Course Contents

Course Contents: 

1. Homotopy and fundamental group. Deformation Retract, Contractible spaces, No-retraction theorem, Brouwer fixed point theorem, Simply connected spaces, fundamental groups of spheres

Van Kampen theorem. Free groups, and Free product of groups. Fundamental groups of  Wedge of circles.

2. Statement of classification of compact surfaces and polygonal representation. Computation of fundamental groups of compact surfaces.

3. Covering spaces, Path lifting, homotopy lifting, general lifting. Examples of covering of

wedge of circles, circles. Universal cover and existence of covering, classification of covering

by subgroups of fundamental groups. Deck transformation, action of fundamental group

on Universal cover, normal covering. Invariance of Domain, Borsuk-Ulam Theorem.

4. Homology: Triangulation, Barycentric Subdivision, Simplicial Approximation; Simplicial ho-

mology: Cycles and Boundaries, Simplicial maps, 

5. CW complex and Homotopy Extension Property (If time permits)