def main():
# Needed to output utf-8
sys.stdout = getwriter('utf-8')(sys.stdout)
# Set up commandline argument parsing
argparser = ArgumentParser(description="Computes induced coproduct on homology")
argparser.add_argument('--hgroups', '-hg', dest='homology_groups',
type=FileType('r'), help="File containing homology groups")
argparser.add_argument('file', type=file, help="LaTeX file to be parsed")
# Read the arguments (if available)
try:
args = argparser.parse_args()
except Exception as e:
print e.message
argparser.print_help()
raise SystemExit
# load file
data = args.file.read()
args.file.close()
# parse file contents
result = CellChainParse.parse(data)
if not result:
raise SystemExit
# construct coalgebra
C = Coalgebra(result["groups"], result["differentials"], result["coproducts"])
# convert coproduct to map form
delta_c = {}
for from_cell, to_cells in C.coproduct.iteritems():
delta_c[from_cell] = [cell for cell, count in to_cells.iteritems() if count % 2]
def delta_c_chain(chain):
return list_mod(reduce(lambda acc, cell: acc + delta_c[cell], chain, []))
"""
Check Coassociativity on Delta_C
"""
# (1 x Delta) Delta and (Delta x 1) Delta
id_x_Delta_Delta = {}
Delta_x_id_Delta = {}
for c, cxc in delta_c.iteritems():
id_x_Delta_Delta[c] = [(l,) + r_cp for (l, r) in cxc for r_cp in delta_c[r]]
Delta_x_id_Delta[c] = [l_cp + (r,) for (l, r) in cxc for l_cp in delta_c[l]]
# DeltaC = (1 x Delta_C + Delta_C x 1) Delta_C
id_x_Delta_Delta_x_id_Delta = add_maps_mod_2(id_x_Delta_Delta, Delta_x_id_Delta)
print DELTA + "_c is co-associative?", not any(id_x_Delta_Delta_x_id_Delta.values())
if any(id_x_Delta_Delta_x_id_Delta.values()):
print u"\n(1 " + OTIMES + " " + DELTA + " + " + DELTA + " " + OTIMES + " 1) " + DELTA + " =",
print format_morphism(id_x_Delta_Delta_x_id_Delta)
print u"\n(1 " + OTIMES + " " + DELTA + " + " + DELTA + " " + OTIMES + " 1) " + DELTA + " (factored) =",
print format_morphism({cell: factorize(factorize_cycles(chain, C), C) for cell, chain in id_x_Delta_Delta_x_id_Delta.iteritems()})
"""
COMPUTE HOMOLOGY
"""
if not args.homology_groups:
# create temporary file for CHomP
scratch = open(TEMP_MAT, 'w+')
differential = {n: C.incidence_matrix(n, sparse=False) for n in range(1, C.topDimension() + 1)}
def compare_incidences(x, y):
return x[1] - y[1] if x[1] != y[1] else x[0] - y[0]
for n, entries in differential.iteritems():
print >> scratch, n - 1
incidences = [(l, r, v % 2) for (l, r), v in entries.iteritems()]
for entry in ["{} {} {}".format(l, r, v) for (l, r, v) in sorted(incidences, cmp=compare_incidences)]:
print >> scratch, entry
scratch.close()
try:
chomp_results = check_output(["chomp-matrix", TEMP_MAT, "-g"])
#print chomp_results
except CalledProcessError as e:
print e.returncode
print e.output
print e.output
finally:
rm(TEMP_MAT) # clean up
lines = chomp_results.splitlines()
dims = [int(k) for k in compile('\d+').findall(lines[0])]
H_gens = {}
offset = 9 + len(dims)
for n, k in enumerate(dims):
H_gens[n] = [[C.groups[n][int(j)] for j in compile('\[(\d+)\]').findall(lines[offset + i])] for i in range(k)]
offset += k + 1
else:
try:
H_gens = eval(args.homology_groups.read())
except Exception as e:
print "Error: Unable to load homology groups!"
argparser.print_help()
raise SystemExit
args.homology_groups.close()
# verify that the provided homology generators are valid for this cell complex
for i, group in H_gens.iteritems():
for cells in group:
if not all([cell in C.groups[i] for cell in cells]):
print "Error: Invalid homology groups provided! Cells in generators not found in the complex."
