155 lines
4.7 KiB
Text
155 lines
4.7 KiB
Text
import homotopy.join homotopy.smash
|
||
|
||
open eq equiv trunc function bool join sphere sphere_index sphere.ops prod
|
||
open pointed sigma smash is_trunc
|
||
|
||
namespace spherical_fibrations
|
||
|
||
/- classifying type of spherical fibrations -/
|
||
definition BG (n : ℕ) : Type₁ :=
|
||
Σ(X : Type₀), ∥ X ≃ S n..-1 ∥
|
||
|
||
definition pointed_BG [instance] [constructor] (n : ℕ) : pointed (BG n) :=
|
||
pointed.mk ⟨ S n..-1 , tr erfl ⟩
|
||
|
||
definition pBG [constructor] (n : ℕ) : Type* := pointed.mk' (BG n)
|
||
|
||
definition G (n : ℕ) : Type₁ :=
|
||
pt = pt :> BG n
|
||
|
||
definition G_char (n : ℕ) : G n ≃ (S n..-1 ≃ S n..-1) :=
|
||
calc
|
||
G n ≃ Σ(p : S n..-1 = S n..-1), _ : sigma_eq_equiv
|
||
... ≃ (S n..-1 = S n..-1) : sigma_equiv_of_is_contr_right
|
||
... ≃ (S n..-1 ≃ S n..-1) : eq_equiv_equiv
|
||
|
||
definition mirror (n : ℕ) : S n..-1 → G n :=
|
||
begin
|
||
intro v, apply to_inv (G_char n),
|
||
exact sorry
|
||
end
|
||
|
||
/-
|
||
Can we give a fibration P : S n → Type, P base = F n = Ω(BF n) = (S. n ≃* S. n)
|
||
and total space sigma P ≃ G (n+1) = Ω(BG (n+1)) = (S n.+1 ≃ S .n+1)
|
||
|
||
Yes, let eval : BG (n+1) → S n be the evaluation map
|
||
-/
|
||
|
||
definition S_of_BG (n : ℕ) : Ω(pBG (n+1)) → S n :=
|
||
λ f, f..1 ▸ base
|
||
|
||
definition BG_succ (n : ℕ) : BG n → BG (n+1) :=
|
||
begin
|
||
intro X, cases X with X p,
|
||
apply sigma.mk (susp X), induction p with f, apply tr,
|
||
apply susp.equiv f
|
||
end
|
||
|
||
/- classifying type of pointed spherical fibrations -/
|
||
definition BF (n : ℕ) : Type₁ :=
|
||
Σ(X : Type*), ∥ X ≃* S* n ∥
|
||
|
||
definition pointed_BF [instance] [constructor] (n : ℕ) : pointed (BF n) :=
|
||
pointed.mk ⟨ S* n , tr pequiv.rfl ⟩
|
||
|
||
definition pBF [constructor] (n : ℕ) : Type* := pointed.mk' (BF n)
|
||
|
||
definition BF_succ (n : ℕ) : BF n → BF (n+1) :=
|
||
begin
|
||
intro X, cases X with X p,
|
||
apply sigma.mk (psusp X), induction p with f, apply tr,
|
||
apply susp.psusp_equiv f
|
||
end
|
||
|
||
definition BF_of_BG {n : ℕ} : BG n → BF n :=
|
||
begin
|
||
intro X, cases X with X p,
|
||
apply sigma.mk (pointed.MK (susp X) susp.north),
|
||
induction p with f, apply tr,
|
||
apply pequiv_of_equiv (susp.equiv f),
|
||
reflexivity
|
||
end
|
||
|
||
definition BG_of_BF {n : ℕ} : BF n → BG (n + 1) :=
|
||
begin
|
||
intro X, cases X with X hX,
|
||
apply sigma.mk (carrier X), induction hX with fX,
|
||
apply tr, exact fX
|
||
