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Finish construction of the LES of homotopy groups without signs

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The maps on every level are just the functorial action of the homotopy groups (possibly composed by a cast), but there are no compositions with path inversion.
There are also some updates in various files after changes in the HoTT library.
This commit is contained in:
Floris van Doorn 2016-04-07 17:28:19 -04:00
parent cb40d9b8fc
commit 37fbe56b8c
7 changed files with 1259 additions and 260 deletions
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homotopy/LES.hlean Normal file
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import .LES_of_homotopy_groups
open eq pointed is_trunc trunc_index trunc group is_equiv equiv algebra prod fin fiber nat
succ_str chain_complex
/--------------
PART 3'
--------------/
namespace chain_complex'
section
universe variable u
parameters {X Y : pType.{u}} (f : X →* Y)
definition homotopy_groups2 [reducible] : +3ℕ → Type*
| (n, fin.mk 0 H) := Ω[n] Y
| (n, fin.mk 1 H) := Ω[n] X
| (n, fin.mk k H) := Ω[n] (pfiber f)
definition homotopy_groups2_add1 (n : ℕ) : Π(x : fin (succ 2)),
homotopy_groups2 (n+1, x) = Ω(homotopy_groups2 (n, x))
| (fin.mk 0 H) := by reflexivity
| (fin.mk 1 H) := by reflexivity
| (fin.mk 2 H) := by reflexivity
| (fin.mk (k+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
definition homotopy_groups_fun2 : Π(n : +3ℕ), homotopy_groups2 (S n) →* homotopy_groups2 n
| (n, fin.mk 0 H) := proof Ω→[n] f qed
| (n, fin.mk 1 H) := proof Ω→[n] (ppoint f) qed
| (n, fin.mk 2 H) :=
proof Ω→[n] (boundary_map f) ∘* pcast (loop_space_succ_eq_in Y n) qed
| (n, fin.mk (k+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
definition homotopy_groups_fun2_add1_0 (n : ℕ) (H : 0 < succ 2)
: homotopy_groups_fun2 (n+1, fin.mk 0 H) ~*
cast proof idp qed ap1 (homotopy_groups_fun2 (n, fin.mk 0 H)) :=
by reflexivity
definition homotopy_groups_fun2_add1_1 (n : ℕ) (H : 1 < succ 2)
: homotopy_groups_fun2 (n+1, fin.mk 1 H) ~*
cast proof idp qed ap1 (homotopy_groups_fun2 (n, fin.mk 1 H)) :=
by reflexivity
definition homotopy_groups_fun2_add1_2 (n : ℕ) (H : 2 < succ 2)
: homotopy_groups_fun2 (n+1, fin.mk 2 H) ~*
cast proof idp qed ap1 (homotopy_groups_fun2 (n, fin.mk 2 H)) :=
begin
esimp,
refine _ ⬝* !ap1_compose⁻¹*,
exact pwhisker_left _ !pcast_ap_loop_space,
end
exit
definition nat_of_str_3S [unfold 2] [reducible]
: Π(x : stratified +ℕ 2), nat_of_str x + 1 = nat_of_str (@S (stratified +ℕ 2) x)
| (n, fin.mk 0 H) := by reflexivity
| (n, fin.mk 1 H) := by reflexivity
| (n, fin.mk 2 H) := by reflexivity
| (n, fin.mk (k+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
definition homotopy_groups2_pequiv : Π(x : +3ℕ),
homotopy_groups (nat_of_str x) ≃* homotopy_groups2 x
| (0, (fin.mk 0 H)) := by reflexivity
| (0, (fin.mk 1 H)) := by reflexivity
| (0, (fin.mk 2 H)) := by reflexivity
| ((n+1), (fin.mk 0 H)) :=
begin
-- uncomment the next two lines to have prettier subgoals
-- esimp, replace (succ 5 * (n + 1) + 0) with (6*n+3+3),
-- rewrite [+homotopy_groups_add3, homotopy_groups2_add1],
apply loop_pequiv_loop,
rexact homotopy_groups2_pequiv (n, fin.mk 0 H)
end
| ((n+1), (fin.mk 1 H)) :=
begin
apply loop_pequiv_loop,
rexact homotopy_groups2_pequiv (n, fin.mk 1 H)
end
| ((n+1), (fin.mk 2 H)) :=
begin
apply loop_pequiv_loop,
rexact homotopy_groups2_pequiv (n, fin.mk 2 H)
end
| (n, (fin.mk (k+3) H)) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
-- | (n, x) := homotopy_groups2_pequiv' n x
/- all cases where n>0 are basically the same -/
definition homotopy_groups_fun2_phomotopy (x : +6ℕ) :
homotopy_groups2_pequiv x ∘* homotopy_groups_fun (nat_of_str x) ~*
(homotopy_groups_fun2 x ∘* homotopy_groups2_pequiv (S x))
∘* pcast (ap (homotopy_groups f) (nat_of_str_6S x)) :=
begin
cases x with n x, cases x with k H,
cases k with k, rotate 1, cases k with k, rotate 1, cases k with k, rotate 1,
cases k with k, rotate 1, cases k with k, rotate 1, cases k with k, rotate 2,
{ /-k=0-/
induction n with n IH,
{ refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹* ⬝* !comp_pid⁻¹*,
reflexivity},
{ refine _ ⬝* !comp_pid⁻¹*,
refine _ ⬝* pwhisker_right _ (!homotopy_groups_fun2_add1_0)⁻¹*,
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
exact IH ⬝* !comp_pid}},
{ /-k=1-/
induction n with n IH,
{ refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹* ⬝* !comp_pid⁻¹*,
reflexivity},
{ refine _ ⬝* !comp_pid⁻¹*,
refine _ ⬝* pwhisker_right _ (!homotopy_groups_fun2_add1_1)⁻¹*,
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
exact IH ⬝* !comp_pid}},
{ /-k=2-/
induction n with n IH,
{ refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹* ⬝* !comp_pid⁻¹*,
refine _ ⬝* !comp_pid⁻¹*,
reflexivity},
{ refine _ ⬝* !comp_pid⁻¹*,
refine _ ⬝* pwhisker_right _ (!homotopy_groups_fun2_add1_2)⁻¹*,
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
exact IH ⬝* !comp_pid}},
{ /-k=3-/
induction n with n IH,
{ refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹* ⬝* !comp_pid⁻¹*,
reflexivity},
{ refine _ ⬝* !comp_pid⁻¹*,
refine _ ⬝* pwhisker_right _ (!homotopy_groups_fun2_add1_3)⁻¹*,
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
exact IH ⬝* !comp_pid}},
{ /-k=4-/
induction n with n IH,
{ refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹* ⬝* !comp_pid⁻¹*,
reflexivity},
{ refine _ ⬝* !comp_pid⁻¹*,
refine _ ⬝* pwhisker_right _ (!homotopy_groups_fun2_add1_4)⁻¹*,
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
exact IH ⬝* !comp_pid}},
{ /-k=5-/
induction n with n IH,
{ refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹*,
refine !comp_pid⁻¹* ⬝* pconcat2 _ _,
{ exact !comp_pid⁻¹*},
{ refine cast (ap (λx, _ ~* loop_pequiv_loop x) !loopn_pequiv_loopn_rfl)⁻¹ _,
refine cast (ap (λx, _ ~* x) !loopn_pequiv_loopn_rfl)⁻¹ _, reflexivity}},
{ refine _ ⬝* !comp_pid⁻¹*,
refine _ ⬝* pwhisker_right _ (!homotopy_groups_fun2_add1_5)⁻¹*,
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
exact IH ⬝* !comp_pid}},
{ /-k=k'+6-/ exfalso, apply lt_le_antisymm H, apply le_add_left}
end
definition type_LES_of_homotopy_groups2 [constructor] : type_chain_complex +6ℕ :=
transfer_type_chain_complex2
(type_LES_of_homotopy_groups f)
!fin_prod_nat_equiv_nat
nat_of_str_6S
@homotopy_groups_fun2
@homotopy_groups2_pequiv
begin
intro m x,
refine homotopy_groups_fun2_phomotopy m x ⬝ _,
apply ap (homotopy_groups_fun2 m), apply ap (homotopy_groups2_pequiv (S m)),
esimp, exact ap010 cast !ap_compose⁻¹ x
end
definition is_exact_type_LES_of_homotopy_groups2 : is_exact_t (type_LES_of_homotopy_groups2) :=
begin
intro n,
apply is_exact_at_transfer2,
apply is_exact_type_LES_of_homotopy_groups
end
definition LES_of_homotopy_groups2 [constructor] : chain_complex +6ℕ :=
trunc_chain_complex type_LES_of_homotopy_groups2
/--------------
PART 4'
--------------/
definition homotopy_groups3 [reducible] : +6ℕ → Set*
| (n, fin.mk 0 H) := π*[2*n] Y
| (n, fin.mk 1 H) := π*[2*n] X
| (n, fin.mk 2 H) := π*[2*n] (pfiber f)
| (n, fin.mk 3 H) := π*[2*n + 1] Y
| (n, fin.mk 4 H) := π*[2*n + 1] X
| (n, fin.mk k H) := π*[2*n + 1] (pfiber f)
definition homotopy_groups3eq2 [reducible]
: Π(n : +6ℕ), ptrunc 0 (homotopy_groups2 n) ≃* homotopy_groups3 n
| (n, fin.mk 0 H) := by reflexivity
| (n, fin.mk 1 H) := by reflexivity
| (n, fin.mk 2 H) := by reflexivity
| (n, fin.mk 3 H) := by reflexivity
| (n, fin.mk 4 H) := by reflexivity
| (n, fin.mk 5 H) := by reflexivity
| (n, fin.mk (k+6) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
definition homotopy_groups_fun3 : Π(n : +6ℕ), homotopy_groups3 (S n) →* homotopy_groups3 n
| (n, fin.mk 0 H) := proof π→*[2*n] f qed
| (n, fin.mk 1 H) := proof π→*[2*n] (ppoint f) qed
| (n, fin.mk 2 H) :=
proof π→*[2*n] (boundary_map f) ∘* pcast (ap (ptrunc 0) (loop_space_succ_eq_in Y (2*n))) qed
| (n, fin.mk 3 H) := proof π→*[2*n + 1] f ∘* tinverse qed
| (n, fin.mk 4 H) := proof π→*[2*n + 1] (ppoint f) ∘* tinverse qed
| (n, fin.mk 5 H) :=
proof (π→*[2*n + 1] (boundary_map f) ∘* tinverse)
∘* pcast (ap (ptrunc 0) (loop_space_succ_eq_in Y (2*n+1))) qed
| (n, fin.mk (k+6) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
