The Fundamental Group and Loops on a Circle · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Counting laps around a circle

Mathematics · Topology · ages 23-24
Name ______________________   Date ____________

  1. Two loops on the circle winding twice each must be homotopic.

    Circle one:   True   False

  2. What is the fundamental group of the circle?

    • The trivial group
    • The integers
    • The real numbers
    • The circle itself

  3. Two loops on the circle wind twice each. A student says they must be homotopic. Is that right?

    Circle one:   True   False

  4. The loop g of t = (cos 6 pi t, sin 6 pi t) for t in 0 to 1 winds how many whole times?

    Answer: ______________

  5. Which loop on the circle is null-homotopic?

    • A loop that goes out and back the same way
    • A loop winding once
    • A loop winding twice
    • A loop winding minus once

  6. A constant loop stays at one point of the circle. A student says its winding number is 0. Is that right?

    Circle one:   True   False

  7. A loop winding twice is followed by one winding -5 times. What is the winding number of the combined loop?

    Answer: ______________

  8. Why can a twice-wound loop never deform to a constant loop?

    • Parametrizations must keep constant speed
    • Winding is invariant and 2 differs from 0
    • Concatenation is not defined for constant loops

  9. Which loop shows a disc and an annulus are genuinely different?

    • A loop around the hole of the annulus, which never shrinks
    • A constant loop sitting inside the disc
    • A loop traversed twice as fast

  10. What is the fundamental group of the figure eight?

    • The integers
    • The integers squared
    • The free group on two generators
    • The trivial group
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Counting laps around a circle W1-mt_zH50_L0nu4-s1

  1. True · On the circle, equal winding numbers mean homotopic.
  2. The integers · Loop classes correspond to winding numbers, and concatenation adds them, giving the integers.
  3. True · On the circle, homotopy classes of loops are exactly winding numbers.
  4. 3 · Angle 6 pi holds three copies of 2 pi.
  5. A loop that goes out and back the same way · Out-and-back sweeps zero net angle, so a loop that goes out and back the same way is null-homotopic.
  6. True · Zero angle swept means winding number 0, the identity class.
  7. -3 · Concatenation adds winding numbers: 2 + (-5) = -3.
  8. Winding is invariant and 2 differs from 0 · Deformation preserves the count, and 2 is not 0.
  9. A loop around the hole of the annulus, which never shrinks · The hole loop carries nonzero winding no disc loop can match.
  10. The free group on two generators · Two independent holes give two free generators with no relations: the free group on two generators.
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