Counting laps around a circle

Mathematics · Topology · ages 23-24

Name ______________________ Date ____________

Two loops in the circle have the same base point and winding number 2. They are homotopic within the circle while keeping that base point fixed.
Circle one: True False
A continuous map H(t,s) into X deforms a loop g to a loop h. Which endpoint condition makes this a based homotopy at b?
- H(0,s)=H(1,s), allowing the common endpoint to move
- H(0,s)=H(1,s)=b for every s
- H(t,0)=H(t,1), imposing no endpoint condition
For t in [0,1], the path g(t)=(cos(pi t), sin(pi t)) is a loop based at (1,0).
Circle one: True False
For t in [0,1], g(t)=(cos(6 pi t), sin(6 pi t)) is a circle loop. What is its winding number, taking counterclockwise as positive?
Answer: ______________
Which nonconstant circle loop is null-homotopic within the circle while keeping its base point fixed?
- A nonconstant path followed by its exact reverse
- Two counterclockwise turns without a return traversal
- One clockwise turn without retracing the original route
- One counterclockwise turn without a clockwise traversal
On [0,1], let g(t)=(cos(2 pi t), sin(2 pi t)) and h(t)=g(t squared). Their different speeds prevent a based homotopy between them in the circle.
Circle one: True False
Two circle loops share a base point. The first has winding -4. What winding must the second have for their concatenation to have winding 1?
Answer: ______________
Why can a circle loop of winding 2 not contract to its base point while staying in the circle?
- Every intermediate loop must keep the original constant speed
- Homotopy preserves winding, and 2 differs from 0
- A constant loop cannot share the original loop's base point
Which loop gives the fundamental-group obstruction to a homeomorphism between a disc and an annulus?
- A loop circling the annulus's hole, with winding 1
- A constant disc loop, which remains at its chosen base point
- A disc loop run faster, with its route and base point unchanged
Let A be the annulus with 1<=||x||<=2. Which deformation retracts A onto its inner circle while remaining in A and fixing that circle throughout?
- Move every point radially to the origin, including all inner-circle points
- Rotate every point through a full turn, keeping its original radius unchanged
- Move points radially to radius 1, fixing all points already at radius 1

Answer key

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

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True · Circle loops at a common base point with equal winding numbers belong to the same based homotopy class.
H(0,s)=H(1,s)=b for every s · Both endpoints must stay at the chosen base point b for every deformation time s.
False · The path starts at (1,0) and ends at (-1,0). Its endpoints differ, so it is not a loop.
3 · The lifted angle increases from 0 to 6 pi. Its change divided by 2 pi is 3.
A nonconstant path followed by its exact reverse · Exact retracing cancels the angle change. The resulting loop has winding 0, so it is null-homotopic in the circle.
False · The continuous parameter change rho(t)=t squared fixes 0 and 1. The timing homotopy connects g to h while fixing their base point.
5 · Write -4+n=1 for the missing winding. Then n=1-(-4)=5.
Homotopy preserves winding, and 2 differs from 0 · Homotopy in the circle preserves winding. A loop of winding 2 cannot become a constant loop of winding 0.
A loop circling the annulus's hole, with winding 1 · The annulus loop represents a nontrivial class. Every disc loop contracts. A homeomorphism would give isomorphic groups, but the integer group and trivial group differ.
Move points radially to radius 1, fixing all points already at radius 1 · The radius (1-s)r+s stays in [1,2], ends at 1, and remains 1 throughout when r=1.