definition homotopy_groups_fun3eq2 [reducible]
: Π(n : +6ℕ), homotopy_groups3eq2 n ∘* ptrunc_functor 0 (homotopy_groups_fun2 n) ~*
homotopy_groups_fun3 n ∘* homotopy_groups3eq2 (S n)
| (n, fin.mk 0 H) := by reflexivity
| (n, fin.mk 1 H) := by reflexivity
| (n, fin.mk 2 H) :=
begin
refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹*,
refine !ptrunc_functor_pcompose ⬝* _,
apply pwhisker_left, apply ptrunc_functor_pcast,
end
| (n, fin.mk 3 H) :=
begin
refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹*,
refine !ptrunc_functor_pcompose ⬝* _,
apply pwhisker_left, apply ptrunc_functor_pinverse
end
| (n, fin.mk 4 H) :=
begin
refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹*,
refine !ptrunc_functor_pcompose ⬝* _,
apply pwhisker_left, apply ptrunc_functor_pinverse
end
| (n, fin.mk 5 H) :=
begin
refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹*,
refine !ptrunc_functor_pcompose ⬝* _,
apply pconcat2,
{ refine !ptrunc_functor_pcompose ⬝* _,
apply pwhisker_left, apply ptrunc_functor_pinverse},
{ apply ptrunc_functor_pcast}
end
| (n, fin.mk (k+6) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
definition LES_of_homotopy_groups3 [constructor] : chain_complex +6ℕ :=
transfer_chain_complex
LES_of_homotopy_groups2
homotopy_groups_fun3
homotopy_groups3eq2
homotopy_groups_fun3eq2
definition is_exact_LES_of_homotopy_groups3 : is_exact (LES_of_homotopy_groups3) :=
begin
intro n,
apply is_exact_at_transfer,
apply is_exact_at_trunc,
apply is_exact_type_LES_of_homotopy_groups2
end
end
open is_trunc
universe variable u
variables {X Y : pType.{u}} (f : X →* Y) (n : ℕ)
include f
/- the carrier of the fiber sequence is definitionally what we want (as pointed sets) -/
example : LES_of_homotopy_groups3 f (str_of_nat 6) = π*[2] Y :> Set* := by reflexivity
example : LES_of_homotopy_groups3 f (str_of_nat 7) = π*[2] X :> Set* := by reflexivity
example : LES_of_homotopy_groups3 f (str_of_nat 8) = π*[2] (pfiber f) :> Set* := by reflexivity
example : LES_of_homotopy_groups3 f (str_of_nat 9) = π*[3] Y :> Set* := by reflexivity
example : LES_of_homotopy_groups3 f (str_of_nat 10) = π*[3] X :> Set* := by reflexivity
example : LES_of_homotopy_groups3 f (str_of_nat 11) = π*[3] (pfiber f) :> Set* := by reflexivity
definition LES_of_homotopy_groups3_0 : LES_of_homotopy_groups3 f (n, 0) = π*[2*n] Y :=
by reflexivity
definition LES_of_homotopy_groups3_1 : LES_of_homotopy_groups3 f (n, 1) = π*[2*n] X :=
by reflexivity
definition LES_of_homotopy_groups3_2 : LES_of_homotopy_groups3 f (n, 2) = π*[2*n] (pfiber f) :=
by reflexivity
definition LES_of_homotopy_groups3_3 : LES_of_homotopy_groups3 f (n, 3) = π*[2*n + 1] Y :=
by reflexivity
definition LES_of_homotopy_groups3_4 : LES_of_homotopy_groups3 f (n, 4) = π*[2*n + 1] X :=
by reflexivity
definition LES_of_homotopy_groups3_5 : LES_of_homotopy_groups3 f (n, 5) = π*[2*n + 1] (pfiber f):=
by reflexivity
/- the functions of the fiber sequence is definitionally what we want (as pointed function).
-/
definition LES_of_homotopy_groups_fun3_0 :
cc_to_fn (LES_of_homotopy_groups3 f) (n, 0) = π→*[2*n] f :=
by reflexivity
definition LES_of_homotopy_groups_fun3_1 :
cc_to_fn (LES_of_homotopy_groups3 f) (n, 1) = π→*[2*n] (ppoint f) :=
by reflexivity
definition LES_of_homotopy_groups_fun3_2 : cc_to_fn (LES_of_homotopy_groups3 f) (n, 2) =
π→*[2*n] (boundary_map f) ∘* pcast (ap (ptrunc 0) (loop_space_succ_eq_in Y (2*n))) :=
by reflexivity
definition LES_of_homotopy_groups_fun3_3 :
cc_to_fn (LES_of_homotopy_groups3 f) (n, 3) = π→*[2*n + 1] f ∘* tinverse :=
by reflexivity
definition LES_of_homotopy_groups_fun3_4 :
cc_to_fn (LES_of_homotopy_groups3 f) (n, 4) = π→*[2*n + 1] (ppoint f) ∘* tinverse :=
by reflexivity
definition LES_of_homotopy_groups_fun3_5 : cc_to_fn (LES_of_homotopy_groups3 f) (n, 5) =
(π→*[2*n + 1] (boundary_map f) ∘* tinverse) ∘*
pcast (ap (ptrunc 0) (loop_space_succ_eq_in Y (2*n+1))) :=
by reflexivity
definition group_LES_of_homotopy_groups3_0 :
Π(k : ℕ) (H : k + 3 < succ 5), group (LES_of_homotopy_groups3 f (0, fin.mk (k+3) H))
| 0 H := begin rexact group_homotopy_group 0 Y end
| 1 H := begin rexact group_homotopy_group 0 X end
| 2 H := begin rexact group_homotopy_group 0 (pfiber f) end
| (k+3) H := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
definition comm_group_LES_of_homotopy_groups3 (n : ℕ) : Π(x : fin (succ 5)),
comm_group (LES_of_homotopy_groups3 f (n + 1, x))
| (fin.mk 0 H) := proof comm_group_homotopy_group (2*n) Y qed
| (fin.mk 1 H) := proof comm_group_homotopy_group (2*n) X qed
| (fin.mk 2 H) := proof comm_group_homotopy_group (2*n) (pfiber f) qed
| (fin.mk 3 H) := proof comm_group_homotopy_group (2*n+1) Y qed
| (fin.mk 4 H) := proof comm_group_homotopy_group (2*n+1) X qed
| (fin.mk 5 H) := proof comm_group_homotopy_group (2*n+1) (pfiber f) qed
| (fin.mk (k+6) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
definition CommGroup_LES_of_homotopy_groups3 (n : +6ℕ) : CommGroup.{u} :=
CommGroup.mk (LES_of_homotopy_groups3 f (pr1 n + 1, pr2 n))
(comm_group_LES_of_homotopy_groups3 f (pr1 n) (pr2 n))
definition homomorphism_LES_of_homotopy_groups_fun3 : Π(k : +6ℕ),
CommGroup_LES_of_homotopy_groups3 f (S k) →g CommGroup_LES_of_homotopy_groups3 f k
| (k, fin.mk 0 H) :=
proof homomorphism.mk (cc_to_fn (LES_of_homotopy_groups3 f) (k + 1, 0))
(phomotopy_group_functor_mul _ _) qed
| (k, fin.mk 1 H) :=
proof homomorphism.mk (cc_to_fn (LES_of_homotopy_groups3 f) (k + 1, 1))
(phomotopy_group_functor_mul _ _) qed
| (k, fin.mk 2 H) :=
begin
apply homomorphism.mk (cc_to_fn (LES_of_homotopy_groups3 f) (k + 1, 2)),
exact abstract begin rewrite [LES_of_homotopy_groups_fun3_2],
refine @is_homomorphism_compose _ _ _ _ _ _ (π→*[2 * (k + 1)] boundary_map f) _ _ _,
{ apply group_homotopy_group ((2 * k) + 1)},
{ apply phomotopy_group_functor_mul},
{ rewrite [▸*, -ap_compose', ▸*],
apply is_homomorphism_cast_loop_space_succ_eq_in} end end
end
| (k, fin.mk 3 H) :=
begin
apply homomorphism.mk (cc_to_fn (LES_of_homotopy_groups3 f) (k + 1, 3)),
exact abstract begin rewrite [LES_of_homotopy_groups_fun3_3],
refine @is_homomorphism_compose _ _ _ _ _ _ (π→*[2 * (k + 1) + 1] f) tinverse _ _,
{ apply group_homotopy_group (2 * (k+1))},
{ apply phomotopy_group_functor_mul},
{ apply is_homomorphism_inverse} end end
end
| (k, fin.mk 4 H) :=
begin
apply homomorphism.mk (cc_to_fn (LES_of_homotopy_groups3 f) (k + 1, 4)),
exact abstract begin rewrite [LES_of_homotopy_groups_fun3_4],
refine @is_homomorphism_compose _ _ _ _ _ _ (π→*[2 * (k + 1) + 1] (ppoint f)) tinverse _ _,
{ apply group_homotopy_group (2 * (k+1))},
{ apply phomotopy_group_functor_mul},
{ apply is_homomorphism_inverse} end end
end
| (k, fin.mk 5 H) :=
begin
apply homomorphism.mk (cc_to_fn (LES_of_homotopy_groups3 f) (k + 1, 5)),
exact abstract begin rewrite [LES_of_homotopy_groups_fun3_5],
refine @is_homomorphism_compose _ _ _ _ _ _
(π→*[2 * (k + 1) + 1] (boundary_map f) ∘ tinverse) _ _ _,
{ refine @is_homomorphism_compose _ _ _ _ _ _
(π→*[2 * (k + 1) + 1] (boundary_map f)) tinverse _ _,
{ apply group_homotopy_group (2 * (k+1))},
{ apply phomotopy_group_functor_mul},
{ apply is_homomorphism_inverse}},
{ rewrite [▸*, -ap_compose', ▸*],
apply is_homomorphism_cast_loop_space_succ_eq_in} end end
end
| (k, fin.mk (l+6) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
--TODO: the maps 3, 4 and 5 are anti-homomorphisms.
end chain_complex'

801
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@ -0,0 +1,801 @@
/-
Copyright (c) 2016 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
We define the fiber sequence of a pointed map f : X →* Y. We mostly follow the proof in section 8.4
of the book.
PART 1:
We define a sequence fiber_sequence as in Definition 8.4.3.
It has types X(n) : Type*
X(0) := Y,
X(1) := X,
X(n+1) := fiber (f(n))
with functions f(n) : X(n+1) →* X(n)
f(0) := f
f(n+1) := point (f(n)) [this is the first projection]
We prove that this is an exact sequence.
Then we prove Lemma 8.4.3, by showing that X(n+3) ≃* Ω(X(n)) and that this equivalence sends
the pointed map f(n+3) to -Ω(f(n)), i.e. the composition of Ω(f(n)) with path inversion.
