chapters
Point-set topology – constructions, connectedness, compactness, separation.
Fundamental group – homotopy and van Kampen.
Covering spaces – lifting and the classification.
CW complexes – cell structures and attaching maps.
Homology – cellular, Mayer-Vietoris, the long exact sequence.
Degree and fixed points – Brouwer, Lefschetz, Borsuk-Ulam.
Surfaces and manifolds – the classification.
Pages collecting material from several chapters:
Review sheet – the statements of the chapters’ theorems, on one page.
The standard spaces – the table, with \(\pi_1\) and \(H_*\).
Counterexamples – filed by the statement each refutes.
Supplementary pages: appendices, resources, and workshops.
Acknowledgements
Mike Usher created the initial PDF of past UGA qualifying exam questions organized by topic.
Syllabus
The University of Georgia study guide for the topology exam, revised May 2006:
The weight of topics on the exam should be about 1/3 general topology and 2/3 algebraic topology.
General topology
Topological spaces, continuous functions, product and quotient topology [Munk00]
Connectedness and compactness [Munk00]
Countability and separation axioms, Urysohn lemma, Tietze theorem [Munk00, except sec. 36]
Complete metric spaces and function spaces [Munk00 and 45]
Algebraic topology
Classification of surfaces [Mass19]
Fundamental group [Mass19]; [Hatc02]
van Kampen’s theorem [Mass19]; [Hatc02]
Classification of covering spaces [Mass19]; [Hatc02]
Homology
Simplicial, singular, cellular; computations and applications [Hatc02]; [Munk84]
Degree of a map \(S^n\to S^n\) [Hatc02]; [Munk84]
Euler characteristic [Hatc02]
Lefschetz fixed point theorem [Hatc02]; [Munk84]
References
[Mass19]
-
- Revisions and corrections Hatcher’s Algebraic Topology page
[Munk84]