Learn with Amar
Teaching video
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Graduate chapters and video availability01 · Read and understand
What you will learn
- Represent the fundamental group of a circle by integer winding.
- Justify the conclusion "π₁(S¹)≅Z, with concatenation corresponding to addition" using the stated assumptions.
Before you start
Continuous paths, homotopy, and the circle covering R→S¹.
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
Compare loops γₘ(t)=exp(2πimt) and γₙ(t)=exp(2πint), based at one.
Why this math matters
Represent the fundamental group of a circle by integer winding. 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:
- Homotopies keep the basepoint fixed.
- Paths remain on the circle.
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.
Measure loop winding through a covering coordinate
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Compare loops γₘ(t)=exp(2πimt) and γₙ(t)=exp(2πint), based at one.
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
A lift starting at zero ends at m or n
The real covering coordinate records accumulated turns without reducing modulo one.
Work through the mathematics
A based homotopy preserves the integer endpoint of the lift
The endpoint cannot move continuously between distinct integers.
Check the conclusion
π₁(S¹)≅Z, with concatenation corresponding to addition
Loops with equal winding are based-homotopic, and each integer is represented by γₘ.
The result
π₁(S¹)≅Z, with concatenation corresponding to addition
Loops with equal winding are based-homotopic, and each integer is represented by γₘ.
Common mistakes to catch
- A loop may contract in the disk even when it cannot contract in its boundary circle.
- Endpoint agreement alone does not imply based homotopy in every space.
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
What winding is obtained by following γ₂ then γ₋₁?
Show a hint
Add the signed turns.
Reveal answer and explanation
One
Concatenation adds lift endpoint increments.
Practice 2
Is γ₁ contractible inside S¹?
Show a hint
Compare its winding with a constant loop.
Reveal answer and explanation
No
A constant loop has winding zero, while γ₁ has winding one.
Take the idea with you
Use winding to describe why a phase loop cannot be unwound without leaving its allowed state space.
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: Count sheets of a circle covering
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