Using this equivalence we get a boundary_map : Ω(Y) → pfiber f.
PART 2:
Now we can define a new fiber sequence X'(n) : Type*, and here we slightly diverge from the book.
We define it as
X'(0) := Y,
X'(1) := X,
X'(2) := fiber f
X'(n+3) := Ω(X'(n))
with maps f'(n) : X'(n+1) →* X'(n)
f'(0) := f
f'(1) := point f
f'(2) := boundary_map
f'(n+3) := Ω(f'(n))
This sequence is not equivalent to the previous sequence. The difference is in the signs.
The sequence f has negative signs (i.e. is composed with the inverse maps) for n ≡ 3, 4, 5 mod 6.
This sign information is captured by e : X'(n) ≃* X'(n) such that
e(k) := 1 for k = 0,1,2,3
e(k+3) := Ω(e(k)) ∘ (-)⁻¹ for k > 0
Now the sequence (X', f' ∘ e) is equivalent to (X, f), Hence (X', f' ∘ e) is an exact sequence.
We then prove that (X', f') is an exact sequence by using that there are other equivalences
eₗ and eᵣ such that
f' = eᵣ ∘ f' ∘ e
f' ∘ eₗ = e ∘ f'.
(this fact is type_chain_complex_cancel_aut and is_exact_at_t_cancel_aut in the file chain_complex)
eₗ and eᵣ are almost the same as e, except that the places where the inverse is taken is
slightly shifted:
eᵣ = (-)⁻¹ for n ≡ 3, 4, 5 mod 6 and eᵣ = 1 otherwise
e = (-)⁻¹ for n ≡ 4, 5, 6 mod 6 (except for n = 0) and e = 1 otherwise
eₗ = (-)⁻¹ for n ≡ 5, 6, 7 mod 6 (except for n = 0, 1) and eₗ = 1 otherwise
PART 3:
We change the type over which the sequence of types and maps are indexed from ℕ to ℕ × 3
(where 3 is the finite type with 3 elements). The reason is that we have that X'(3n) = Ωⁿ(Y), but
this equality is not definitionally true. Hence we cannot even state whether f'(3n) = Ωⁿ(f) without
using transports. This gets ugly. However, if we use as index type ℕ × 3, we can do this. We can
define
Y : ℕ × 3 → Type* as
Y(n, 0) := Ωⁿ(Y)
Y(n, 1) := Ωⁿ(X)
Y(n, 2) := Ωⁿ(fiber f)
with maps g(n) : Y(S n) →* Y(n) (where the successor is defined in the obvious way)
g(n, 0) := Ωⁿ(f)
g(n, 1) := Ωⁿ(point f)
g(n, 2) := Ωⁿ(boundary_map) ∘ cast
Here "cast" is the transport over the equality Ωⁿ⁺¹(Y) = Ωⁿ(Ω(Y)). We show that the sequence
(ℕ, X', f') is equivalent to (ℕ × 3, Y, g).
PART 4:
We get the long exact sequence of homotopy groups by taking the set-truncation of (Y, g).
-/
import .chain_complex algebra.homotopy_group
open eq pointed sigma fiber equiv is_equiv sigma.ops is_trunc nat trunc algebra function sum
section MOVE
-- TODO: MOVE
open group chain_complex
definition pinverse_pinverse (A : Type*) : pinverse ∘* pinverse ~* pid (Ω A) :=
begin
fapply phomotopy.mk,
{ apply inv_inv},
{ reflexivity}
end
definition to_pmap_pequiv_of_pmap {A B : Type*} (f : A →* B) (H : is_equiv f)
: pequiv.to_pmap (pequiv_of_pmap f H) = f :=
by cases f; reflexivity
definition to_pmap_pequiv_trans {A B C : Type*} (f : A ≃* B) (g : B ≃* C)
: pequiv.to_pmap (f ⬝e* g) = g ∘* f :=
!to_pmap_pequiv_of_pmap
definition pequiv_pinverse (A : Type*) : Ω A ≃* Ω A :=
pequiv_of_pmap pinverse !is_equiv_eq_inverse
definition tr_mul_tr {A : Type*} (n : ℕ) (p q : Ω[n + 1] A) :
tr p *[πg[n+1] A] tr q = tr (p ⬝ q) :=
by reflexivity
definition is_homomorphism_cast_loop_space_succ_eq_in {A : Type*} (n : ℕ) :
is_homomorphism
(cast (ap (trunc 0 ∘ pointed.carrier) (loop_space_succ_eq_in A (succ n)))
: πg[n+1+1] A → πg[n+1] Ω A) :=
begin
intro g h, induction g with g, induction h with h,
xrewrite [tr_mul_tr, - + fn_cast_eq_cast_fn _ (λn, tr), tr_mul_tr, ↑cast, -tr_compose,
loop_space_succ_eq_in_concat, - + tr_compose],
end
definition is_homomorphism_inverse (A : Type*) (n : ℕ)
: is_homomorphism (λp, p⁻¹ : πag[n+2] A → πag[n+2] A) :=
begin
intro g h, rewrite mul.comm,
induction g with g, induction h with h,
exact ap tr !con_inv
end
end MOVE
/--------------
PART 1
--------------/
namespace chain_complex
definition fiber_sequence_helper [constructor] (v : Σ(X Y : Type*), X →* Y)
: Σ(Z X : Type*), Z →* X :=
⟨pfiber v.2.2, v.1, ppoint v.2.2⟩
definition fiber_sequence_helpern (v : Σ(X Y : Type*), X →* Y) (n : ℕ)
: Σ(Z X : Type*), Z →* X :=
iterate fiber_sequence_helper n v
section
universe variable u
parameters {X Y : pType.{u}} (f : X →* Y)
include f
definition fiber_sequence_carrier (n : ℕ) : Type* :=
(fiber_sequence_helpern ⟨X, Y, f⟩ n).2.1
definition fiber_sequence_fun (n : ℕ)
: fiber_sequence_carrier (n + 1) →* fiber_sequence_carrier n :=
(fiber_sequence_helpern ⟨X, Y, f⟩ n).2.2
/- Definition 8.4.3 -/
definition fiber_sequence : type_chain_complex.{0 u} +ℕ :=
begin
fconstructor,
{ exact fiber_sequence_carrier},
{ exact fiber_sequence_fun},
{ intro n x, cases n with n,
{ exact point_eq x},
{ exact point_eq x}}
end
definition is_exact_fiber_sequence : is_exact_t fiber_sequence :=
λn x p, fiber.mk (fiber.mk x p) rfl
/- (generalization of) Lemma 8.4.4(i)(ii) -/
definition fiber_sequence_carrier_equiv (n : ℕ)
: fiber_sequence_carrier (n+3) ≃ Ω(fiber_sequence_carrier n) :=
calc
fiber_sequence_carrier (n+3) ≃ fiber (fiber_sequence_fun (n+1)) pt : erfl
... ≃ Σ(x : fiber_sequence_carrier _), fiber_sequence_fun (n+1) x = pt
: fiber.sigma_char
... ≃ Σ(x : fiber (fiber_sequence_fun n) pt), fiber_sequence_fun _ x = pt
: erfl
... ≃ Σ(v : Σ(x : fiber_sequence_carrier _), fiber_sequence_fun _ x = pt),
fiber_sequence_fun _ (fiber.mk v.1 v.2) = pt
: by exact sigma_equiv_sigma !fiber.sigma_char (λa, erfl)
... ≃ Σ(v : Σ(x : fiber_sequence_carrier _), fiber_sequence_fun _ x = pt),
v.1 = pt
: erfl
... ≃ Σ(v : Σ(x : fiber_sequence_carrier _), x = pt),
fiber_sequence_fun _ v.1 = pt
: sigma_assoc_comm_equiv
... ≃ fiber_sequence_fun _ !center.1 = pt
: @(sigma_equiv_of_is_contr_left _) !is_contr_sigma_eq'
... ≃ fiber_sequence_fun _ pt = pt
: erfl
... ≃ pt = pt
: by exact !equiv_eq_closed_left !respect_pt
... ≃ Ω(fiber_sequence_carrier n) : erfl
/- computation rule -/
definition fiber_sequence_carrier_equiv_eq (n : ℕ)
(x : fiber_sequence_carrier (n+1)) (p : fiber_sequence_fun n x = pt)
(q : fiber_sequence_fun (n+1) (fiber.mk x p) = pt)
: fiber_sequence_carrier_equiv n (fiber.mk (fiber.mk x p) q)
= !respect_pt⁻¹ ⬝ ap (fiber_sequence_fun n) q⁻¹ ⬝ p :=
begin
refine _ ⬝ !con.assoc⁻¹,
apply whisker_left,
refine transport_eq_Fl _ _ ⬝ _,
apply whisker_right,
refine inverse2 !ap_inv ⬝ !inv_inv ⬝ _,
refine ap_compose (fiber_sequence_fun n) pr₁ _ ⬝
ap02 (fiber_sequence_fun n) !ap_pr1_center_eq_sigma_eq',
end
definition fiber_sequence_carrier_equiv_inv_eq (n : ℕ)
(p : Ω(fiber_sequence_carrier n)) : (fiber_sequence_carrier_equiv n)⁻¹ᵉ p =
fiber.mk (fiber.mk pt (respect_pt (fiber_sequence_fun n) ⬝ p)) idp :=
begin
apply inv_eq_of_eq,
refine _ ⬝ !fiber_sequence_carrier_equiv_eq⁻¹, esimp,
exact !inv_con_cancel_left⁻¹
end
definition fiber_sequence_carrier_pequiv (n : ℕ)
: fiber_sequence_carrier (n+3) ≃* Ω(fiber_sequence_carrier n) :=
pequiv_of_equiv (fiber_sequence_carrier_equiv n)
begin
esimp,
apply con.left_inv
end
definition fiber_sequence_carrier_pequiv_eq (n : ℕ)
(x : fiber_sequence_carrier (n+1)) (p : fiber_sequence_fun n x = pt)
(q : fiber_sequence_fun (n+1) (fiber.mk x p) = pt)
: fiber_sequence_carrier_pequiv n (fiber.mk (fiber.mk x p) q)
= !respect_pt⁻¹ ⬝ ap (fiber_sequence_fun n) q⁻¹ ⬝ p :=
fiber_sequence_carrier_equiv_eq n x p q
definition fiber_sequence_carrier_pequiv_inv_eq (n : ℕ)
(p : Ω(fiber_sequence_carrier n)) : (fiber_sequence_carrier_pequiv n)⁻¹ᵉ* p =
fiber.mk (fiber.mk pt (respect_pt (fiber_sequence_fun n) ⬝ p)) idp :=
by rexact fiber_sequence_carrier_equiv_inv_eq n p
/- Lemma 8.4.4(iii) -/
definition fiber_sequence_fun_eq_helper (n : ℕ)
