Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Graduate chapters and video availability01 · Read and understand
What you will learn
- Relate a covering's degree to its action on fundamental groups.
- Justify the conclusion "p* sends n↦3n on π₁(S¹)≅Z" using the stated assumptions.
Before you start
Fundamental groups and complex unit-circle notation.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Analyze p:S¹→S¹ given by p(z)=z³.
Why this math matters
Relate a covering's degree to its action on fundamental groups. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The covering degree is positive three.
- Basepoints are chosen at one.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Count sheets of a circle covering
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Analyze p:S¹→S¹ given by p(z)=z³.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
p(e^(iθ))=e^(3iθ)
The map triples the angular coordinate.
Work through the mathematics
Each point has three preimages separated by 2π/3
Locally each short target arc lifts to three disjoint arcs.
Check the conclusion
p* sends n↦3n on π₁(S¹)≅Z
A single trip around the source winds three times around the target, giving subgroup 3Z of index three.
The result
p* sends n↦3n on π₁(S¹)≅Z
A single trip around the source winds three times around the target, giving subgroup 3Z of index three.
Common mistakes to catch
- A covering is locally a homeomorphism, not necessarily globally injective.
- A lifted path need not close merely because its projection closes.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Does a once-winding target loop lift to a closed loop starting at one?
Show a hint
Its lifted endpoint is e^(2πi/3).
Reveal answer and explanation
No
The lift ends at another point in the fiber.
Practice 2
Which target winding numbers lift to closed loops?
Show a hint
Require the lifted angular change to be a multiple of 2π.
Reveal answer and explanation
Multiples of three
Dividing the target winding by three must produce an integer.
Take the idea with you
Interpret phase ambiguity when a measurement records only the cube of a unit complex state.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Compute torus homology from a cell model
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