module Homotopy.Space.Sphere where

# The -1 and 0 Spheres🔗

In classical topology, the *topological space*
$S^n$
is typically defined as the subspace of
$\mathbb{R}^{n+1}$
consisting of all points at unit distance from the origin. We see from
this definition that the
$0$-sphere
is the discrete two point space
$\{-1, 1\} \subset \mathbb{R}$,
and that the
$-1$
sphere is the empty subspace
$\varnothing \subset \{0\}$.
We will recycle existing types and define:

S⁻¹ : Type S⁻¹ = ⊥ S⁰ : Type S⁰ = Bool

We note that `S⁰`

may be identified with
`Susp S⁻¹`

. Since the pattern matching checker can prove that
`merid x i`

is impossible when `x : ⊥`

, the case
for this constructor does not need to be written, this makes the proof
look rather tautologous.

open is-iso SuspS⁻¹≃S⁰ : Susp S⁻¹ ≡ S⁰ SuspS⁻¹≃S⁰ = ua (SuspS⁻¹→S⁰ , is-iso→is-equiv iso-pf) where SuspS⁻¹→S⁰ : Susp S⁻¹ → S⁰ SuspS⁻¹→S⁰ N = true SuspS⁻¹→S⁰ S = false S⁰→SuspS⁻¹ : S⁰ → Susp S⁻¹ S⁰→SuspS⁻¹ true = N S⁰→SuspS⁻¹ false = S iso-pf : is-iso SuspS⁻¹→S⁰ iso-pf .inv = S⁰→SuspS⁻¹ iso-pf .rinv false = refl iso-pf .rinv true = refl iso-pf .linv N = refl iso-pf .linv S = refl

# n-Spheres🔗

The spheres of higher dimension can be defined inductively: $S^{n + 1} = \Sigma S^n$, that is, suspending the $n$-sphere constructs the $n+1$-sphere.

The spheres are essentially indexed by the natural numbers, except that we want to start at $-1$ instead of $0$. To remind ourselves of this, we name our spheres with a superscript $^{n-1}$:

Sⁿ⁻¹ : Nat → Type Sⁿ⁻¹ zero = S⁻¹ Sⁿ⁻¹ (suc n) = Susp (Sⁿ⁻¹ n)

A slightly less trivial example of definitions lining up is the
verification that `Sⁿ⁻¹ 2`

is equivalent to our previous
definition of `S¹`

:

SuspS⁰≃S¹ : Sⁿ⁻¹ 2 ≡ S¹ SuspS⁰≃S¹ = ua (SuspS⁰→S¹ , is-iso→is-equiv iso-pf) where

In `Sⁿ⁻¹ 2`

, we have two point constructors joined by two
paths, while in `S¹`

we have a single point constructor and a
loop. Geometrically, we can picture a morphing `Sⁿ⁻¹ 2`

into
`S¹`

by squashing one of the meridians down to a point, thus
bringing `N`

and `S`

together. This intuition
leads to a definition:

SuspS⁰→S¹ : Sⁿ⁻¹ 2 → S¹ SuspS⁰→S¹ N = base SuspS⁰→S¹ S = base SuspS⁰→S¹ (merid N i) = loop i SuspS⁰→S¹ (merid S i) = base

In the other direction, we send `base`

to `N`

,
and then need to send `loop`

to some path `N ≡ N`

.
Since this map should define an equivalence, we make it such that loop
wraps around the space `Sⁿ 2`

by way of traversing both
meridians.

S¹→SuspS⁰ : S¹ → Sⁿ⁻¹ 2 S¹→SuspS⁰ base = N S¹→SuspS⁰ (loop i) = (merid N ∙ sym (merid S)) i

##
We then verify that these maps are inverse equivalences. One of the
steps involves a slightly tricky `hcomp`

, although this can
be avoided by working with transports instead of dependent paths, and
then by using lemmas on transport in pathspaces.

iso-pf : is-iso SuspS⁰→S¹ iso-pf .inv = S¹→SuspS⁰ iso-pf .rinv base = refl iso-pf .rinv (loop i) = ap (λ p → p i) (ap SuspS⁰→S¹ (merid N ∙ sym (merid S)) ≡⟨ ap-∙ SuspS⁰→S¹ (merid N) (sym (merid S))⟩≡ loop ∙ refl ≡⟨ ∙-idr loop ⟩≡ loop ∎) iso-pf .linv N = refl iso-pf .linv S = merid S iso-pf .linv (merid N i) j = hcomp (∂ i ∨ ∂ j) λ where k (k = i0) → merid N i k (i = i0) → N k (i = i1) → merid S (j ∨ ~ k) k (j = i0) → ∙-filler (merid N) (sym (merid S)) k i k (j = i1) → merid N i iso-pf .linv (merid S i) j = merid S (i ∧ j)