For the qualifying exam in topology and/or differential geometry the candidate is to prepare a syllabus by selecting topics from the list below. The total amount of material on the syllabus should be roughly equal to that covered in a standard one semester graduate course. Once you have made your selections discuss them with the professor who will examine you.
Basic topological notions: Path connectivity, connectivity, product topology, quotient topology.
The fundamental group, computation of the fundamental group, van Kampen's theorem, covering spaces.
Homology: Singular chains, chain complexes, homotopy invariance, relationship between the first homology and the fundamental group, relative homology, the long exact sequence of relative homology, the Mayer-Vietoris sequence, applications to computing the homology of surfaces, projective spaces, etc.
Topological manifolds, differentiable manifolds.
References: Harper and Greenberg, Algebraic Topology, a First Course, parts I and II
The orthogonal group in 2 and 3 dimensions, the Serret-Frenet frame of a space curve; the Gauss map and the Weingarten equation for a surface in Euclidean 3-space, the Gauss curvature equation and the Codazzi-Mainardi equation for a surface in Euclidean 3-space; the surfaces in Euclidean 3-space of zero Gauss curvature; the fundamental existence and rigidity theorem for surfaces in Euclidean space; the Gauss-Bonnet formula for surfaces in Euclidean 3-space.
References: M. do Carmo, Differential Geometry of Curves and Surfaces
Riemannian metrics and connections; geodesics and the first and second variational formulas; completeness and the Hopf-Rinow theorem; the Riemann curvature tensor, sectional curvature, Ricci curvature, and scalar curvature; the theorems of Hadamard and Bonnet-Myers; the Jacobi equation; the geometry of submanifolds --- the second fundamental form, equations of Gauss, Ricci, and Codazzi; spaces of constant curvature.
References:
M. do Carmo, Riemannian Geometry
M. Spivak, A Comprehensive Introduction to Differential Geometry
S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces
S. Sternberg, Lectures on Differential Geometry, 2nd ed.