Cohomology Calculations

Algebraic Topology · Axiom Academy

EXAMPLE Cohomology Calculations Computing cohomology groups for fundamental topological spaces Excellent work! You've completed cohomology calculations for four fundamental spaces. Here's what we learned: Universal Coefficient Theorem: Provides a systematic way to compute cohomology from homology, with exact sequences Künneth Formula: For product spaces like the torus, cohomology decomposes as Torsion Detection: The projective plane RP^2 exhibits Z/2Z torsion in H^1, distinguishing it from simply-connected spaces Pattern Recognition: Spheres have cohomology concentrated in dimensions 0 and n, while products like T^2 show richer intermediate-dimensional structure These calculations form the foundation for understanding more complex spaces through cellular decompositions and spectral sequences!

This is the written version of the interactive lesson above. See the full Algebraic Topology course.