Homotopy and coverings: constructions, obstructions and classification
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 240 minutes, as an unvalidated planning estimate. This collection contains 8 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Attempt these public mixed problems before revealing their guidance. Use first-α-then-β path multiplication and state every basepoint, endpoint condition and covering hypothesis.
Written arguments require human mathematical review. Saved work and exposed solutions are practice evidence, not secure independent examination results.
Work is untimed. Separate exact computations from general proofs and identify any further theorem dependency.
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0 of 8 tasks have recorded responses. No overall mastery score is assigned.
Construct a strong deformation retraction of the planar annulus 3≤||x||≤8 onto its radius-five circle. Explain why it induces an isomorphism on fundamental groups. If c and d are paths from x0 to x1, compare the two maps sending a loop α at x1 to cαc̄ and dαd̄ at x0. Derive the exact conjugating loop at x0 and explain why one cannot commute an arbitrary connecting path with α merely because a fundamental group is abelian. Include explicit identity and inverse homotopies for loop classes.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For p:S1→S1, p(z)=z^6, lift α(t)=exp(2πi(5t+t²)) starting at exp(2πi·2/6). Compute the lifted endpoint and decide closure. Explain why every endpoint-homotopic base loop has the same lifted endpoint from that start. Prove the necessary path and square lifting claims by local sheets. Contrast this with exp(2πit):(0,3)→S1, showing why that surjective local homeomorphism is not a covering.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For f(exp(2πit))=exp(2πi(4t+(1/7)sin(4πt))) and g(z)=conjugate(z)^3, find the degrees of f, g and g∘f. Decide whether each map extends continuously from the boundary to a circle-valued map on the disk. Prove the degree obstruction and its converse. Then give the no-retraction proof of the two-dimensional Brouwer fixed-point theorem, including why the boundary-ray construction is continuous for a hypothetical fixed-point-free disk self-map.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A finite connected graph has seven vertices and eleven edges. Choose a maximal tree and determine its free-group rank. After contracting that tree, label the generators a,b,c,d,e. Attach two disks along the words aba^(-1)b^(-1) and cd. Give a presentation and simplify it to an explicit free product. Prove why attaching a disk imposes a normal-closure relation rather than merely deleting one word. State the open-cover and intersection assumptions behind van Kampen, and explain how its subdivided-homotopy argument controls the kernel.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For the torus S1×S1, let p(z,w)=(z³,w²). A based circle map into the torus is f(exp(2πit))=(exp(2πi·6t),exp(2πi·5t)). Decide whether f lifts through p with starting point (1,1). Replace 5 by 4 and construct a lift. Derive the subgroup criterion in this example and explain the sufficiency proof for a general map from a path-connected locally path-connected source. State why local path connectedness appears in the continuity argument.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For the circle covering z↦z^8, label the fiber over 1 by j=0,...,7. Compute the permutation induced by a loop of winding 3, its cycle decomposition and its order. Determine all deck transformations and prove there are no others. Contrast a right monodromy action on the fiber with a deck transformation of the whole covering space. For the general comparison, assume B is path connected and locally path connected and p:E→B is a path-connected covering. Explain what changes when its subgroup H is nonnormal.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
State the hypotheses under which connected covering spaces are classified by subgroups. Give the based and unbased classifications of connected covers of the circle, including the universal cover and the number of sheets. Construct the topology on the universal path-class cover for a general admissible base and explain why semilocal simple connectedness makes its local neighborhoods sheets. Why do infinitely many small essential loops at a point obstruct this construction?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A connected space X strongly deformation retracts onto a bouquet of two circles and satisfies the local hypotheses for covering classification. Prove X cannot be homotopy equivalent to a torus. Can a connected simply connected cover of X have finitely many sheets? Explain using subgroup index, and explain why agreement of the abelianizations Z² would not settle the homotopy-equivalence question. Finally state two major topics not established by this fundamental-group bridge.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
Construct a strong deformation retraction of the planar annulus 3≤||x||≤8 onto its radius-five circle. Explain why it induces an isomorphism on fundamental groups. If c and d are paths from x0 to x1, compare the two maps sending a loop α at x1 to cαc̄ and dαd̄ at x0. Derive the exact conjugating loop at x0 and explain why one cannot commute an arbitrary connecting path with α merely because a fundamental group is abelian. Include explicit identity and inverse homotopies for loop classes.
For p:S1→S1, p(z)=z^6, lift α(t)=exp(2πi(5t+t²)) starting at exp(2πi·2/6). Compute the lifted endpoint and decide closure. Explain why every endpoint-homotopic base loop has the same lifted endpoint from that start. Prove the necessary path and square lifting claims by local sheets. Contrast this with exp(2πit):(0,3)→S1, showing why that surjective local homeomorphism is not a covering.
For f(exp(2πit))=exp(2πi(4t+(1/7)sin(4πt))) and g(z)=conjugate(z)^3, find the degrees of f, g and g∘f. Decide whether each map extends continuously from the boundary to a circle-valued map on the disk. Prove the degree obstruction and its converse. Then give the no-retraction proof of the two-dimensional Brouwer fixed-point theorem, including why the boundary-ray construction is continuous for a hypothetical fixed-point-free disk self-map.
A finite connected graph has seven vertices and eleven edges. Choose a maximal tree and determine its free-group rank. After contracting that tree, label the generators a,b,c,d,e. Attach two disks along the words aba^(-1)b^(-1) and cd. Give a presentation and simplify it to an explicit free product. Prove why attaching a disk imposes a normal-closure relation rather than merely deleting one word. State the open-cover and intersection assumptions behind van Kampen, and explain how its subdivided-homotopy argument controls the kernel.
For the torus S1×S1, let p(z,w)=(z³,w²). A based circle map into the torus is f(exp(2πit))=(exp(2πi·6t),exp(2πi·5t)). Decide whether f lifts through p with starting point (1,1). Replace 5 by 4 and construct a lift. Derive the subgroup criterion in this example and explain the sufficiency proof for a general map from a path-connected locally path-connected source. State why local path connectedness appears in the continuity argument.
For the circle covering z↦z^8, label the fiber over 1 by j=0,...,7. Compute the permutation induced by a loop of winding 3, its cycle decomposition and its order. Determine all deck transformations and prove there are no others. Contrast a right monodromy action on the fiber with a deck transformation of the whole covering space. For the general comparison, assume B is path connected and locally path connected and p:E→B is a path-connected covering. Explain what changes when its subgroup H is nonnormal.
State the hypotheses under which connected covering spaces are classified by subgroups. Give the based and unbased classifications of connected covers of the circle, including the universal cover and the number of sheets. Construct the topology on the universal path-class cover for a general admissible base and explain why semilocal simple connectedness makes its local neighborhoods sheets. Why do infinitely many small essential loops at a point obstruct this construction?
A connected space X strongly deformation retracts onto a bouquet of two circles and satisfies the local hypotheses for covering classification. Prove X cannot be homotopy equivalent to a torus. Can a connected simply connected cover of X have finitely many sheets? Explain using subgroup index, and explain why agreement of the abelianizations Z² would not settle the homotopy-equivalence question. Finally state two major topics not established by this fundamental-group bridge.
