**MTH 507: Introduction to Algebraic Topology (4)**

**Pre-requisites**: *MTH 301 Groups and Rings, MTH 304 Metric Spaces and Topology*

*Suggested Books*:

This is a first course in algebraic topology. The subject revolves around finding and computing invariants associated with topological spaces. The first such invariant is the fundamental group of a pointed topological space which we'll study in detail along with the classification of covering spaces using fundamental group actions.

*Suggested Books*:

*The Fundamental Group:* Homotopy, Fundamental Group, Introduction to Covering Spaces, The Fundamental Group of the circle **S ^{1}**, Retractions and fixed points, Application to the Fundamental Theorem of Algebra, The Borsuk-Ulam Theorem, Homotopy Equivalence and Deformation Retractions, Fundamental group of a product of spaces, and Fundamental group the torus

**T**=

^{2}**S**×

^{1}**S**, n-sphere

^{1}**S**, and the real projective n-space

^{n}**RP**.

^{n} *Van Kampen’s Theorem:* Free Products of Groups, The Van Kampen Theorem, Fundamental Group of a Wedge of Circles, Definition and construction of Cell Complexes, Application to Van Kampen Theorem to Cell Complexes, Statement of the Classification Theorem for Surfaces, and Fundamental groups of the closed orientable and non-orientable surfaces of genus g.

*Covering Spaces:* Universal Cover and its existence, Unique Lifting Property, Galois Correspondence of covering spaces and their Fundamental Groups, Representing Covering Spaces by Permutations – Deck Transformations, Group Actions, Covering Space Actions, Normal or Regular Covering Spaces, and Application of Covering Spaces to Cayley Complexes.

*Suggested Books*:

- J. R. Munkres, Topology (2nd Edition), Pearson Publishing Inc, 2000
- Hatcher, Algebraic Topology, Cambridge University Press, 2002
- M. A. Amstrong, Basic Topology, Springer International Edition, 2004
- W. S. Massey, Algebraic Topology: An Introduction , Springer, 1977
- J. J. Rotman, An Introduction to Algebraic Topology, Springer, 1988
- M. J. Greenberg and J. R. Harper, Algebraic Topology: A First Course, Perseus Books Publishing, 1981
- E. H. Spanier, Algebraic Topology, Springer, 1994

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