raise SystemExit
d_cells = list_mod(derivative(cells, C))
if d_cells:
print "Error: Group generator in homology is not a cycle in the cell complex."
print "Dimension =", i, "; Generator =", cells
print "Derivative =", d_cells
raise SystemExit
H = {dim: ["h{}_{}".format(dim, i) for i, gen in enumerate(gens)] for dim, gens in H_gens.iteritems()}
print "\nH = H*(C) = ", H
"""
DEFINE MAPS BETWEEN CHAINS AND HOMOLOGY
"""
# Define g
g = {}
for dim, gens in H_gens.iteritems():
for i, gen in enumerate(gens):
if gen:
g["h{}_{}".format(dim, i)] = gen
print
print "g = ", format_morphism(g)
def g_tensor(tp):
return tuple(map(lambda c: g[c], list(tp)))
# generate f: C -> H
f, integrate = generate_f_integral(C, g)
def f_tensor(tp):
return tuple(map(f, list(tp)))
"""
COMPUTE Delta_2, g^2
"""
# define Delta g
delta_g = {h: delta_c_chain(cycle) for h, cycle in g.iteritems()}
print
print DELTA + u"g =", format_morphism(delta_g)
# print
# print DELTA + u"g (unsimplified) =", delta_g
factored_delta_g = {cell: factorize_cycles(chain, C) for cell, chain in delta_g.iteritems()}
print
print DELTA + u"g (factored) =", format_morphism(factored_delta_g)
delta2 = {}
for cell, chain in factored_delta_g.iteritems():
delta2[cell] = [f_tensor(cxc) for cxc in chain]
# flatten delta2 and remove any empty elements
delta2 = chain_map_mod(expand_map_all(delta2))
print
print DELTA + u"_2 =", format_morphism(delta2)
# however, we still need all keys to be available
for h in g.iterkeys():
if h not in delta2:
delta2[h] = []
# (g x g) Delta2
gxgDelta = {}
for cell, chain in delta2.iteritems():
gxgDelta[cell] = [g_tensor(cxc) for cxc in chain]
gxgDelta = chain_map_mod(expand_map_all(gxgDelta))
print
print u"(g " + OTIMES + " g)" + DELTA + "^2 =", format_morphism(gxgDelta)
# nabla g^2
nabla_g2 = add_maps_mod_2(gxgDelta, delta_g)
print
print u"(g " + OTIMES + " g)" + DELTA + "^2 + " + DELTA + "g =", format_morphism(nabla_g2)
# g^2
g2 = {cell: integrate(chain) for cell, chain in nabla_g2.iteritems()}
print
print u"g^2 =", format_morphism(g2)
"""
VERIFY CONSISTENCY OF DELTA2, g^2 RESULTS
"""
nabla_g2_computed = {cell: derivative(chain, C) for cell, chain in g2.iteritems() if chain}
nabla_g2_computed = chain_map_mod(expand_map_all(nabla_g2_computed))
print
print NABLA + u" g^2 =", format_morphism(nabla_g2_computed)
print u"\n(g " + OTIMES + " g)" + DELTA + "^2 + " + DELTA + "g + " + NABLA + "g^2 = 0 ? ",
print not any(add_maps_mod_2(nabla_g2_computed, nabla_g2).values())
"""
VERIFY DELTA2 COASSOCIATIVITY
"""
print "\nChecking coassociativity...\n"
# (1 x Delta_2) Delta_2
# (Delta_2 x 1) Delta_2
id_x_Delta2_Delta2 = {}
Delta2_x_id_Delta2 = {}
for h, hxhs in delta2.iteritems():
id_x_Delta2_Delta2[h] = [(l,) + r_cp for (l, r) in hxhs for r_cp in delta2[r]]
