TOPOLOGY
Qualifying Examination Syllabus

The qualifying examination in Topology covers topics from general (point-set) topology and algebraic topology. Specifically:

  1. General Topology.
    1. Definition and examples of topological space, basis, sub-basis;
    2. Closed sets, closure, limit points;
    3. Compactness, Tychonoff theorem;
    4. Connectivity, path connectivity;
    5. Relative topology, product spaces, quotient spaces;
    6. Continuity, homeomorphisms;
    7. Separation axioms, normal spaces, regular spaces;
    8. Metric spaces, convergence;
    9. Urysohn's lemma, Tietze theorem, Lebesgue lemma.
  2. Algebraic Topology.
    1. Compact, connected surfaces, classification in terms of Euler characteristic and genus;
    2. Homotopy, retractions, deformation retractions;
    3. Definition and computation of the fundamental group of graphs, surfaces and other familiar spaces, Seifert-Van Kampen Theorem;
    4. Homotopy equivalence, simple connectivity, contractibility;
    5. Covering spaces, lifting theorems, quotient spaces of properly discontinuous group actions, classification of covering spaces;
    6. Applications of homotopy such as Brouwer fixed point theorem and Borsuk-Ulam theorem (dimension two);
    7. Definition of higher homotopy groups, suspension and loop spaces;
    8. Definition of singular homology groups, exact sequence of a pair, homotopy invariance, excision property, Mayer-Vietoris sequence;
    9. Eilenberg-Steenrod Axioms;
    10. Computation of homology groups for graphs, compact surfaces and other familiar spaces;
    11. Applications of homology such as the Jordan-Brouwer separation theorem and Brouwer fixed point theorem (dimensions greater than 2), existence and non-existence of non-vanishing vector fields on spheres.

    References

    • M.C. Gemigiani, Elementary Topology, Addison Wesley.
    • J. Munkres, Topology, A First Course, Prentice Hall, Inc.
    • W. Massey, A Basic Course in Algebraic Topology, Springer-Verlag, Chapters 1-8.
    • Greenberg and J. Harper, Algebraic Topology: A First Course Benjamin Press, Parts I and II.
    • J. Vick, Homology Theory: An Introduction to Algebraic Topology, Springer Verlag, Chapters 1-4.