Mathematics / advanced
Homology, cohomology, and topological invariants
Use chains, exact sequences, products, and duality to calculate topological information beyond the fundamental group.
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Inside this module
Module 01 advanced
Homology, cohomology, and topological invariants
Use chains, exact sequences, products, and duality to calculate topological information beyond the fundamental group.
- 01
Simplicial chains and boundary operators
Build oriented chain groups, verify boundary cancellation, and distinguish cycles from the boundaries that make them trivial.
35 min - 02
Singular homology and homotopy invariance
Replace triangulation choices by continuous simplex maps and use chain homotopies to prove deformation invariance.
36 min - 03
Relative homology and long exact sequences
Quotient chains in a subspace and track how relative cycles connect to ordinary boundaries one degree lower.
38 min - 04
Excision and Mayer–Vietoris
Localize relative homology under valid excision hypotheses and compute a space from an open cover with its overlap maps.
39 min - 05
Cellular homology and computations
Use skeleton-relative generators and attaching-map degrees to compute homology while preserving integral torsion.
38 min - 06
Cochains and the universal coefficient theorem
Dualize a chain complex correctly and retain the Ext term and noncanonical splitting in cohomology calculations.
40 min - 07
Cup products and cohomology rings
Add multiplicative structure to cohomology, track graded signs, and distinguish spaces with identical additive groups.
40 min - 08
Künneth formula and tensor products
Compute product homology with degree bookkeeping, tensor terms, Tor corrections, and noncanonical splitting.
41 min - 09
Orientations and Poincaré duality
Use local orientations and a fundamental class to relate complementary dimensions, with boundary, compactness, and torsion caveats.
41 min - 10
De Rham cohomology and the integration pairing
Relate closed differential forms to real singular cohomology through smooth-chain integration while retaining torsion and global-exactness limits.
40 min