Delta2_x_id_Delta2[h] = [l_cp + (r,) for (l, r) in hxhs for l_cp in delta2[l]]
id_x_Delta2_Delta2 = chain_map_mod(expand_map_all(id_x_Delta2_Delta2))
Delta2_x_id_Delta2 = chain_map_mod(expand_map_all(Delta2_x_id_Delta2))
print u"\n(1 " + OTIMES + " " + DELTA + "^2) " + DELTA + "^2 =", format_morphism(id_x_Delta2_Delta2)
print u"\n(" + DELTA + "^2 " + OTIMES + " 1) " + DELTA + "^2 =", format_morphism(Delta2_x_id_Delta2)
# z_1 = (1 x Delta_2 + Delta_2 x 1) Delta_2
z1 = add_maps_mod_2(id_x_Delta2_Delta2, Delta2_x_id_Delta2)
print u"\n" + DELTA + "^2 is co-associative?", not any(z1.values())
if any(z1.values()):
print "\nz1 = " + NABLA + "(" + DELTA + "^3) =",
print format_morphism(z1)
print "\nz1 = " + NABLA + "(" + DELTA + "^3) (factored) =",
print format_morphism({cell: factorize(factorize_cycles(chain, C), C) for cell, chain in z1.iteritems()})
"""
COMPUTE DELTA_C_3
"""
# Delta_c3
delta_c3 = integrate(id_x_Delta_Delta_x_id_Delta)
delta_c3 = chain_map_mod(expand_map_all(delta_c3))
print
print DELTA + u"_C3 =", format_morphism(delta_c3)
# however, we still need all keys to be available
for cell in delta_c.iterkeys():
if cell not in delta_c3:
delta_c3[cell] = []
# verify consistency
nabla_delta_c3_computed = derivative(delta_c3, C)
nabla_delta_c3_computed = chain_map_mod(expand_map_all(nabla_delta_c3_computed))
print
print NABLA + DELTA + u"_C3 =", format_morphism(nabla_delta_c3_computed)
print u"\n(1 " + OTIMES + " " + DELTA + " + " + DELTA + " " + OTIMES + " 1) " + DELTA + " + " + NABLA + DELTA + u"_C3 = 0 ? ",
print not any(add_maps_mod_2(id_x_Delta_Delta_x_id_Delta, nabla_delta_c3_computed).values())
if any(add_maps_mod_2(id_x_Delta_Delta_x_id_Delta, nabla_delta_c3_computed).values()):
print u"\n(1 " + OTIMES + " " + DELTA + " + " + DELTA + " " + OTIMES + " 1) " + DELTA + " + " + NABLA + DELTA + u"_C3 =",
print format_morphism(add_maps_mod_2(id_x_Delta_Delta_x_id_Delta, nabla_delta_c3_computed))
"""
COMPUTE DELTA3, g^3
"""
# (1 x Delta) g^2 and (Delta x 1) g^2
id_x_Delta_g2 = {}
Delta_x_id_g2 = {}
for h, cxcs in g2.iteritems():
id_x_Delta_g2[h] = [(l,) + r_cp for (l, r) in cxcs for r_cp in delta_c[r]]
Delta_x_id_g2[h] = [l_cp + (r,) for (l, r) in cxcs for l_cp in delta_c[l]]
id_x_Delta_g2 = chain_map_mod(id_x_Delta_g2)
Delta_x_id_g2 = chain_map_mod(Delta_x_id_g2)
print u"\n(1 " + OTIMES + " " + DELTA + ") g^2 =", format_morphism(id_x_Delta_g2)
print u"\n(" + DELTA + " " + OTIMES + " 1) g^2 =", format_morphism(Delta_x_id_g2)
# (g x g^2) Delta_2 and (g^2 x g) Delta_2
g_x_g2_Delta2 = {}
g2_x_g_Delta2 = {}
for h, hxhs in delta2.iteritems():
g_x_g2_Delta2[h] = [(l_cp,) + r_cp for l, r in hxhs for l_cp in g[l] for r_cp in g2[r]]
g2_x_g_Delta2[h] = [l_cp + (r_cp,) for l, r in hxhs for l_cp in g2[l] for r_cp in g[r]]
