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Algebraic Topology
Explore paths and coverings; homology; cohomology and applications. Concepts and connections are available independently of video coverage.
Prerequisites and connections
Topological spaces
A topology specifies open sets closed under arbitrary unions and finite intersections.
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Paths and coverings
Homotopy
A homotopy continuously deforms one map into another.
Video explanations to be addedFundamental group
Based loops modulo endpoint-fixed homotopy form a group under concatenation.
Video explanations to be addedCovering spaces
A covering locally maps disjoint sheets homeomorphically onto an open neighborhood.
Video explanations to be addedHomotopy equivalence
Maps with inverse compositions homotopic to identities describe spaces with the same homotopy type.
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Homology
Simplicial complexes
Simplices fitted together along faces provide combinatorial models of spaces.
Video explanations to be addedChain complexes
Successive boundary maps compose to zero in a graded sequence of groups or modules.
Video explanations to be addedHomology groups
Cycles modulo boundaries measure holes through a chain complex.
Video explanations to be addedEuler characteristic
Alternating cell or homology-rank counts define a topological invariant in suitable finite settings.
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Cohomology and applications
Cohomology
Dual cochain constructions yield invariants and additional operations such as cup products.
Video explanations to be addedExact sequences
At each position, the incoming image equals the outgoing kernel.
Video explanations to be addedHigher homotopy groups
Based maps from spheres modulo homotopy measure higher-dimensional homotopy structure.
Video explanations to be addedPersistent homology
Homological features are tracked across a nested family of spaces or data scales.
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