Mathematics / advanced
Homotopy, coverings, and fundamental groups
Extract global information from paths and deformations through covering spaces, lifting, and higher homotopy structure.
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Inside this module
Module 01 advanced
Homotopy, coverings, and fundamental groups
Extract global information from paths and deformations through covering spaces, lifting, and higher homotopy structure.
- 01
Homotopies and homotopy equivalence
Distinguish continuous deformation of maps from homeomorphism and verify a deformation retraction explicitly.
21 min - 02
Fundamental groups and based loops
Organize endpoint-fixed loop deformations into a group and track winding, induced maps, and changes of basepoint.
22 min - 03
Van Kampen's theorem and presentations
Compute fundamental groups by gluing open pieces and translate cell attachments into generators and normal relations.
23 min - 04
Covering spaces and path lifting
Use evenly covered neighborhoods to unwrap paths uniquely and test whether a general based map admits a lift.
23 min - 05
Universal covers and deck transformations
State the existence hypotheses for a simply connected cover and identify its symmetries over the base.
24 min - 06
Classification of connected coverings
Translate pointed covers into subgroups, unpointed covers into conjugacy classes, and sheet counts into subgroup indices.
25 min - 07
Higher homotopy groups and sphere maps
Replace loops by based spheres, explain commutativity above dimension one, and separate higher homotopy from fundamental-group data.
24 min - 08
Fibrations and homotopy lifting
Generalize covering-space lifting to families whose fibers need not be discrete, while separating existence from uniqueness.
24 min - 09
Long exact homotopy sequences
Read the fibration sequence degree by degree and handle its low-degree component terms as pointed sets.
26 min - 10
CW complexes and cellular approximation
Build spaces from attaching maps and simplify homotopy problems skeleton by skeleton without discarding their gluing data.
25 min