1.5 KiB
Comparison maps between Ω S¹ base and ℤ
The ℤ-bundle helix
We want to make a ℤ-bundle over S¹ by
'copying ℤ across the loop via sucℤPath'.
In Quest2.agda locate
helix : S¹ → Type
helix = {!!}
Try to imitate the definition of doubleCover to define the bunlde helix.
You should compare your definition to ours in Quest2Solutions.agda.
Note that we have called this helix, since the picture of this ℤ-bundle
looks like
Counting loops
Now we can do what was originally difficult - constructing an inverse map
(over all points).
Now we want to be able to count how many times a path base ≡ base loops around
S¹, which we can do now using helix and finding end points of 'lifted' paths.
For example the path loop should loop around once,
counted by looking at the end point of 'lifted' loop, starting at 0.
Hence try to define
spinCountBase : base ≡ base → helix base
spinCountBase = {!!}
Try computing a few values using C-c C-n,
you can try it on refl, loop, loop 3 times, loop (- 1) times and so on.
Generalising
The function spinCountBase
can actually be improved without any extra work to a function on all of S¹
spinCount : (x : S¹) → base ≡ x → helix x
spinCount = {!!}
We will show that this and a general version of loop_times are
inverses of each other over S¹, in particular obtaining an isomorphism
between base ≡ base and ℤ.