## Course Syllabus

MA 5102 Basic Algebraic Topology

### Course Text

M. J. Greenberg and J. R. Harper, Algebraic Topology, Benjamin, 1981.

W. Fulton, Algebraic topology: A First Course, Springer-Verlag, 1995.

A. Hatcher, Algebraic Topology, Cambridge Univ. Press, Cambridge, 2002

W. Vick, Homology Theory: An Introduction to Algebraic Topology, 2nd Edition, Springer-Verlag, 1994.

### Course Content

Paths and homotopy, homotopy equivalence, contractibility, deformation retracts. Basic constructions: cones, mapping cones, mapping cylinders, suspension.

Fundamental groups. Examples (including the fundamental group of the circle) and applications (including Fundamental Theorem of Algebra, Brouwer Fixed Point Theorem and Borsuk-Ulam Theorem, both in dimension two). Van Kampen`s Theorem.

Covering spaces, lifting properties, deck transformations. universal coverings (existence theorem optional).

Singular Homology. Mayer-Vietoris Sequences. Long exact sequence of pairs and triples. Homotopy invariance and excision theorem (without proof).

Applications of homology: Jordan-Brouwer separation theorem, invariance of dimension, Hopf`s Theorem for commutative division algebras with identity, Borsuk-Ulam Theorem, Lefschetz Fixed Point Theorem.

Prerequisite: This course assumes knowledge of general topology and basic algebra.

### Grading

Quizzes, Midsem | : 60% |

Final | : 40% |

### Homeworks

Homeworks will be assigned reguarly .

Homeworks are meant to help you learn the material regularly and unless you do them by yourself, you will not be able to do well on the tests. Its also good to do as many practice problems as possible.

### Exams

Dates for the quizzes will be announced in the first week of class. No make up quizzes will be offered.

Make up exams will be offered only when you have a valid excuse.