Mathematics · unit version 1

Homotopy, fundamental groups and covering spaces

A proof-based undergraduate bridge from point-set topology to algebraic topology: explicit loop and lifting arguments, circle degree, gluing, graph and surface groups, and covering classification. Homology, cohomology and higher homotopy are further courses; written solutions require mathematical review.

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Readiness and preparation

Check your starting point

Try these short questions on paper, then compare your reasoning. They are optional self-checks, not a placement score or a navigation requirement. Use a targeted review when useful and return with your browser’s Back control.

What makes a map continuous in the open-set definition?

Lifts and homotopies must be jointly continuous, not merely pointwise formulas.

Compare reasoning and find a refresher

The inverse image of every open set in the target must be open in the domain. A formula defined on pieces also needs an appropriate pasting or quotient argument.

Review continuity

When does a rule on representatives define a map from a quotient?

Loop and path classes must not depend on the chosen representative.

Compare reasoning and find a refresher

The rule must take equivalent representatives to the same output. Continuity of a descended map then follows from the quotient topology when its composite with the quotient projection is continuous.

Review quotient maps

Why must H be normal to multiply its cosets as a quotient group?

Covering classification uses arbitrary subgroups, while regular coverings involve normal subgroups.

Compare reasoning and find a refresher

Normality makes multiplication independent of coset representatives. An arbitrary subgroup still has a coset set and an index, but that set need not inherit the quotient group multiplication.

Review normal subgroups and quotients

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Homotopy and coverings: constructions, obstructions and classification

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Lesson sequence

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Homotopies and fixed endpoints
Lesson details and objectives
Distinguish free homotopy from homotopy relative to a subspace.
- Distinguish free homotopy from homotopy relative to a subspace.
- Construct endpoint-preserving path deformations.
- Check continuity using pasting and product topology.
6 practice tasks.
Build the fundamental group
Lesson details and objectives
Prove concatenation descends to endpoint-homotopy classes.
- Prove concatenation descends to endpoint-homotopy classes.
- Construct associativity, identity and inverse homotopies.
- Distinguish equality of parametrized loops from equality of classes.
6 practice tasks.
Basepoints and induced homomorphisms
Lesson details and objectives
Construct the homomorphism induced by a based continuous map.
- Construct the homomorphism induced by a based continuous map.
- Track the conjugation caused by moving a basepoint.
- Prove based homotopy invariance and functoriality.
6 practice tasks.
Deformation retracts and homotopy equivalence
Lesson details and objectives
Verify a strong deformation retraction by formula.
- Verify a strong deformation retraction by formula.
- Derive fundamental-group invariance under homotopy equivalence.
- Distinguish a retraction from a deformation retraction.
6 practice tasks.
Covering maps and local sheets
Lesson details and objectives
Check the full evenly-covered-neighborhood definition.
- Check the full evenly-covered-neighborhood definition.
- Construct finite and infinite circle covers.
- Separate local homeomorphisms from covering maps.
6 practice tasks.
Lift paths one sheet at a time
Lesson details and objectives
Construct a lift using a finite subdivision of the interval.
- Construct a lift using a finite subdivision of the interval.
- Prove uniqueness from local sheets and connectedness.
- Calculate lifted endpoints in circle covers.
6 practice tasks.
Lift homotopies and detect injectivity
Lesson details and objectives
Construct a lifted square by a finite grid of sheet inverses.
- Construct a lifted square by a finite grid of sheet inverses.
- Show endpoint-homotopic paths have lifts with common endpoints.
- Prove a covering induces an injective fundamental-group map.
6 practice tasks.
Compute the fundamental group of the circle
Lesson details and objectives
Associate an integer displacement to every based circle loop.
- Associate an integer displacement to every based circle loop.
- Prove displacement classifies endpoint homotopy classes.
- Verify concatenation adds winding integers.
6 practice tasks.
Winding numbers and circle degree
Lesson details and objectives
Define degree through lifted arguments.
- Define degree through lifted arguments.
- Prove degree is invariant under homotopy and multiplicative under composition.
- Compute signed winding without assuming a globally defined angle.
6 practice tasks.
No retraction and the disk fixed-point theorem
Lesson details and objectives
Use induced groups to rule out a disk retraction.
- Use induced groups to rule out a disk retraction.
- Construct a boundary retraction from a hypothetical fixed-point-free map.
- Identify which higher-dimensional statements need new invariants.
6 practice tasks.
Van Kampen and gluing loops
Lesson details and objectives
State the open-cover and path-connected-intersection hypotheses.
- State the open-cover and path-connected-intersection hypotheses.
- Construct generators by subdividing loops and inserting connector paths.
- Explain kernel relations using a subdivided null homotopy.
6 practice tasks.
Graph loops and free groups
Lesson details and objectives
Reduce graph loops using a maximal tree.
- Reduce graph loops using a maximal tree.
- Construct the reduced-word tree covering a bouquet.
- Compute the rank E−V+1 for a finite connected graph.
6 practice tasks.
Cell attachments and surface groups
Lesson details and objectives
Derive the normal-closure relation imposed by attaching a disk.
- Derive the normal-closure relation of a two-cell attachment.
- Compute surface groups with explicit nontriviality checks.
- Distinguish fundamental-group calculations from classification of spaces.
6 practice tasks.
The covering lifting criterion
Lesson details and objectives
Prove necessity of the subgroup inclusion for a based lift.
- Prove the based subgroup-containment criterion for lifts.
- Use local path connectedness to establish continuity.
- Compute circle-map lift obstructions and based uniqueness.
6 practice tasks.
Monodromy and deck transformations
Lesson details and objectives
Construct the action of loop classes on a covering fiber.
- Construct the right monodromy action on a fiber.
- Derive right-coset coordinates and the deck normalizer quotient.
- Distinguish regular from nonregular connected covers.
6 practice tasks.
Classify connected covers
Lesson details and objectives
Construct path-class neighborhoods for a universal cover.
- Construct covering neighborhoods in the universal path-class space.
- Prove simple connectedness using closed lifts and injectivity.
- Distinguish subgroup classification, conjugacy and sheet index.
6 practice tasks.