Course Identification

Quotient Spaces, Cell Complexes, simplicial complexes
Homotopy,  Homotopy Equivalence, Deformation Retractions,
Classification of Surfaces
Fundamental Group, Covering Spaces,
Simplicial homology, .Singular homology, Cellular Homology, Relative Homology, long exact sequence of a pair, excision, Mayer–Vietoris, geometric meaning via pseudomanifolds, Brouwer fixed-point theorem, euler characteristic, invariance of dimension, Cohomology, Cup product, Universal coefficient theorem. 

Time permitting, we will get through as much of this as we can.

By the end of the course, students have:

• Conceptual intuition:

  • When you walk down the street and see the same bus twice, you will consider the possibility that you are not in the universal cover.

  • You will be comfortable with the idea of homotopy type, recognizing when spaces are quasi “the same” for topological purposes.

  • You will see and understand cycles, both geometrically and algebraically.

• Core skills:

  • Compute fundamental groups of standard spaces (spheres, tori, surfaces, CW complexes).

  • Use covering space theory to classify covers and connect them with subgroups of the fundamental group.

  • Calculate homology and cohomology groups of key examples (spheres, projective spaces, surfaces) using tools such as exact sequences, excision, Mayer–Vietoris, and cellular homology.

  • Apply invariants (Euler characteristic, homology, cohomology ring) to distinguish spaces and solve problems.

• Applications & broader vision:

  • Understand the classification of compact surfaces.

  • Appreciate major results (Brouwer fixed-point theorem, invariance of dimension) as consequences of algebraic topology.

  • Develop geometric insight into abstract algebraic structures, bridging computation and visualization.

Algebraic Topology, by Allen Hatcher
https://pi.math.cornell.edu/~hatcher/AT/AT.pdf
(covering spaces, fundamental group, homology, cohomology...)

Topology, (2nd edition) by James R. Munkries 
(contains fundamental group, covering spaces, and good reference for point set topology which we will refer to but not cover)