g_x_g2_Delta2 = chain_map_mod(g_x_g2_Delta2)
g2_x_g_Delta2 = chain_map_mod(g2_x_g_Delta2)
print u"\n( g " + OTIMES + " g^2 ) " + DELTA + "^2 =", format_morphism(g_x_g2_Delta2)
print u"\n( g^2 " + OTIMES + " g ) " + DELTA + "^2 =", format_morphism(g2_x_g_Delta2)
# delta_c3_g
delta_c3_g = {}
for h, chain in g.iteritems():
delta_c3_g[h] = []
for cell in chain:
if cell in delta_c3:
delta_c3_g[h] += delta_c3[cell]
delta_c3_g = chain_map_mod(delta_c3_g)
print u"\n" + DELTA + "_c3 g =", format_morphism(delta_c3_g)
# phi_1 = (1 x Delta + Delta x 1) g^2 + (g x g^2 + g^2 x g) Delta_2
phi_1 = reduce(add_maps_mod_2, [g_x_g2_Delta2, g2_x_g_Delta2, id_x_Delta_g2, Delta_x_id_g2, delta_c3_g], {})
print u"\n" + PHI + u"_1 = (g " + OTIMES + " g^2 + g^2 " + OTIMES + " g) " + DELTA + "^2 +",
print u"(1 " + OTIMES + " " + DELTA + " + " + DELTA + " " + OTIMES + " 1) g^2 + " + DELTA + "_c3 g =",
print format_morphism(phi_1)
# Nabla phi_1 == 0 ? (Verify consistency)
nabla_phi_1 = {}
for h, chain in phi_1.iteritems():
nabla_phi_1[h] = derivative(chain, C)
nabla_phi_1 = chain_map_mod(expand_map_all(nabla_phi_1))
print "\n" + NABLA + PHI + u"_1 =", format_morphism(nabla_phi_1)
# factor phi_1
factored_phi_1 = {h: factorize_cycles(chain, C) for h, chain in phi_1.iteritems() if chain}
print "\n" + PHI + u"_1 (factored) =", format_morphism(factored_phi_1)
delta3 = {h: [f_tensor(cxcxc) for cxcxc in cycles] for h, cycles in factored_phi_1.iteritems()}
# flatten delta3 and remove any empty elements (mod 2)
delta3 = chain_map_mod(expand_map_all(delta3))
print "\n" + DELTA + u"_3 =", format_morphism(delta3)
# however, we still need all keys to be available
for h in g.iterkeys():
if h not in delta3:
delta3[h] = []
"""
Check coassociativity of Delta^3
"""
print "\nChecking coassociativity...\n"
# (Delta_3 x 1 + 1 x Delta_3) Delta_2
delta3_x_id_delta2 = {}
id_x_delta3_delta2 = {}
for h, hxhs in delta2.iteritems():
delta3_x_id_delta2[h] = [l_cp + (r, ) for (l, r) in hxhs for l_cp in delta3[l]]
id_x_delta3_delta2[h] = [(l, ) + r_cp for (l, r) in hxhs for r_cp in delta3[r]]
delta3_x_id_delta2 = chain_map_mod(expand_map_all(delta3_x_id_delta2))
id_x_delta3_delta2 = chain_map_mod(expand_map_all(id_x_delta3_delta2))
print u"\n( " + DELTA + "^3 " + OTIMES + " 1 ) " + DELTA + "^2 =", format_morphism(delta3_x_id_delta2)
print u"\n( 1 " + OTIMES, DELTA + "^3 ) " + DELTA + "^2 =", format_morphism(id_x_delta3_delta2)
# (1 x 1 x Delta_c2) Delta_c3 ## (1 x 1 x Delta_c2) Delta_c3 ## (1 x 1 x Delta_c2) Delta_c3 #
id_x_id_x_delta2_delta3 = {}
id_x_delta2_x_id_delta3 = {}
delta2_x_id_x_id_delta3 = {}
for h, hxhxhs in delta3.iteritems():
id_x_id_x_delta2_delta3[h] = [(l, m) + r_cp for (l, m, r) in hxhxhs for r_cp in delta2[r]]
