definition of cohomology groups

Signed-off-by: Ali Caglayan <alizter@gmail.com>

<!-- ps-id: 576d0fe0-7342-497e-bd1e-e42b31c19ba8 -->

Author: Alizter

theories/Homotopy/Cohomology.v

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+Require Import Basics Types Pointed WildCat.Core WildCat.Equiv.
+Require Import Truncations.Core.
+Require Import Homotopy.EMSpace.
+Require Import Homotopy.HSpace.Core Homotopy.HSpace.Pointwise Homotopy.HSpace.HGroup Homotopy.HSpace.Coherent.
+Require Import Algebra.AbGroups.AbelianGroup.
+Require Import Homotopy.Suspension.
+Require Import Spheres HomotopyGroup.
+
+Local Open Scope pointed_scope.
+
+(** * Cohomology groups *)
+
+(** Reduced cohomology groups are defined as the homotopy classes of pointed maps from the space to an Eilenberg-MacLane space. The group structure comes from the H-space structure on [K(G, n)]. *)
+Definition Cohomology `{Univalence} (n : nat) (X : pType) (G : AbGroup) : AbGroup.
+Proof.
+  snrapply Build_AbGroup'.
+  1: exact (Tr 0 (X ->** K(G, n))).
+  1-3: shelve.
+  nrapply isabgroup_ishabgroup_tr.
+  nrapply ishabgroup_hspace_pmap. 
+  apply iscohhabgroup_em.
+Defined.
+
+(** ** Cohomology of suspension *)
+
+(** The (n+1)th cohomology of a suspension is the nth cohomology of the original space. *) 
+(* TODO: show this preserves the operation somehow and is therefore a group iso *)
+Definition cohomology_susp `{Univalence} n (X : pType) G
+  : Cohomology n.+1 (psusp X) G <~> Cohomology n X G.
+Proof.
+  apply Trunc_functor_equiv.
+  nrefine (_ oE (loop_susp_adjoint _ _)).
+  rapply pequiv_pequiv_postcompose.
+  symmetry.
+  apply pequiv_loops_em_em.
+Defined.
+
+(** ** Cohomology of spheres *)
+
+(* TODO: improve this to a group isomorphism once cohomology can be easily checked to be op preserving. *)
+Definition cohomology_sn `{Univalence} (n : nat) (G : AbGroup)
+  : Cohomology n.+1 (psphere n.+1) G <~> G.
+Proof.
+  nrefine (_ oE (equiv_tr 0 _)^-1).
+  1: refine ?[goal1].
+  2: { rapply istrunc_equiv_istrunc. symmetry. apply ?goal1. }
+  nrefine (_ oE pmap_from_psphere_iterated_loops _ _).
+  symmetry.
+  rapply pequiv_loops_em_g.
+Defined.
+