(p : Ω(fiber_sequence_carrier (n + 1))) :
fiber_sequence_carrier_pequiv n
(fiber_sequence_fun (n + 3)
((fiber_sequence_carrier_pequiv (n + 1))⁻¹ᵉ* p)) =
ap1 (fiber_sequence_fun n) p⁻¹ :=
begin
refine ap (λx, fiber_sequence_carrier_pequiv n (fiber_sequence_fun (n + 3) x))
(fiber_sequence_carrier_pequiv_inv_eq (n+1) p) ⬝ _,
/- the following three lines are rewriting some reflexivities: -/
-- replace (n + 3) with (n + 2 + 1),
-- refine ap (fiber_sequence_carrier_pequiv n)
-- (fiber_sequence_fun_eq1 (n+2) _ idp) ⬝ _,
refine fiber_sequence_carrier_pequiv_eq n pt (respect_pt (fiber_sequence_fun n)) _ ⬝ _,
esimp,
apply whisker_right,
apply whisker_left,
apply ap02, apply inverse2, apply idp_con,
end
theorem fiber_sequence_carrier_pequiv_eq_point_eq_idp (n : ℕ) :
fiber_sequence_carrier_pequiv_eq n
(Point (fiber_sequence_carrier (n+1)))
(respect_pt (fiber_sequence_fun n))
(respect_pt (fiber_sequence_fun (n + 1))) = idp :=
begin
apply con_inv_eq_idp,
refine ap (λx, whisker_left _ (_ ⬝ x)) _ ⬝ _,
{ reflexivity},
{ reflexivity},
refine ap (whisker_left _)
(transport_eq_Fl_idp_left (fiber_sequence_fun n)
(respect_pt (fiber_sequence_fun n))) ⬝ _,
apply whisker_left_idp_con_eq_assoc
end
theorem fiber_sequence_fun_phomotopy_helper (n : ℕ) :
(fiber_sequence_carrier_pequiv n ∘*
fiber_sequence_fun (n + 3)) ∘*
(fiber_sequence_carrier_pequiv (n + 1))⁻¹ᵉ* ~*
ap1 (fiber_sequence_fun n) ∘* pinverse :=
begin
fapply phomotopy.mk,
{ exact chain_complex.fiber_sequence_fun_eq_helper f n},
{ esimp, rewrite [idp_con], refine _ ⬝ whisker_left _ !idp_con⁻¹,
apply whisker_right,
apply whisker_left,
exact chain_complex.fiber_sequence_carrier_pequiv_eq_point_eq_idp f n}
end
theorem fiber_sequence_fun_eq (n : ℕ) : Π(x : fiber_sequence_carrier (n + 4)),
fiber_sequence_carrier_pequiv n (fiber_sequence_fun (n + 3) x) =
ap1 (fiber_sequence_fun n) (fiber_sequence_carrier_pequiv (n + 1) x)⁻¹ :=
begin
apply homotopy_of_inv_homotopy_pre (fiber_sequence_carrier_pequiv (n + 1)),
apply fiber_sequence_fun_eq_helper n
end
theorem fiber_sequence_fun_phomotopy (n : ℕ) :
fiber_sequence_carrier_pequiv n ∘*
fiber_sequence_fun (n + 3) ~*
(ap1 (fiber_sequence_fun n) ∘* pinverse) ∘* fiber_sequence_carrier_pequiv (n + 1) :=
begin
apply phomotopy_of_pinv_right_phomotopy,
apply fiber_sequence_fun_phomotopy_helper
end
definition boundary_map : Ω Y →* pfiber f :=
fiber_sequence_fun 2 ∘* (fiber_sequence_carrier_pequiv 0)⁻¹ᵉ*
/--------------
PART 2
--------------/
/- Now we are ready to define the long exact sequence of homotopy groups.
First we define its carrier -/
definition loop_spaces : ℕ → Type*
| 0 := Y
| 1 := X
| 2 := pfiber f
| (k+3) := Ω (loop_spaces k)
/- The maps between the homotopy groups -/
definition loop_spaces_fun
: Π(n : ℕ), loop_spaces (n+1) →* loop_spaces n
| 0 := proof f qed
| 1 := proof ppoint f qed
| 2 := proof boundary_map qed
| (k+3) := proof ap1 (loop_spaces_fun k) qed
definition loop_spaces_fun_add3 [unfold_full] (n : ℕ) :
loop_spaces_fun (n + 3) = ap1 (loop_spaces_fun n) :=
proof idp qed
definition fiber_sequence_pequiv_loop_spaces :
Πn, fiber_sequence_carrier n ≃* loop_spaces n
| 0 := by reflexivity
| 1 := by reflexivity
| 2 := by reflexivity
| (k+3) :=
begin
refine fiber_sequence_carrier_pequiv k ⬝e* _,
apply loop_pequiv_loop,
exact fiber_sequence_pequiv_loop_spaces k
end
definition fiber_sequence_pequiv_loop_spaces_add3 (n : ℕ)
: fiber_sequence_pequiv_loop_spaces (n + 3) =
ap1 (fiber_sequence_pequiv_loop_spaces n) ∘* fiber_sequence_carrier_pequiv n :=
by reflexivity
definition fiber_sequence_pequiv_loop_spaces_3_phomotopy
: fiber_sequence_pequiv_loop_spaces 3 ~* proof fiber_sequence_carrier_pequiv nat.zero qed :=
begin
refine pwhisker_right _ ap1_id ⬝* _,
apply pid_comp
end
definition pid_or_pinverse : Π(n : ℕ), loop_spaces n ≃* loop_spaces n
| 0 := pequiv.rfl
| 1 := pequiv.rfl
| 2 := pequiv.rfl
| 3 := pequiv.rfl
| (k+4) := !pequiv_pinverse ⬝e* loop_pequiv_loop (pid_or_pinverse (k+1))
definition pid_or_pinverse_add4 (n : ℕ)
: pid_or_pinverse (n + 4) = !pequiv_pinverse ⬝e* loop_pequiv_loop (pid_or_pinverse (n + 1)) :=
by reflexivity
definition pid_or_pinverse_add4_rev : Π(n : ℕ),
pid_or_pinverse (n + 4) ~* pinverse ∘* Ω→(pid_or_pinverse (n + 1))
| 0 := begin rewrite [pid_or_pinverse_add4, + to_pmap_pequiv_trans],
replace pid_or_pinverse (0 + 1) with pequiv.refl X,
rewrite [loop_pequiv_loop_rfl, ▸*], refine !pid_comp ⬝* _,
exact !comp_pid⁻¹* ⬝* pwhisker_left _ !ap1_id⁻¹* end
| 1 := begin rewrite [pid_or_pinverse_add4, + to_pmap_pequiv_trans],
replace pid_or_pinverse (1 + 1) with pequiv.refl (pfiber f),
rewrite [loop_pequiv_loop_rfl, ▸*], refine !pid_comp ⬝* _,
exact !comp_pid⁻¹* ⬝* pwhisker_left _ !ap1_id⁻¹* end
| 2 := begin rewrite [pid_or_pinverse_add4, + to_pmap_pequiv_trans],
replace pid_or_pinverse (2 + 1) with pequiv.refl (Ω Y),
rewrite [loop_pequiv_loop_rfl, ▸*], refine !pid_comp ⬝* _,
exact !comp_pid⁻¹* ⬝* pwhisker_left _ !ap1_id⁻¹* end
| (k+3) :=
begin
replace (k + 3 + 1) with (k + 4),
rewrite [+ pid_or_pinverse_add4, + to_pmap_pequiv_trans],
refine _ ⬝* pwhisker_left _ !ap1_compose⁻¹*,
refine _ ⬝* !passoc,
apply pconcat2,
{ refine ap1_phomotopy (pid_or_pinverse_add4_rev k) ⬝* _,
refine !ap1_compose ⬝* _, apply pwhisker_right, apply ap1_pinverse},
{ refine !ap1_pinverse⁻¹*}
end
theorem fiber_sequence_phomotopy_loop_spaces : Π(n : ℕ),
fiber_sequence_pequiv_loop_spaces n ∘* fiber_sequence_fun n ~*
(loop_spaces_fun n ∘* pid_or_pinverse (n + 1)) ∘* fiber_sequence_pequiv_loop_spaces (n + 1)
| 0 := proof proof phomotopy.rfl qed ⬝* pwhisker_right _ !comp_pid⁻¹* qed
| 1 := by reflexivity
| 2 :=
begin
refine !pid_comp ⬝* _,
replace loop_spaces_fun 2 with boundary_map,
refine _ ⬝* pwhisker_left _ fiber_sequence_pequiv_loop_spaces_3_phomotopy⁻¹*,
apply phomotopy_of_pinv_right_phomotopy,
exact !pid_comp⁻¹*
end
| (k+3) :=
begin
replace (k + 3 + 1) with (k + 1 + 3),
rewrite [fiber_sequence_pequiv_loop_spaces_add3 k,
fiber_sequence_pequiv_loop_spaces_add3 (k+1)],
refine !passoc ⬝* _,
refine pwhisker_left _ (fiber_sequence_fun_phomotopy k) ⬝* _,
refine !passoc⁻¹* ⬝* _ ⬝* !passoc,
apply pwhisker_right,
replace (k + 1 + 3) with (k + 4),
xrewrite [loop_spaces_fun_add3, pid_or_pinverse_add4, to_pmap_pequiv_trans],
refine _ ⬝* !passoc⁻¹*,
refine _ ⬝* pwhisker_left _ !passoc⁻¹*,
refine _ ⬝* pwhisker_left _ (pwhisker_left _ !ap1_compose_pinverse),
refine !passoc⁻¹* ⬝* _ ⬝* !passoc ⬝* !passoc,
apply pwhisker_right,
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose ⬝* pwhisker_right _ !ap1_compose,
apply ap1_phomotopy,
exact fiber_sequence_phomotopy_loop_spaces k
end
definition pid_or_pinverse_right : Π(n : ℕ), loop_spaces n →* loop_spaces n
| 0 := !pid
| 1 := !pid
| 2 := !pid
| (k+3) := Ω→(pid_or_pinverse_right k) ∘* pinverse
definition pid_or_pinverse_left : Π(n : ℕ), loop_spaces n →* loop_spaces n
| 0 := pequiv.rfl
| 1 := pequiv.rfl
| 2 := pequiv.rfl
| 3 := pequiv.rfl
| 4 := pequiv.rfl
| (k+5) := Ω→(pid_or_pinverse_left (k+2)) ∘* pinverse
definition pid_or_pinverse_right_add3 (n : ℕ)
: pid_or_pinverse_right (n + 3) = Ω→(pid_or_pinverse_right n) ∘* pinverse :=
by reflexivity
definition pid_or_pinverse_left_add5 (n : ℕ)
: pid_or_pinverse_left (n + 5) = Ω→(pid_or_pinverse_left (n+2)) ∘* pinverse :=
by reflexivity
theorem pid_or_pinverse_commute_right : Π(n : ℕ),
loop_spaces_fun n ~* pid_or_pinverse_right n ∘* loop_spaces_fun n ∘* pid_or_pinverse (n + 1)