id_x_delta2_x_id_delta3[h] = [(l, ) + m_cp + (r, ) for (l, m, r) in hxhxhs for m_cp in delta2[m]]
delta2_x_id_x_id_delta3[h] = [l_cp + (m, r) for (l, m, r) in hxhxhs for l_cp in delta2[l]]
id_x_id_x_delta2_delta3 = chain_map_mod(expand_map_all(id_x_id_x_delta2_delta3))
id_x_delta2_x_id_delta3 = chain_map_mod(expand_map_all(id_x_delta2_x_id_delta3))
delta2_x_id_x_id_delta3 = chain_map_mod(expand_map_all(delta2_x_id_x_id_delta3))
print u"\n( 1 " + OTIMES + " 1 " + OTIMES, DELTA + "^2 ) " + DELTA + "^3 =", format_morphism(id_x_id_x_delta2_delta3)
print u"\n( 1 " + OTIMES, DELTA + "^2 " + OTIMES + " 1 ) " + DELTA + "^3 =", format_morphism(id_x_delta2_x_id_delta3)
print u"\n( " + DELTA + "^2 " + OTIMES + " 1 " + OTIMES + " 1 ) " + DELTA + "^3 =", format_morphism(delta2_x_id_x_id_delta3)
z2 = reduce(add_maps_mod_2, [
delta3_x_id_delta2, id_x_delta3_delta2,
id_x_id_x_delta2_delta3, id_x_delta2_x_id_delta3, delta2_x_id_x_id_delta3], {})
# DeltaC = (1 x Delta_C + Delta_C x 1) Delta_C
print "\n" + DELTA + "^3 is co-associative?", not any(z2.values())
if any(z2.values()):
print "\n" + NABLA + "(" + DELTA + "^4) =",
print format_morphism(z2)
print "\n" + NABLA + "(" + DELTA + "^4) (factored) =",
print format_morphism({cell: factorize(factorize_cycles(chain, C), C) for cell, chain in z2.iteritems()})
"""
Compute g^3
"""
# (g x g x g) Delta3
gxgxg_delta3 = {}
for h, chain in delta3.iteritems():
gxgxg_delta3[h] = [g_tensor(hxhxh) for hxhxh in chain]
gxgxg_delta3 = chain_map_mod(expand_map_all(gxgxg_delta3))
print u"\n(g " + OTIMES + " g " + OTIMES + " g)" + DELTA + "^3 =", format_morphism(gxgxg_delta3)
# nabla g^3
nabla_g3 = add_maps_mod_2(gxgxg_delta3, phi_1)
print u"\n(g " + OTIMES + " g " + OTIMES + " g)" + DELTA + "^3 + " + PHI + "_1 =", format_morphism(nabla_g3)
# g^3
g3 = {h: chain_integrate(chain, C) for h, chain in nabla_g3.iteritems()}
g3 = chain_map_mod(expand_map_all(g3))
print u"\ng^3 =", format_morphism(g3)
# however, we still need all keys to be available
for h in g.iterkeys():
if h not in g3:
g3[h] = []
"""
VERIFY CONSISTENCY OF phi_1, Delta^3, and g^3
"""
nabla_g3_computed = {h: derivative(chain, C) for h, chain in g3.iteritems() if chain}
nabla_g3_computed = chain_map_mod(expand_map_all(nabla_g3_computed))
print "\n" + NABLA + u" g^3 =", format_morphism(nabla_g3_computed)
print u"(g " + OTIMES + " g " + OTIMES + " g)" + DELTA + "^3 + " + PHI + "_1 + " + NABLA + "g^3 = 0 ? ",
print not any(reduce(add_maps_mod_2, [gxgxg_delta3, nabla_g3_computed, phi_1], {}).values())
if any(reduce(add_maps_mod_2, [gxgxg_delta3, nabla_g3_computed, phi_1], {}).values()):
print "\t", reduce(add_maps_mod_2, [gxgxg_delta3, nabla_g3_computed, phi_1], {})
"""
COMPUTE \Delta_C4
"""
print "\n\nComputing", DELTA + u"_C4...\n\n"
# (Delta_c3 x 1 + 1 x Delta_c3) Delta_c2
delta_c3_x_id_delta_c = {}