end
|
||
|
||
definition BG_mul {n m : ℕ} (X : BG n) (Y : BG m) : BG (n + m) :=
|
||
begin
|
||
cases X with X pX, cases Y with Y pY,
|
||
apply sigma.mk (join X Y),
|
||
induction pX with fX, induction pY with fY,
|
||
apply tr, rewrite add_sub_one,
|
||
exact (join.equiv_closed fX fY) ⬝e (join.spheres n..-1 m..-1)
|
||
end
|
||
|
||
definition BF_mul {n m : ℕ} (X : BF n) (Y : BF m) : BF (n + m) :=
|
||
begin
|
||
cases X with X hX, cases Y with Y hY,
|
||
apply sigma.mk (psmash X Y),
|
||
induction hX with fX, induction hY with fY, apply tr,
|
||
exact sorry -- needs smash.spheres : psmash (S. n) (S. m) ≃ S. (n + m)
|
||
end
|
||
|
||
definition BF_of_BG_mul (n m : ℕ) (X : BG n) (Y : BG m)
|
||
: BF_of_BG (BG_mul X Y) = BF_mul (BF_of_BG X) (BF_of_BG Y) :=
|
||
sorry
|
||
|
||
-- Thom spaces
|
||
namespace thom
|
||
variables {X : Type} {n : ℕ} (α : X → BF n)
|
||
|
||
-- the canonical section of an F-object
|
||
protected definition sec (x : X) : carrier (sigma.pr1 (α x)) :=
|
||
Point _
|
||
|
||
open pushout sigma
|
||
|
||
definition thom_space : Type :=
|
||
pushout (λx : X, ⟨x , thom.sec α x⟩) (const X unit.star)
|
||
end thom
|
||
|
||
/-
|
||
Things to do:
|
||
- Orientability and orientations
|
||
* Thom class u ∈ ~Hⁿ(Tξ)
|
||
* eventually prove Thom-Isomorphism (Rudyak IV.5.7)
|
||
- define BG∞ and BF∞ as colimits of BG n and BF n
|
||
- Ω(BF n) = ΩⁿSⁿ₁ + ΩⁿSⁿ₋₁ (self-maps of degree ±1)
|
||
- succ_BF n is (n - 2) connected (from Freudenthal)
|
||
- pfiber (BG_of_BF n) ≃* S. n
|
||
- π₁(BF n)=π₁(BG n)=ℤ/2ℤ
|
||
- double covers BSG and BSF
|
||
- O : BF n → BG 1 = Σ(A : Type), ∥ A = bool ∥
|
||
- BSG n = sigma O
|
||
- π₁(BSG n)=π₁(BSF n)=O
|
||
- BSO(n),
|
||
- find BF' n : Type₀ with BF' n ≃ BF n etc.
|
||
- canonical bundle γₙ : ℝP(n) → ℝP∞=BO(1) → Type₀
|
||
prove T(γₙ) = ℝP(n+1)
|
||
- BG∞ = BF∞ (in fact = BGL₁(S), the group of units of the sphere spectrum)
|
||
- clutching construction:
|
||
any f : S n → SG(n) gives S n.+1 → BSG(n) (mut.mut. for O(n),SO(n),etc.)
|
||
- all bundles on S 3 are trivial, incl. tangent bundle
|
||
- Adams' result on vector fields on spheres:
|
||
there are maximally ρ(n)-1 indep.sections of the tangent bundle of S (n-1)
|
||
where ρ(n) is the n'th Radon-Hurwitz number.→
|
||
-/
|
||
|
||
-- tangent bundle on S 2:
|
||
|
||
namespace two_sphere
|
||
|
||
definition tau : S 2 → BG 2 :=
|
||
begin
|
||
intro v, induction v with x, do 2 exact pt,
|
||
exact sorry
|
||
end
|
||
|
||
end two_sphere
|
||
|
||
end spherical_fibrations
|