| 0 := proof !comp_pid⁻¹* ⬝* !pid_comp⁻¹* qed
| 1 := proof !comp_pid⁻¹* ⬝* !pid_comp⁻¹* qed
| 2 := proof !comp_pid⁻¹* ⬝* !pid_comp⁻¹* qed
| (k+3) :=
begin
replace (k + 3 + 1) with (k + 4),
rewrite [pid_or_pinverse_right_add3, loop_spaces_fun_add3],
refine _ ⬝* pwhisker_left _ (pwhisker_left _ !pid_or_pinverse_add4_rev⁻¹*),
refine ap1_phomotopy (pid_or_pinverse_commute_right k) ⬝* _,
refine !ap1_compose ⬝* _ ⬝* !passoc⁻¹*,
apply pwhisker_left,
refine !ap1_compose ⬝* _ ⬝* !passoc ⬝* !passoc,
apply pwhisker_right,
refine _ ⬝* pwhisker_right _ !ap1_compose_pinverse,
refine _ ⬝* !passoc⁻¹*,
refine !comp_pid⁻¹* ⬝* pwhisker_left _ _,
symmetry, apply pinverse_pinverse
end
theorem pid_or_pinverse_commute_left : Π(n : ℕ),
loop_spaces_fun n ∘* pid_or_pinverse_left (n + 1) ~* pid_or_pinverse n ∘* loop_spaces_fun n
| 0 := proof !comp_pid ⬝* !pid_comp⁻¹* qed
| 1 := proof !comp_pid ⬝* !pid_comp⁻¹* qed
| 2 := proof !comp_pid ⬝* !pid_comp⁻¹* qed
| 3 := proof !comp_pid ⬝* !pid_comp⁻¹* qed
| (k+4) :=
begin
replace (k + 4 + 1) with (k + 5),
rewrite [pid_or_pinverse_left_add5, pid_or_pinverse_add4, to_pmap_pequiv_trans],
replace (k + 4) with (k + 1 + 3),
rewrite [loop_spaces_fun_add3],
refine !passoc⁻¹* ⬝* _ ⬝* !passoc⁻¹*,
refine _ ⬝* pwhisker_left _ !ap1_compose_pinverse,
refine _ ⬝* !passoc,
apply pwhisker_right,
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose,
exact ap1_phomotopy (pid_or_pinverse_commute_left (k+1))
end
definition LES_of_loop_spaces' [constructor] : type_chain_complex +ℕ :=
transfer_type_chain_complex
fiber_sequence
(λn, loop_spaces_fun n ∘* pid_or_pinverse (n + 1))
fiber_sequence_pequiv_loop_spaces
fiber_sequence_phomotopy_loop_spaces
definition LES_of_loop_spaces [constructor] : type_chain_complex +ℕ :=
type_chain_complex_cancel_aut
LES_of_loop_spaces'
loop_spaces_fun
pid_or_pinverse
pid_or_pinverse_right
(λn x, idp)
pid_or_pinverse_commute_right
definition is_exact_LES_of_loop_spaces : is_exact_t LES_of_loop_spaces :=
begin
intro n,
refine is_exact_at_t_cancel_aut n pid_or_pinverse_left _ _ pid_or_pinverse_commute_left _,
apply is_exact_at_t_transfer,
apply is_exact_fiber_sequence
end
open prod succ_str fin
/--------------
PART 3
--------------/
definition loop_spaces2 [reducible] : +3ℕ → Type*
| (n, fin.mk 0 H) := Ω[n] Y
| (n, fin.mk 1 H) := Ω[n] X
| (n, fin.mk k H) := Ω[n] (pfiber f)
definition loop_spaces2_add1 (n : ℕ) : Π(x : fin (nat.succ 2)),
loop_spaces2 (n+1, x) = Ω (loop_spaces2 (n, x))
| (fin.mk 0 H) := by reflexivity
| (fin.mk 1 H) := by reflexivity
| (fin.mk 2 H) := by reflexivity
| (fin.mk (k+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
definition loop_spaces_fun2 : Π(n : +3ℕ), loop_spaces2 (S n) →* loop_spaces2 n
| (n, fin.mk 0 H) := proof Ω→[n] f qed
| (n, fin.mk 1 H) := proof Ω→[n] (ppoint f) qed
| (n, fin.mk 2 H) := proof Ω→[n] boundary_map ∘* pcast (loop_space_succ_eq_in Y n) qed
| (n, fin.mk (k+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
definition loop_spaces_fun2_add1_0 (n : ℕ) (H : 0 < succ 2)
: loop_spaces_fun2 (n+1, fin.mk 0 H) ~*
cast proof idp qed ap1 (loop_spaces_fun2 (n, fin.mk 0 H)) :=
by reflexivity
definition loop_spaces_fun2_add1_1 (n : ℕ) (H : 1 < succ 2)
: loop_spaces_fun2 (n+1, fin.mk 1 H) ~*
cast proof idp qed ap1 (loop_spaces_fun2 (n, fin.mk 1 H)) :=
by reflexivity
definition loop_spaces_fun2_add1_2 (n : ℕ) (H : 2 < succ 2)
: loop_spaces_fun2 (n+1, fin.mk 2 H) ~*
cast proof idp qed ap1 (loop_spaces_fun2 (n, fin.mk 2 H)) :=
begin
esimp,
refine _ ⬝* !ap1_compose⁻¹*,
apply pwhisker_left,
apply pcast_ap_loop_space
end
definition nat_of_str [unfold 2] [reducible] {n : ℕ} : ℕ × fin (succ n) → ℕ :=
λx, succ n * pr1 x + val (pr2 x)
definition str_of_nat {n : ℕ} : ℕ → ℕ × fin (succ n) :=
λm, (m / (succ n), mk_mod n m)
definition nat_of_str_3S [unfold 2] [reducible]
: Π(x : stratified +ℕ 2), nat_of_str x + 1 = nat_of_str (@S (stratified +ℕ 2) x)
| (n, fin.mk 0 H) := by reflexivity
| (n, fin.mk 1 H) := by reflexivity
| (n, fin.mk 2 H) := by reflexivity
| (n, fin.mk (k+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
definition fin_prod_nat_equiv_nat [constructor] (n : ℕ) : ℕ × fin (succ n) ≃ ℕ :=
equiv.MK nat_of_str str_of_nat
abstract begin
intro m, unfold [nat_of_str, str_of_nat, mk_mod],
refine _ ⬝ (eq_div_mul_add_mod m (succ n))⁻¹,
rewrite [mul.comm]
end end
abstract begin
intro x, cases x with m k,
cases k with k H,
apply prod_eq: esimp [str_of_nat],
{ rewrite [add.comm, add_mul_div_self_left _ _ (!zero_lt_succ), ▸*,
div_eq_zero_of_lt H, zero_add]},
{ apply eq_of_veq, esimp [mk_mod],
rewrite [add.comm, add_mul_mod_self_left, ▸*, mod_eq_of_lt H]}
end end
/-
note: in the following theorem the (n+1) case is 3 times the same,
so maybe this can be simplified
-/
definition loop_spaces2_pequiv' : Π(n : ℕ) (x : fin (nat.succ 2)),
loop_spaces (nat_of_str (n, x)) ≃* loop_spaces2 (n, x)
| 0 (fin.mk 0 H) := by reflexivity
| 0 (fin.mk 1 H) := by reflexivity
| 0 (fin.mk 2 H) := by reflexivity
| (n+1) (fin.mk 0 H) :=
begin
apply loop_pequiv_loop,
rexact loop_spaces2_pequiv' n (fin.mk 0 H)
end
| (n+1) (fin.mk 1 H) :=
begin
apply loop_pequiv_loop,
rexact loop_spaces2_pequiv' n (fin.mk 1 H)
end
| (n+1) (fin.mk 2 H) :=
begin
apply loop_pequiv_loop,
rexact loop_spaces2_pequiv' n (fin.mk 2 H)
end
| n (fin.mk (k+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
definition loop_spaces2_pequiv : Π(x : +3ℕ),
loop_spaces (nat_of_str x) ≃* loop_spaces2 x
| (n, x) := loop_spaces2_pequiv' n x
local attribute loop_pequiv_loop [reducible]
/- all cases where n>0 are basically the same -/
definition loop_spaces_fun2_phomotopy (x : +3ℕ) :
loop_spaces2_pequiv x ∘* loop_spaces_fun (nat_of_str x) ~*
(loop_spaces_fun2 x ∘* loop_spaces2_pequiv (S x))
∘* pcast (ap (loop_spaces) (nat_of_str_3S x)) :=
begin
cases x with n x, cases x with k H,
cases k with k, rotate 1, cases k with k, rotate 1, cases k with k, rotate 2,
{ /-k=0-/
induction n with n IH,
{ refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹* ⬝* !comp_pid⁻¹*,
reflexivity},
{ refine _ ⬝* !comp_pid⁻¹*,
refine _ ⬝* pwhisker_right _ !loop_spaces_fun2_add1_0⁻¹*,
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
exact IH ⬝* !comp_pid}},
{ /-k=1-/
induction n with n IH,
{ refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹* ⬝* !comp_pid⁻¹*,
reflexivity},
{ refine _ ⬝* !comp_pid⁻¹*,
refine _ ⬝* pwhisker_right _ !loop_spaces_fun2_add1_1⁻¹*,
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
exact IH ⬝* !comp_pid}},
{ /-k=2-/
induction n with n IH,
{ refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹*,
refine !comp_pid⁻¹* ⬝* pconcat2 _ _,
{ exact (comp_pid (chain_complex.boundary_map f))⁻¹*},
{ refine cast (ap (λx, _ ~* x) !loop_pequiv_loop_rfl)⁻¹ _, reflexivity}},
{ refine _ ⬝* !comp_pid⁻¹*,
refine _ ⬝* pwhisker_right _ !loop_spaces_fun2_add1_2⁻¹*,
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
exact IH ⬝* !comp_pid}},
{ /-k=k'+3-/ exfalso, apply lt_le_antisymm H, apply le_add_left}
end
definition LES_of_loop_spaces2 [constructor] : type_chain_complex +3ℕ :=
transfer_type_chain_complex2
LES_of_loop_spaces
!fin_prod_nat_equiv_nat
nat_of_str_3S
@loop_spaces_fun2
@loop_spaces2_pequiv
begin
intro m x,
refine loop_spaces_fun2_phomotopy m x ⬝ _,
apply ap (loop_spaces_fun2 m), apply ap (loop_spaces2_pequiv (S m)),
esimp, exact ap010 cast !ap_compose⁻¹ x
end
definition is_exact_LES_of_loop_spaces2 : is_exact_t LES_of_loop_spaces2 :=
