Ph.D. Mathematics Preliminary Exam Syllabi

Syllabus for the Preliminary Examination in Topology

It is assumed that students have a working knowledge of the equivalent of a one semester course in general topology (for example, see the appended syllabus for the undergraduate course M367K). For the semester in differential topology, it will also be assumed that students know the basic material from an undergraduate linear algebra course.
 
Algebraic Topology

1. Manifolds:     Identification (quotient) spaces and identification (quotient) maps; topological n-manifolds, including surfaces, Sⁿ, RPⁿ, CPⁿ, and lens spaces.

2. Triangulated manifolds:     Representation of triangulated, closed 2-manifolds as connected sums of tori or projective planes.

3. Fundamental group and covering spaces: Fundamental group, functoriality, retract, deformation retract; Van Kampen's Theorem, classification of surfaces by abelianizing the fundamental group, covering spaces, path lifting, homotopy lifting, uniqueness of lifts, general lifting theorem for maps, covering transformations, regular covers, correspondence between subgroups of the fundamental group and covering spaces, computing the fundamental group of the circle, RPⁿ, lens spaces via covering spaces.

4. Simplicial homology:     Homology groups, functoriality, topological invariance, Mayer-Vietoris sequence; applications, including Euler characteristic, classification of closed triangulated surfaces via homology and via Euler characteristic and orientability; degree of a map between oriented manifolds, Lefschetz number, Brouwer Fixed Point Theorem.
 
 References:
        Armstrong, Basic Topology, Springer, 1983  (principal text).
        Greenberg, Lectures on Algebraic Topology, W.A. Benjamin, 1967.
        Massey, Algebraic Topology, an Introduction, 4th corrected printing, Springer, 1977.
        Munkres, Elements of Algebraic Topology, Addison-Wesley, 1984.
 
Differential Topology

1.  Smooth mappings:     Inverse Function Theorem, Local Submersion Theorem (Implicit Function Theorem).

2. Differentiable manifolds:     Differentiable manifolds and submanifolds; examples, including  surfaces, Sⁿ, RPⁿ, CPⁿ  and lens spaces; tangent bundles; Sard's Theorem and its applications; differentiable transversality; orientation.

3. Vector fields and differential forms:     Integrating vector fields; degree of a map, Brouwer Fixed Point Theorem, No Retraction Theorem, Poincare-Hopf Theorem; differential forms, Stokes Theorem.
 
References:
        Guillemin  Pollack, Differential Topology, Prentice-Hall, 1974 (basic reference).
        Hirsch, Differential Topology, Springer, 1976.
        Milnor, Topology from the Differentiable Viewpoint, University of Virginia Press, 1965.
        Spivak, Calculus on Manifolds, Benjamin, 1965 (differentiation, Inverse Function  Theorem, Stokes Theorem).

For the examples indicated we refer to the books of Greenberg, Hirsch and Munkres.
 

Syllabus for M367K -- Topology I

Cardinality:     1-1 correspondence, countability, and uncountability.

Definitions of topological space:     Basis, sub-basis, metric space.

Countability properties:     Dense sets, countable basis, local basis.

Separation properties:     Hausdorff, regular, normal.

Covering properties:     Compact, countably compact, Lindelof.

Continuity and homeomorphisms:     Properties preserved by continuous functions, Urysohn's Lemma, Tietze Extension Theorem.

Connectedness:     Definition, examples, invariance under continuous functions.
 
Reference:     Munkres, Topology: a First Course, Prentice-Hall, 1975.