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test_cup_aplk_cochain_validations.py

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+from sage.all import *
+import itertools
+load('aplK.py')
+load("minimalModel_parallel.py")
+
+S = simplicial_complexes.K3Surface()
+S.set_immutable()
+cochainComplex = S.chain_complex(base_ring=QQ).dual()
+A = AlgebraAplK(S)
+smodel = load("Sullivan_model_K3_deg3.sobj")
+smorphism = load("model_morphism_K3_deg3.sobj")
+#cohomology_correspondence = load("cohomology_correspondence_K3_deg3.sobj")
+#smodel_basis_3 = smodel.basis(3)
+smodel_basis_2 = smodel.basis(2)
+#H = S.cohomology_ring(QQ)
+
+"""
+for g in smodel_basis_3:
+    a = smorphism[3][str(g)]
+    dg = g.differential()
+    print("AplK Differential validation: generator "+str(g), flush=True)
+    success = phi_polinomial(dg,smorphism,A) == a.differential()
+    if success:
+        print(success, flush=True)
+    else:
+        raise Exception("the differential and the model morphism do not commute") 
+    ####################  
+    cf = A.cochain_to_vector(a.integral(),3)
+    cg = A.cochain_to_vector(a.differential().integral(),4)
+    print("Cochain Complex Differential validation of element associated to generator "+str(g), flush=True)
+    success = cochainComplex.differential(3)*cf == cg
+    if success:
+        print(success, flush=True)
+    else:
+        raise Exception("the aplk and cochain complex differentials do not commute")
+"""
+from sage.homology.algebraic_topological_model import algebraic_topological_model
+phi, M = algebraic_topological_model(S, QQ)
+# https://doc.sagemath.org/pdf/en/reference/homology/homology.pdf
+
+"""
+H = S.cohomology_ring(base_ring=QQ)
+g20 = H.basis()[(2,0)]
+g21 = H.basis()[(2,1)]
+g20.cup_product(g20)
+2*h^{4,0}
+sage: g21.cup_product(g21)
+2*h^{4,0}
+
+for i in range(22):
+    for j in range(i,22):
+        g1 = H.basis()[(2,i)]
+        g2 = H.basis()[(2,j)]
+        hsuccess = g1.cup_product(g2) == g2.cup_product(g1)
+        if not hsuccess:
+        print("producto diferente: (", str(i)+","+str(j)+")", flush=True)
+
+"""
+
+"""
+sage: S = simplicial_complexes.K3Surface()
+sage: S
+Minimal triangulation of the K3 surface
+sage: C = S.chain_complex().dual()
+sage: H = S.cohomology_ring(base_ring=QQ)
+sage: H.contraction?
+sage: phi = H.contraction().dual()
+sage: p = phi.pi()
+sage: iota = phi.iota()
+sage: M = iota.domain()
+sage: p2 = p.to_matrix(2)
+sage: p2
+22 x 560 dense matrix over Rational Field (use the '.str()' method to see the entries)
+sage: g20 = H.basis()[(2,0)]
+sage: c20 = g20.to_cycle()
+sage: v20 = c20.to_vector()
+sage: m20 = p2*v20
+sage: m20
+(58, 14, 19, 31, -18, -6, -60, 61, 40, 2, 58, -26, -11, -14, 15, -110, 133, 20, -28, 68, -14, 15)
+sage: g21 = H.basis()[(2,1)]
+sage: c21 = g21.to_cycle()
+sage: v21 = c21.to_vector()
+sage: m21 = p2*v21
+sage: m = matrix(QQ,[m20,m21])
+sage: m.rank()
+2
+"""
+
+"""
+# coordinates of the cohomology generator h^(2,0)
+# equivalent to h^(2,0).to_cycle().to_vector()
+sage: vec = H.contraction().iota().in_degree(2).column(0) 
+sage: chains = S.n_chains(2, H.base_ring(), cochains=H._cohomology)
+sage: chains.from_vector(vec) 
+sage: chains.from_vector(vec) == g20.to_cycle()
+True
+"""
+
+"""
+sage: V = H.contraction().iota().in_degree(2).column_space()
+sage: V.coordinate_vector(g20.to_cycle().to_vector())
+(-1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0)
+##################################################################################
+sage: p = H.contraction().pi()
+sage: m = p.to_matrix(2)
+sage: m
+22 x 560 dense matrix over Rational Field (use the '.str()' method to see the entries)
+sage: v20 = g20.to_cycle().to_vector()
+sage: m*v20
+(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
+sage: v21 = c21.to_vector()
+sage: m*v21
+(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
+
+
+
+"""
+
+C2 = S.n_chains(2,QQ,cochains=True)
+basis2 = C2.basis()
+C4 = S.n_chains(4,QQ,cochains=True)
+basis4 = C4.basis()
+
+for i in range(len(smodel_basis_2)):
+    for j in range(i, len(smodel_basis_2)):
+        g1 = smodel_basis_2[i]
+        g2 = smodel_basis_2[j]
+        a1 = smorphism[2][str(g1)]
+        a2 = smorphism[2][str(g2)]
+        
+        cA1 = a1.integral()
+        dA1 = C2.zero()
+        for k in basis2.keys():
+            dA1 += basis2[k]*cA1[k.tuple()]
+        cA2 = a2.integral()
+        dA2 = C2.zero()
+        for k in basis2.keys():
+            dA2 += basis2[k]*cA2[k.tuple()]
+        dA12 = dA1.cup_product(dA2)
+        
+        a4 = a1*a2
+        dA4 = C4.zero()
+        cA4 = a4.integral()
+        for k in basis4.keys():
+            dA4 += basis4[k]*cA4[k.tuple()]
+        success = dA12 == dA4
+        if success:
+            print("cup product commutes for elements " + str(g1) + " and " + str(g2), flush=True)
+            print(success, flush=True)
+        else:
+            print("dA12", flush=True)
+            print(dA12)
+            print("dA4", flush=True)
+            print(dA4)
+            raise Exception("the aplk and cup products do not commute")
+