begin
intro n,
apply is_exact_at_transfer2,
apply is_exact_LES_of_loop_spaces
end
definition LES_of_homotopy_groups' [constructor] : chain_complex +3ℕ :=
trunc_chain_complex LES_of_loop_spaces2
/--------------
PART 4
--------------/
definition homotopy_groups [reducible] : +3ℕ → Set*
| (n, fin.mk 0 H) := π*[n] Y
| (n, fin.mk 1 H) := π*[n] X
| (n, fin.mk k H) := π*[n] (pfiber f)
definition homotopy_groups_pequiv_loop_spaces2 [reducible]
: Π(n : +3ℕ), ptrunc 0 (loop_spaces2 n) ≃* homotopy_groups n
| (n, fin.mk 0 H) := by reflexivity
| (n, fin.mk 1 H) := by reflexivity
| (n, fin.mk 2 H) := by reflexivity
| (n, fin.mk (k+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
definition homotopy_groups_fun : Π(n : +3ℕ), homotopy_groups (S n) →* homotopy_groups n
| (n, fin.mk 0 H) := proof π→*[n] f qed
| (n, fin.mk 1 H) := proof π→*[n] (ppoint f) qed
| (n, fin.mk 2 H) :=
proof π→*[n] boundary_map ∘* pcast (ap (ptrunc 0) (loop_space_succ_eq_in Y n)) qed
| (n, fin.mk (k+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
definition homotopy_groups_fun_phomotopy_loop_spaces_fun2 [reducible]
: Π(n : +3ℕ), homotopy_groups_pequiv_loop_spaces2 n ∘* ptrunc_functor 0 (loop_spaces_fun2 n) ~*
homotopy_groups_fun n ∘* homotopy_groups_pequiv_loop_spaces2 (S n)
| (n, fin.mk 0 H) := by reflexivity
| (n, fin.mk 1 H) := by reflexivity
| (n, fin.mk 2 H) :=
begin
refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹*,
refine !ptrunc_functor_pcompose ⬝* _,
apply pwhisker_left, apply ptrunc_functor_pcast,
end
| (n, fin.mk (k+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
definition LES_of_homotopy_groups [constructor] : chain_complex +3ℕ :=
transfer_chain_complex
LES_of_homotopy_groups'
homotopy_groups_fun
homotopy_groups_pequiv_loop_spaces2
homotopy_groups_fun_phomotopy_loop_spaces_fun2
definition is_exact_LES_of_homotopy_groups : is_exact LES_of_homotopy_groups :=
begin
intro n,
apply is_exact_at_transfer,
apply is_exact_at_trunc,
apply is_exact_LES_of_loop_spaces2
end
variable (n : ℕ)
/- the carrier of the fiber sequence is definitionally what we want (as pointed sets) -/
example : LES_of_homotopy_groups (str_of_nat 6) = π*[2] Y :> Set* := by reflexivity
example : LES_of_homotopy_groups (str_of_nat 7) = π*[2] X :> Set* := by reflexivity
example : LES_of_homotopy_groups (str_of_nat 8) = π*[2] (pfiber f) :> Set* := by reflexivity
example : LES_of_homotopy_groups (str_of_nat 9) = π*[3] Y :> Set* := by reflexivity
example : LES_of_homotopy_groups (str_of_nat 10) = π*[3] X :> Set* := by reflexivity
example : LES_of_homotopy_groups (str_of_nat 11) = π*[3] (pfiber f) :> Set* := by reflexivity
definition LES_of_homotopy_groups_0 : LES_of_homotopy_groups (n, 0) = π*[n] Y :=
by reflexivity
definition LES_of_homotopy_groups_1 : LES_of_homotopy_groups (n, 1) = π*[n] X :=
by reflexivity
definition LES_of_homotopy_groups_2 : LES_of_homotopy_groups (n, 2) = π*[n] (pfiber f) :=
by reflexivity
/- the functions of the fiber sequence is definitionally what we want (as pointed function).
-/
definition LES_of_homotopy_groups_fun_0 :
cc_to_fn LES_of_homotopy_groups (n, 0) = π→*[n] f :=
by reflexivity
definition LES_of_homotopy_groups_fun_1 :
cc_to_fn LES_of_homotopy_groups (n, 1) = π→*[n] (ppoint f) :=
by reflexivity
definition LES_of_homotopy_groups_fun_2 : cc_to_fn LES_of_homotopy_groups (n, 2) =
π→*[n] boundary_map ∘* pcast (ap (ptrunc 0) (loop_space_succ_eq_in Y n)) :=
by reflexivity
open group
definition group_LES_of_homotopy_groups_0
: Π(x : fin (succ 2)), group (LES_of_homotopy_groups (1, x))
| (fin.mk 0 H) := begin rexact group_homotopy_group 0 Y end
| (fin.mk 1 H) := begin rexact group_homotopy_group 0 X end
| (fin.mk 2 H) := begin rexact group_homotopy_group 0 (pfiber f) end
| (fin.mk (k+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
definition comm_group_LES_of_homotopy_groups (n : ℕ) : Π(x : fin (succ 2)),
comm_group (LES_of_homotopy_groups (n + 2, x))
| (fin.mk 0 H) := proof comm_group_homotopy_group n Y qed
| (fin.mk 1 H) := proof comm_group_homotopy_group n X qed
| (fin.mk 2 H) := proof comm_group_homotopy_group n (pfiber f) qed
| (fin.mk (k+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
definition CommGroup_LES_of_homotopy_groups (n : +3ℕ) : CommGroup.{u} :=
CommGroup.mk (LES_of_homotopy_groups (pr1 n + 2, pr2 n))
(comm_group_LES_of_homotopy_groups (pr1 n) (pr2 n))
definition homomorphism_LES_of_homotopy_groups_fun : Π(k : +3ℕ),
CommGroup_LES_of_homotopy_groups (S k) →g CommGroup_LES_of_homotopy_groups k
| (k, fin.mk 0 H) :=
proof homomorphism.mk (cc_to_fn LES_of_homotopy_groups (k + 2, 0))
(phomotopy_group_functor_mul _ _) qed
| (k, fin.mk 1 H) :=
proof homomorphism.mk (cc_to_fn LES_of_homotopy_groups (k + 2, 1))
(phomotopy_group_functor_mul _ _) qed
| (k, fin.mk 2 H) :=
begin
apply homomorphism.mk (cc_to_fn LES_of_homotopy_groups (k + 2, 2)),
exact abstract begin rewrite [LES_of_homotopy_groups_fun_2],
refine @is_homomorphism_compose _ _ _ _ _ _ (π→*[k + 2] boundary_map) _ _ _,
{ apply group_homotopy_group (k + 1)},
{ apply phomotopy_group_functor_mul},
{ rewrite [▸*, -ap_compose', ▸*],
apply is_homomorphism_cast_loop_space_succ_eq_in} end end
end
| (k, fin.mk (l+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
end
end chain_complex

8
homotopy/LES_applications.hlean
View file

@ -1,3 +1,9 @@
/-
Copyright (c) 2016 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import .LES_of_homotopy_groups homotopy.connectedness homotopy.homotopy_group homotopy.join
open eq is_trunc pointed is_conn is_equiv fiber equiv trunc nat chain_complex prod fin algebra
group trunc_index function join pushout
@ -49,7 +55,7 @@ namespace is_conn
(@pgroup_of_group _ (comm_group_LES_of_homotopy_groups3 f k 1) idp)
(homomorphism.struct (homomorphism_LES_of_homotopy_groups_fun3 f (k, 0)))},
{ /- k = 1 -/
exact sorry},
exact sorry}, -- need some more facts about anti-homomorphisms
{ /- k > 1 odd -/
have H2' : 2 * succ k ≤ n, from le.trans !self_le_succ H2,
have H3 : is_equiv (π→*[2*(succ k) + 1] f ∘* tinverse), from

258
homotopy/LES_of_homotopy_groups.hlean
View file

@ -3,41 +3,11 @@ Copyright (c) 2016 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
We define the fiber sequence of a pointed map f : X →* Y. We follow the proof in section 8.4 of
the book closely. First we define a sequence fiber_sequence as in Definition 8.4.3.
It has types X(n) : Type*
X(0) := Y,
X(1) := X,
X(n+1) := pfiber (f(n))
with functions f(n) : X(n+1) →* X(n)
f(0) := f
f(n+1) := ppoint f(n)
We prove that this is an exact sequence.
Then we prove Lemma 8.4.3, by showing that X(n+3) ≃* Ω(X(n)) and that this equivalence sends
the map f(n+3) to -Ω(f(n)), i.e. the composition of Ω(f(n)) with path inversion.
This is the hardest part of this formalization, because we need to show that they are the same
as pointed maps (we define a pointed homotopy between them).
Using this equivalence we get a boundary_map : Ω(Y) → pfiber f and we can define a new
fiber sequence X'(n) : Type*
X'(0) := Y,
X'(1) := X,
X'(2) := pfiber f
X'(n+3) := Ω(X'(n))
and maps f'(n) : X'(n+1) →* X'(n)
f'(0) := f
f'(1) := ppoint f
f'(2) := boundary_map f
f'(3) := -Ω(f)
f'(4) := -Ω(ppoint f)
f'(5) := -Ω(boundary_map f)
f'(n+6) := Ω²(f'(n))
We can show that these sequences are equivalent, hence the sequence (X',f') is an exact
sequence. Now we get the fiber sequence by taking the set-truncation of this sequence.
