Elementary point set
topology:
This material is from our undergraduate course. Although this material
is technically not part of the graduate syllabus it may be on the exam
because it is a prerequisite.
Topological Spaces and Continuous Functions
Connectedness and Compactness
Countability and Separation Axioms
Fundamental Group and Covering Spaces
Reference:
Munkres, Topology, first 4 Chapters and Chapters 9 and 13
Algebraic Topology
The graduate level Algebraic Topology material is basic Homology Theory.
Sinplicial Complexes, Simplicial Homology
Betti numbers, Euler characteristic
Singular Homology and Cohomology, with Coefficients
CW-complexes, cellular homology
Excision, Mayer-Vietoris exact sequences
Eilenberg-Steenrod Axioms for Homology and Cohomology
Tor, Ext, Universal Coefficient Theorems, Kunneth Theorem
Manifolds, orientation, Elementary Poincare Duality
References:
Hatcher, Algebraic Topology
Greenberg and Harper, Algebraic Topology: A First Course
Spanier, Algebraic Topology
Dold, Lectures on Algebraic Topology
Munkres, Elements of Algebraic Topology