id_x_delta_c3_delta_c = {}
for cell, cxcs in delta_c.iteritems():
delta_c3_x_id_delta_c[cell] = [l_cp + (r, ) for (l, r) in cxcs for l_cp in delta_c3[l]]
id_x_delta_c3_delta_c[cell] = [(l, ) + r_cp for (l, r) in cxcs for r_cp in delta_c3[r]]
delta_c3_x_id_delta_c = chain_map_mod(expand_map_all(delta_c3_x_id_delta_c))
id_x_delta_c3_delta_c = chain_map_mod(expand_map_all(id_x_delta_c3_delta_c))
print u"\n( " + DELTA + "_c3 " + OTIMES + " 1 ) " + DELTA + "_c2 =", format_morphism(delta_c3_x_id_delta_c)
print u"\n( 1 " + OTIMES, DELTA + "_c3 ) " + DELTA + "_c2 =", format_morphism(id_x_delta_c3_delta_c)
# (1 x 1 x Delta_c2) Delta_c3 ## (1 x 1 x Delta_c2) Delta_c3 ## (1 x 1 x Delta_c2) Delta_c3 #
id_x_id_x_delta_c2_delta_c3 = {}
id_x_delta_c2_x_id_delta_c3 = {}
delta_c2_x_id_x_id_delta_c3 = {}
for cell, cxcxcs in delta_c3.iteritems():
id_x_id_x_delta_c2_delta_c3[cell] = [(l, m) + r_cp for (l, m, r) in cxcxcs for r_cp in delta_c[r]]
id_x_delta_c2_x_id_delta_c3[cell] = [(l, ) + m_cp + (r, ) for (l, m, r) in cxcxcs for m_cp in delta_c[m]]
delta_c2_x_id_x_id_delta_c3[cell] = [l_cp + (m, r) for (l, m, r) in cxcxcs for l_cp in delta_c[l]]
id_x_id_x_delta_c2_delta_c3 = chain_map_mod(expand_map_all(id_x_id_x_delta_c2_delta_c3))
id_x_delta_c2_x_id_delta_c3 = chain_map_mod(expand_map_all(id_x_delta_c2_x_id_delta_c3))
delta_c2_x_id_x_id_delta_c3 = chain_map_mod(expand_map_all(delta_c2_x_id_x_id_delta_c3))
print u"\n( 1 " + OTIMES + " 1 " + OTIMES, DELTA + "_c2 ) " + DELTA + "_c3 =", format_morphism(id_x_id_x_delta_c2_delta_c3)
print u"\n( 1 " + OTIMES, DELTA + "_c2 " + OTIMES + " 1 ) " + DELTA + "_c3 =", format_morphism(id_x_delta_c2_x_id_delta_c3)
print u"\n( " + DELTA + "_c2 " + OTIMES + " 1 " + OTIMES + " 1 ) " + DELTA + "_c3 =", format_morphism(delta_c2_x_id_x_id_delta_c3)
nabla_delta_c4 = reduce(add_maps_mod_2, [
delta_c3_x_id_delta_c, id_x_delta_c3_delta_c,
id_x_id_x_delta_c2_delta_c3, id_x_delta_c2_x_id_delta_c3, delta_c2_x_id_x_id_delta_c3], {})
# DeltaC = (1 x Delta_C + Delta_C x 1) Delta_C
print DELTA + "_c3 is co-associative?", not any(nabla_delta_c4.values())
if any(nabla_delta_c4.values()):
print "\n" + NABLA + "(" + DELTA + "_C4) =",
print format_morphism(nabla_delta_c4)
print "\n" + NABLA + "(" + DELTA + "_C4) (factored) =",
print format_morphism({cell: factorize(factorize_cycles(chain, C), C) for cell, chain in nabla_delta_c4.iteritems()})
# Delta_c4
delta_c4 = integrate(nabla_delta_c4)
delta_c4 = chain_map_mod(expand_map_all(delta_c4))
print
print DELTA + u"_C4 =", format_morphism(delta_c4)
# verify consistency
nabla_delta_c4_computed = derivative(delta_c4, C)
nabla_delta_c4_computed = chain_map_mod(expand_map_all(nabla_delta_c4_computed))
print
print NABLA + DELTA + u"_C4 =", format_morphism(nabla_delta_c4_computed)