The old formalization of the LES of homotopy groups, where all the odd levels have a composition
with negation
-/
import .chain_complex algebra.homotopy_group
import .LES2
open eq pointed sigma fiber equiv is_equiv sigma.ops is_trunc nat trunc algebra function
@ -47,183 +17,11 @@ open eq pointed sigma fiber equiv is_equiv sigma.ops is_trunc nat trunc algebra
namespace chain_complex
definition fiber_sequence_helper [constructor] (v : Σ(X Y : Type*), X →* Y)
: Σ(Z X : Type*), Z →* X :=
⟨pfiber v.2.2, v.1, ppoint v.2.2⟩
definition fiber_sequence_helpern (v : Σ(X Y : Type*), X →* Y) (n : ℕ)
: Σ(Z X : Type*), Z →* X :=
iterate fiber_sequence_helper n v
namespace old
universe variable u
variables {X Y : pType.{u}} (f : X →* Y) (n : ℕ)
include f
definition fiber_sequence_carrier : Type* :=
(fiber_sequence_helpern ⟨X, Y, f⟩ n).2.1
definition fiber_sequence_fun
: fiber_sequence_carrier f (n + 1) →* fiber_sequence_carrier f n :=
(fiber_sequence_helpern ⟨X, Y, f⟩ n).2.2
/- Definition 8.4.3 -/
definition fiber_sequence : type_chain_complex.{0 u} +ℕ :=
begin
fconstructor,
{ exact fiber_sequence_carrier f},
{ exact fiber_sequence_fun f},
{ intro n x, cases n with n,
{ exact point_eq x},
{ exact point_eq x}}
end
definition is_exact_fiber_sequence : is_exact_t (fiber_sequence f) :=
λn x p, fiber.mk (fiber.mk x p) rfl
/- (generalization of) Lemma 8.4.4(i)(ii) -/
definition fiber_sequence_carrier_equiv
: fiber_sequence_carrier f (n+3) ≃ Ω(fiber_sequence_carrier f n) :=
calc
fiber_sequence_carrier f (n+3) ≃ fiber (fiber_sequence_fun f (n+1)) pt : erfl
... ≃ Σ(x : fiber_sequence_carrier f _), fiber_sequence_fun f (n+1) x = pt
: fiber.sigma_char
... ≃ Σ(x : fiber (fiber_sequence_fun f n) pt), fiber_sequence_fun f _ x = pt
: erfl
... ≃ Σ(v : Σ(x : fiber_sequence_carrier f _), fiber_sequence_fun f _ x = pt),
fiber_sequence_fun f _ (fiber.mk v.1 v.2) = pt
: by exact sigma_equiv_sigma !fiber.sigma_char (λa, erfl)
... ≃ Σ(v : Σ(x : fiber_sequence_carrier f _), fiber_sequence_fun f _ x = pt),
v.1 = pt
: erfl
... ≃ Σ(v : Σ(x : fiber_sequence_carrier f _), x = pt),
fiber_sequence_fun f _ v.1 = pt
: sigma_assoc_comm_equiv
... ≃ fiber_sequence_fun f _ !center.1 = pt
: @(sigma_equiv_of_is_contr_left _) !is_contr_sigma_eq'
... ≃ fiber_sequence_fun f _ pt = pt
: erfl
... ≃ pt = pt
: by exact !equiv_eq_closed_left !respect_pt
... ≃ Ω(fiber_sequence_carrier f n) : erfl
/- computation rule -/
definition fiber_sequence_carrier_equiv_eq
(x : fiber_sequence_carrier f (n+1)) (p : fiber_sequence_fun f n x = pt)
(q : fiber_sequence_fun f (n+1) (fiber.mk x p) = pt)
: fiber_sequence_carrier_equiv f n (fiber.mk (fiber.mk x p) q)
= !respect_pt⁻¹ ⬝ ap (fiber_sequence_fun f n) q⁻¹ ⬝ p :=
begin
refine _ ⬝ !con.assoc⁻¹,
apply whisker_left,
refine transport_eq_Fl _ _ ⬝ _,
apply whisker_right,
refine inverse2 !ap_inv ⬝ !inv_inv ⬝ _,
refine ap_compose (fiber_sequence_fun f n) pr₁ _ ⬝
ap02 (fiber_sequence_fun f n) !ap_pr1_center_eq_sigma_eq',
end
definition fiber_sequence_carrier_equiv_inv_eq
(p : Ω(fiber_sequence_carrier f n)) : (fiber_sequence_carrier_equiv f n)⁻¹ᵉ p =
fiber.mk (fiber.mk pt (respect_pt (fiber_sequence_fun f n) ⬝ p)) idp :=
begin
apply inv_eq_of_eq,
refine _ ⬝ !fiber_sequence_carrier_equiv_eq⁻¹, esimp,
exact !inv_con_cancel_left⁻¹
end
definition fiber_sequence_carrier_pequiv
: fiber_sequence_carrier f (n+3) ≃* Ω(fiber_sequence_carrier f n) :=
pequiv_of_equiv (fiber_sequence_carrier_equiv f n)
begin
esimp,
apply con.left_inv
end
definition fiber_sequence_carrier_pequiv_eq
(x : fiber_sequence_carrier f (n+1)) (p : fiber_sequence_fun f n x = pt)
(q : fiber_sequence_fun f (n+1) (fiber.mk x p) = pt)
: fiber_sequence_carrier_pequiv f n (fiber.mk (fiber.mk x p) q)
= !respect_pt⁻¹ ⬝ ap (fiber_sequence_fun f n) q⁻¹ ⬝ p :=
fiber_sequence_carrier_equiv_eq f n x p q
definition fiber_sequence_carrier_pequiv_inv_eq
(p : Ω(fiber_sequence_carrier f n)) : (fiber_sequence_carrier_pequiv f n)⁻¹ᵉ* p =
fiber.mk (fiber.mk pt (respect_pt (fiber_sequence_fun f n) ⬝ p)) idp :=
fiber_sequence_carrier_equiv_inv_eq f n p
attribute pequiv._trans_of_to_pmap [unfold 3]
/- Lemma 8.4.4(iii) -/
definition fiber_sequence_fun_eq_helper
(p : Ω(fiber_sequence_carrier f (n + 1))) :
fiber_sequence_carrier_pequiv f n
(fiber_sequence_fun f (n + 3)
((fiber_sequence_carrier_pequiv f (n + 1))⁻¹ᵉ* p)) =
ap1 (fiber_sequence_fun f n) p⁻¹ :=
begin
refine ap (λx, fiber_sequence_carrier_pequiv f n (fiber_sequence_fun f (n + 3) x))
(fiber_sequence_carrier_pequiv_inv_eq f (n+1) p) ⬝ _,
/- the following three lines are rewriting some reflexivities: -/
-- replace (n + 3) with (n + 2 + 1),
-- refine ap (fiber_sequence_carrier_pequiv f n)
-- (fiber_sequence_fun_eq1 f (n+2) _ idp) ⬝ _,
refine fiber_sequence_carrier_pequiv_eq f n pt (respect_pt (fiber_sequence_fun f n)) _ ⬝ _,
esimp,
apply whisker_right,
apply whisker_left,
apply ap02, apply inverse2, apply idp_con,
end
theorem fiber_sequence_carrier_pequiv_eq_point_eq_idp :
fiber_sequence_carrier_pequiv_eq f n
(Point (fiber_sequence_carrier f (n+1)))
(respect_pt (fiber_sequence_fun f n))
(respect_pt (fiber_sequence_fun f (n + 1))) = idp :=
begin
apply con_inv_eq_idp,
refine ap (λx, whisker_left _ (_ ⬝ x)) _ ⬝ _,
{ reflexivity},
{ reflexivity},
esimp,
refine ap (whisker_left _)
(transport_eq_Fl_idp_left (fiber_sequence_fun f n)
(respect_pt (fiber_sequence_fun f n))) ⬝ _,
apply whisker_left_idp_con_eq_assoc
end
theorem fiber_sequence_fun_phomotopy_helper :
(fiber_sequence_carrier_pequiv f n ∘*
fiber_sequence_fun f (n + 3)) ∘*
(fiber_sequence_carrier_pequiv f (n + 1))⁻¹ᵉ* ~*
ap1 (fiber_sequence_fun f n) ∘* pinverse :=
begin
fapply phomotopy.mk,
{ exact (fiber_sequence_fun_eq_helper f n)},
{ esimp, rewrite [idp_con], refine _ ⬝ whisker_left _ !idp_con⁻¹,
apply whisker_right,
apply whisker_left,
exact fiber_sequence_carrier_pequiv_eq_point_eq_idp f n}
end
theorem fiber_sequence_fun_eq : Π(x : fiber_sequence_carrier f (n + 4)),
fiber_sequence_carrier_pequiv f n (fiber_sequence_fun f (n + 3) x) =
ap1 (fiber_sequence_fun f n) (fiber_sequence_carrier_pequiv f (n + 1) x)⁻¹ :=
begin
apply homotopy_of_inv_homotopy_pre (fiber_sequence_carrier_pequiv f (n + 1)),
apply fiber_sequence_fun_eq_helper f n
end
theorem fiber_sequence_fun_phomotopy :
fiber_sequence_carrier_pequiv f n ∘*
fiber_sequence_fun f (n + 3) ~*
(ap1 (fiber_sequence_fun f n) ∘* pinverse) ∘* fiber_sequence_carrier_pequiv f (n + 1) :=
begin
apply phomotopy_of_pinv_right_phomotopy,
apply fiber_sequence_fun_phomotopy_helper
end
definition boundary_map : Ω Y →* pfiber f :=
fiber_sequence_fun f 2 ∘* (fiber_sequence_carrier_pequiv f 0)⁻¹ᵉ*
/--------------
PART 2
@ -324,7 +122,7 @@ namespace chain_complex
| (k+3) :=
begin
refine fiber_sequence_carrier_pequiv f k ⬝e* _,
apply loop_space_pequiv,
apply loop_pequiv_loop,
exact fiber_sequence_pequiv_homotopy_groups k
end
@ -486,7 +284,7 @@ namespace chain_complex
definition comm_group_LES_of_homotopy_groups (n : ℕ) : comm_group (LES_of_homotopy_groups f (n + 6)) :=
comm_group_homotopy_group 0 (homotopy_groups f n)
end chain_complex
end old end chain_complex
open group prod succ_str fin
@ -494,30 +292,7 @@ open group prod succ_str fin
PART 3
--------------/
namespace chain_complex
--TODO: move
definition tr_mul_tr {A : Type*} (n : ℕ) (p q : Ω[n + 1] A) :
tr p *[πg[n+1] A] tr q = tr (p ⬝ q) :=
by reflexivity
definition is_homomorphism_cast_loop_space_succ_eq_in {A : Type*} (n : ℕ) :
is_homomorphism
(cast (ap (trunc 0 ∘ pointed.carrier) (loop_space_succ_eq_in A (succ n)))
: πg[n+1+1] A → πg[n+1] Ω A) :=
begin
intro g h, induction g with g, induction h with h,
xrewrite [tr_mul_tr, - + fn_cast_eq_cast_fn _ (λn, tr), tr_mul_tr, ↑cast, -tr_compose,
loop_space_succ_eq_in_concat, - + tr_compose],
end
definition is_homomorphism_inverse (A : Type*) (n : ℕ)
: is_homomorphism (λp, p⁻¹ : πag[n+2] A → πag[n+2] A) :=
begin
intro g h, rewrite mul.comm,
induction g with g, induction h with h,
exact ap tr !con_inv
end
namespace chain_complex namespace old
section
universe variable u
@ -651,32 +426,32 @@ namespace chain_complex
-- uncomment the next two lines to have prettier subgoals
-- esimp, replace (succ 5 * (n + 1) + 0) with (6*n+3+3),
-- rewrite [+homotopy_groups_add3, homotopy_groups2_add1],
apply loop_space_pequiv, apply loop_space_pequiv,
apply loop_pequiv_loop, apply loop_pequiv_loop,
rexact homotopy_groups2_pequiv' n (fin.mk 0 H)
end
| (n+1) (fin.mk 1 H) :=
begin
apply loop_space_pequiv, apply loop_space_pequiv,
apply loop_pequiv_loop, apply loop_pequiv_loop,
rexact homotopy_groups2_pequiv' n (fin.mk 1 H)
end
| (n+1) (fin.mk 2 H) :=
begin
apply loop_space_pequiv, apply loop_space_pequiv,
apply loop_pequiv_loop, apply loop_pequiv_loop,
rexact homotopy_groups2_pequiv' n (fin.mk 2 H)
end
| (n+1) (fin.mk 3 H) :=
begin
apply loop_space_pequiv, apply loop_space_pequiv,
apply loop_pequiv_loop, apply loop_pequiv_loop,
rexact homotopy_groups2_pequiv' n (fin.mk 3 H)
end
| (n+1) (fin.mk 4 H) :=
begin
apply loop_space_pequiv, apply loop_space_pequiv,
apply loop_pequiv_loop, apply loop_pequiv_loop,
rexact homotopy_groups2_pequiv' n (fin.mk 4 H)
end
| (n+1) (fin.mk 5 H) :=
begin
apply loop_space_pequiv, apply loop_space_pequiv,
apply loop_pequiv_loop, apply loop_pequiv_loop,
rexact homotopy_groups2_pequiv' n (fin.mk 5 H)
end
| n (fin.mk (k+6) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
@ -744,9 +519,9 @@ namespace chain_complex
induction n with n IH,
{ refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹*,
refine !comp_pid⁻¹* ⬝* pconcat2 _ _,
{ exact !comp_pid⁻¹*},
{ refine cast (ap (λx, _ ~* loop_space_pequiv x) !loop_space_pequiv_rfl)⁻¹ _,
refine cast (ap (λx, _ ~* x) !loop_space_pequiv_rfl)⁻¹ _, reflexivity}},
{ exact (comp_pid (ap1 (boundary_map f) ∘* pinverse))⁻¹*},
{ refine cast (ap (λx, _ ~* loop_pequiv_loop x) !loop_pequiv_loop_rfl)⁻¹ _,
refine cast (ap (λx, _ ~* x) !loop_pequiv_loop_rfl)⁻¹ _, reflexivity}},
{ refine _ ⬝* !comp_pid⁻¹*,
refine _ ⬝* pwhisker_right _ (!homotopy_groups_fun2_add1_5)⁻¹*,
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
@ -988,4 +763,5 @@ namespace chain_complex
--TODO: the maps 3, 4 and 5 are anti-homomorphisms.
