85 lines
3.2 KiB
Text
85 lines
3.2 KiB
Text
/-
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Copyright (c) 2015 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn
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homotopy groups of a pointed space
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-/
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import types.pointed .trunc_group .hott types.trunc
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open nat eq pointed trunc is_trunc algebra
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namespace eq
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definition homotopy_group [reducible] (n : ℕ) (A : Type*) : Type :=
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trunc 0 (Ω[n] A)
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notation `π[`:95 n:0 `] `:0 A:95 := homotopy_group n A
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definition pointed_homotopy_group [instance] [constructor] (n : ℕ) (A : Type*)
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: pointed (π[n] A) :=
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pointed.mk (tr rfln)
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definition group_homotopy_group [instance] [constructor] (n : ℕ) (A : Type*)
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: group (π[succ n] A) :=
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trunc_group concat inverse idp con.assoc idp_con con_idp con.left_inv
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definition comm_group_homotopy_group [constructor] (n : ℕ) (A : Type*)
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: comm_group (π[succ (succ n)] A) :=
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trunc_comm_group concat inverse idp con.assoc idp_con con_idp con.left_inv eckmann_hilton
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local attribute comm_group_homotopy_group [instance]
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definition Pointed_homotopy_group [constructor] (n : ℕ) (A : Type*) : Type* :=
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Pointed.mk (π[n] A)
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definition Group_homotopy_group [constructor] (n : ℕ) (A : Type*) : Group :=
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Group.mk (π[succ n] A) _
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definition CommGroup_homotopy_group [constructor] (n : ℕ) (A : Type*) : CommGroup :=
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CommGroup.mk (π[succ (succ n)] A) _
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definition fundamental_group [constructor] (A : Type*) : Group :=
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Group_homotopy_group zero A
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notation `πP[`:95 n:0 `] `:0 A:95 := Pointed_homotopy_group n A
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notation `πG[`:95 n:0 ` +1] `:0 A:95 := Group_homotopy_group n A
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notation `πaG[`:95 n:0 ` +2] `:0 A:95 := CommGroup_homotopy_group n A
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prefix `π₁`:95 := fundamental_group
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open equiv unit
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theorem trivial_homotopy_of_is_hset (A : Type*) [H : is_hset A] (n : ℕ) : πG[n+1] A = G0 :=
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begin
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apply trivial_group_of_is_contr,
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apply is_trunc_trunc_of_is_trunc,
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apply is_contr_loop_of_is_trunc,
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apply is_trunc_succ_succ_of_is_hset
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end
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definition homotopy_group_succ_out (A : Type*) (n : ℕ) : πG[ n +1] A = π₁ Ω[n] A := idp
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definition homotopy_group_succ_in (A : Type*) (n : ℕ) : πG[succ n +1] A = πG[n +1] Ω A :=
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begin
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fapply Group_eq,
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{ apply equiv_of_eq, exact ap (λ(X : Type*), trunc 0 X) (loop_space_succ_eq_in A (succ n))},
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{ exact abstract [irreducible] begin refine trunc.rec _, intro p, refine trunc.rec _, intro q,
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rewrite [▸*,-+tr_eq_cast_ap, +trunc_transport, ↑[group_homotopy_group, group.to_monoid,
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monoid.to_semigroup, semigroup.to_has_mul, trunc_mul], trunc_transport], apply ap tr,
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apply loop_space_succ_eq_in_concat end end},
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end
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definition homotopy_group_add (A : Type*) (n m : ℕ) : πG[n+m +1] A = πG[n +1] Ω[m] A :=
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begin
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revert A, induction m with m IH: intro A,
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{ reflexivity},
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{ esimp [Iterated_loop_space, nat.add], refine !homotopy_group_succ_in ⬝ _, refine !IH ⬝ _,
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exact ap (Group_homotopy_group n) !loop_space_succ_eq_in⁻¹}
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end
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theorem trivial_homotopy_of_is_hset_loop_space {A : Type*} {n : ℕ} (m : ℕ) (H : is_hset (Ω[n] A))
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: πG[m+n+1] A = G0 :=
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!homotopy_group_add ⬝ !trivial_homotopy_of_is_hset
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end eq
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