INTRODUCTION TO ALGEBRAIC TOPOLOGY
Fundamental Group: Homotopy of maps, The fundamental group, Fundamental group of the circle, Covering spaces, Borsuk-Ulam and Ham-sandwich theorems, A lifting criterion, Seifert-van kampen theorem, Brouwer fixed point theorem and other applications.
Piecewise-Linear Topology: Polyhedra, PL maps, PL manifolds, Cell complexes, Subdivisions, Simplicial complexes, Simplicial maps, Triangulations, Derived subdivisions, Pseudomanifolds, Abstract simplicial complexes, isomorphism.
Simplicial Homology: Orientation of complexes, Chains, Cycles and boundaries, Homology groups, Euler-Poincare formula, Barycentric subdivision, Simplicial approximation, Induced homomorphism, Degree and Lefschetz number fixed-point theorem.
Books
- Croom, F. H., Basic Concepts of Algebraic Topology, Springer-Verlag (UTM), 1978.
- Kosniowski, C. A., First course in Algebraic Topology, Cambridge Univ. Press, 1980.
- Munkres, J. R., Elements of Algebraic Topology, The Benjamin Cummings Pub., Co., 1984.
- Rourke, C. P. and Sanderson, B. J., Introduction to Piecewise-Linear Topology, Springer-Verlag, 1982.