**MTH ****516: Topology II**** ****(4)**

**Pre-requisites (Desirable)**: MTH 507 or MTH 605 and MTH 302 or MTH 601

*Learning Objectives*:

This is an advanced course in Topology.

*Course Contents*:

**Simplicial Homology**: Simplicial Complexes, Barycentric Subdivision, and Simplicial Homology with examples

**Singular and Cellular Homology**: Definition with examples, Homotopy Invariance, Exact Sequence of Relative Homology, Excision, Mayer-Vietoris Sequence, Degree of Maps, and Cellular Homology, Jordan-Brouwer Separation Theorem, Invariance of domain and dimension, Borsuk-Ulam Theorem, Lefschetz-Hopf Fixed Point Theorem, Axioms for homology, Fundamental group and homology, and Simplicial Approximation Theorem

**Cohomology**: Universal Coefficient Theorem, Künneth Formula, Cup Product and the Cohomology Ring, Cap Product, Orientations on Manifolds, and Poincaré Duality

**Higher Homotopy Groups**: Definition with examples, Aspherical Spaces, Relative Homotopy Groups, Long Exact Sequence of a triple, n-connected spaces, and Whitehead's Theorem

*Suggested Books*:

- A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.
- E. H. Spanier, Algebraic Topology, Springer, 1994.
- J. R. Munkres, Elements of Algebraic Topology, Westview Press, 1996.
- J. J. Rotman, An Introduction to Algebraic Topology, Springer, 1988.
- M. J. Greenberg & J. R. Harper, Algebraic Topology: A First Course, Perseus Books Publishing, 1981.
- W. S. Massey, A Basic Course in Algebraic Topology, Springer International Edition, 2007.
- G. Bredon, Topology and Geometry, Springer International Edition.

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