end old
end chain_complex

40
homotopy/chain_complex.hlean
View file

@ -5,7 +5,7 @@ Authors: Floris van Doorn
-/
import types.int types.pointed2 types.trunc algebra.hott ..group_theory.basic .fin
import types.int types.pointed types.trunc algebra.hott ..group_theory.basic .fin
open eq pointed int unit is_equiv equiv is_trunc trunc function algebra group sigma.ops
sum prod nat bool fin
@ -127,7 +127,7 @@ namespace chain_complex
-- : is_exact_at_t (type_chain_complex_from_left X) n :=
-- H
definition transfer_type_chain_complex [constructor]
definition transfer_type_chain_complex [constructor] /- X -/
{Y : N → Type*} (g : Π{n : N}, Y (S n) →* Y n) (e : Π{n}, X n ≃* Y n)
(p : Π{n} (x : X (S n)), e (tcc_to_fn X n x) = g (e x)) : type_chain_complex N :=
type_chain_complex.mk Y @g
@ -158,6 +158,41 @@ namespace chain_complex
apply right_inv
end
-- cancel automorphisms inside a long exact sequence
definition type_chain_complex_cancel_aut [constructor] /- X -/
(g : Π{n : N}, X (S n) →* X n) (e : Π{n}, X n ≃* X n)
(r : Π{n}, X n →* X n)
(p : Π{n : N} (x : X (S n)), g (e x) = tcc_to_fn X n x)
(pr : Π{n : N} (x : X (S n)), g x = r (g (e x))) : type_chain_complex N :=
type_chain_complex.mk X @g
abstract begin
have p' : Π{n : N} (x : X (S n)), g x = tcc_to_fn X n (e⁻¹ x),
from λn, homotopy_inv_of_homotopy_pre e _ _ p,
intro n x,
refine ap g !p' ⬝ !pr ⬝ _,
refine ap r !p ⬝ _,
refine ap r (tcc_is_chain_complex X n _) ⬝ _,
apply respect_pt
end end
theorem is_exact_at_t_cancel_aut {X : type_chain_complex N}
{g : Π{n : N}, X (S n) →* X n} {e : Π{n}, X n ≃* X n}
{r : Π{n}, X n →* X n} (l : Π{n}, X n →* X n)
(p : Π{n : N} (x : X (S n)), g (e x) = tcc_to_fn X n x)
(pr : Π{n : N} (x : X (S n)), g x = r (g (e x)))
(pl : Π{n : N} (x : X (S n)), g (l x) = e (g x))
(H : is_exact_at_t X n) : is_exact_at_t (type_chain_complex_cancel_aut X @g @e @r @p @pr) n :=
begin
intro y q, esimp at *,
have H2 : tcc_to_fn X n (e⁻¹ y) = pt,
from (homotopy_inv_of_homotopy_pre e _ _ p _)⁻¹ ⬝ q,
cases (H _ H2) with x s,
refine fiber.mk (l (e x)) _,
refine !pl ⬝ _,
refine ap e (!p ⬝ s) ⬝ _,
apply right_inv
end
-- move to init.equiv. This is inv_commute for A ≡ unit
definition inv_commute1' {B C : Type} (f : B → C) [is_equiv f] (h : B → B) (h' : C → C)
(p : Π(b : B), f (h b) = h' (f b)) (c : C) : f⁻¹ (h' c) = h (f⁻¹ c) :=
@ -167,6 +202,7 @@ namespace chain_complex
(p : Π(b : B), f (h b) = h' (f b)) (c : C) : f⁻¹ (h' c) = h (f⁻¹ c) :=
inv_commute1' (to_fun f) h h' p c
-- move
definition fn_cast_eq_cast_fn {A : Type} {P Q : A → Type} {x y : A} (p : x = y)
(f : Πx, P x → Q x) (z : P x) : f y (cast (ap P p) z) = cast (ap Q p) (f x z) :=
by induction p; reflexivity

7
homotopy/sec86.hlean
View file

@ -267,10 +267,3 @@ namespace sphere
-- end
end sphere
/-
changes in book:
proof 8.6.15: also mention that we ignore multiplication
proof 8.4.4: respects points
proof 8.4.8: do k=0 separately
-/

16
homotopy/spectrum.hlean
View file

@ -5,7 +5,7 @@ Authors: Michael Shulman
-/
import types.int types.pointed2 types.trunc homotopy.susp algebra.homotopy_group .chain_complex cubical
import types.int types.pointed types.trunc homotopy.susp algebra.homotopy_group .chain_complex cubical
open eq nat int susp pointed pmap sigma is_equiv equiv fiber algebra trunc trunc_index pi
/-----------------------------------------
@ -136,7 +136,7 @@ namespace pointed
definition pequiv_postcompose {A B B' : Type*} (f : A →* B) (g : B ≃* B') : pfiber (g ∘* f) ≃* pfiber f :=
begin
fapply pequiv_of_equiv, esimp,
fapply pequiv_of_equiv, esimp,
refine ((transport_fiber_equiv (g ∘* f) (respect_pt g)⁻¹) ⬝e (@fiber.equiv_postcompose A B f (Point B) B' g _)),
esimp, apply (ap (fiber.mk (Point A))), rewrite con.assoc, apply inv_con_eq_of_eq_con,
rewrite [con.assoc, con.right_inv, con_idp, -ap_compose'], apply ap_con_eq_con
@ -144,7 +144,7 @@ namespace pointed
definition pequiv_precompose {A A' B : Type*} (f : A →* B) (g : A' ≃* A) : pfiber (f ∘* g) ≃* pfiber f :=
begin
fapply pequiv_of_equiv, esimp,
fapply pequiv_of_equiv, esimp,
refine (@fiber.equiv_precompose A B f (Point B) A' g _),
esimp, apply (eq_of_fn_eq_fn (fiber.sigma_char _ _)), fapply sigma_eq: esimp,
{ apply respect_pt g },
@ -157,13 +157,11 @@ namespace pointed
... ≃* pfiber (g ∘* h) : pfiber_equiv_of_phomotopy s
... ≃* pfiber g : pequiv_precompose
definition ppi {A : Type} (P : A → Type*) : Type* :=
pointed.mk' (Πa, P a)
definition loop_ppi_commute {A : Type} (B : A → Type*) : Ω(ppi B) ≃* (ppi (λa, Ω (B a))) :=
definition loop_ppi_commute {A : Type} (B : A → Type*) : Ω(ppi B) ≃* Π*a, Ω (B a) :=
pequiv_of_equiv eq_equiv_homotopy rfl
definition equiv_ppi_right {A : Type} {P Q : A → Type*} (g : Πa, P a ≃* Q a) : ppi P ≃* ppi Q :=
definition equiv_ppi_right {A : Type} {P Q : A → Type*} (g : Πa, P a ≃* Q a)
: (Π*a, P a) ≃* (Π*a, Q a) :=
pequiv_of_equiv (pi_equiv_pi_right g)
begin esimp, apply eq_of_homotopy, intros a, esimp, exact (respect_pt (g a)) end
@ -206,7 +204,7 @@ namespace spectrum
definition glue {{N : succ_str}} := @gen_prespectrum.glue N
--definition glue := (@gen_prespectrum.glue +ℤ)
definition equiv_glue {N : succ_str} (E : gen_prespectrum N) [H : is_spectrum E] (n:N) : (E n) ≃* (Ω (E (S n))) :=
definition equiv_glue {N : succ_str} (E : gen_prespectrum N) [H : is_spectrum E] (n:N) : (E n) ≃* (Ω (E (S n))) :=
pequiv_of_pmap (glue E n) (is_spectrum.is_equiv_glue E n)
-- Sometimes an ℕ-indexed version does arise naturally, however, so