print u"\n(1 " + OTIMES + " " + DELTA + " + " + DELTA + " " + OTIMES + " 1) " + DELTA + " + " + NABLA + DELTA + u"_C3 = 0 ? ",
print not any(add_maps_mod_2(nabla_delta_c4, nabla_delta_c4_computed).values())
if any(add_maps_mod_2(nabla_delta_c4, nabla_delta_c4_computed).values()):
print u"\n(1 " + OTIMES + " " + DELTA + " + " + DELTA + " " + OTIMES + " 1) " + DELTA + " + " + NABLA + DELTA + u"_C3 =",
print format_morphism(add_maps_mod_2(nabla_delta_c4, nabla_delta_c4_computed))
"""
COMPUTE \Phi_2, \Delta4
"""
#####################
# Facets of J_4
#####################
# (Delta_C4) g #
delta_c4_g = {}
for h, chain in g.iteritems():
delta_c4_g[h] = []
for cell in chain:
if cell in delta_c4:
delta_c4_g[h] += delta_c4[cell]
delta_c4_g = chain_map_mod(delta_c4_g)
print u"\n" + DELTA + "_c4 g =", format_morphism(delta_c4_g)
# (1 x Delta_C3) g^2 ## (Delta_C3 x 1) g^2 #
id_x_Delta_c3_g2 = {}
Delta_c3_x_id_g2 = {}
for h, cxcs in g2.iteritems():
id_x_Delta_c3_g2[h] = [(l, ) + r_cp for (l, r) in cxcs for r_cp in delta_c3[r]]
Delta_c3_x_id_g2[h] = [l_cp + (r, ) for (l, r) in cxcs for l_cp in delta_c3[l]]
id_x_Delta_c3_g2 = chain_map_mod(id_x_Delta_c3_g2)
Delta_c3_x_id_g2 = chain_map_mod(Delta_c3_x_id_g2)
print u"\n(1 " + OTIMES + " " + DELTA + "_C3) g^2 =", format_morphism(id_x_Delta_c3_g2)
print u"\n( " + DELTA + "_C3 " + OTIMES + " 1) g^2 =", format_morphism(Delta_c3_x_id_g2)
# (1 x 1 x Delta) g^3 ## (1 x Delta x 1) g^3 ## (Delta x 1 x 1) g^3 #
id_x_id_x_Delta_g3 = {}
id_x_Delta_x_id_g3 = {}
Delta_x_id_x_id_g3 = {}
for h, cxcxcs in g3.iteritems():
id_x_id_x_Delta_g3[h] = [(l, m) + r_cp for (l, m, r) in cxcxcs for r_cp in delta_c[r]]
id_x_Delta_x_id_g3[h] = [(l, ) + m_cp + (r, ) for (l, m, r) in cxcxcs for m_cp in delta_c[m]]
Delta_x_id_x_id_g3[h] = [l_cp + (m, r) for (l, m, r) in cxcxcs for l_cp in delta_c[l]]
id_x_id_x_Delta_g3 = chain_map_mod(id_x_id_x_Delta_g3)
id_x_Delta_x_id_g3 = chain_map_mod(id_x_Delta_x_id_g3)
Delta_x_id_x_id_g3 = chain_map_mod(Delta_x_id_x_id_g3)
print u"\n(1 " + OTIMES + " 1 " + OTIMES + " " + DELTA + ") g^3 =", format_morphism(id_x_id_x_Delta_g3)
print u"\n(1 " + OTIMES + " " + DELTA + " " + OTIMES + " 1) g^3 =", format_morphism(id_x_Delta_x_id_g3)
print u"\n(" + DELTA + " " + OTIMES + " 1 " + OTIMES + " 1) g^3 =", format_morphism(Delta_x_id_x_id_g3)
# (g x g^3) Delta_2 ## (g^2 x g^2) Delta_2 ## (g^3 x g) Delta_2 #
g_x_g3_Delta2 = {}
g2_x_g2_Delta2 = {}
g3_x_g_Delta2 = {}
for h, hxhs in delta2.iteritems():
g_x_g3_Delta2[h] = [(l_cp, ) + r_cp for l, r in hxhs for l_cp in g[l] for r_cp in g3[r]]
g2_x_g2_Delta2[h] = [l_cp + r_cp for l, r in hxhs for l_cp in g2[l] for r_cp in g2[r]]
g3_x_g_Delta2[h] = [l_cp + (r_cp, ) for l, r in hxhs for l_cp in g3[l] for r_cp in g[r]]
g_x_g3_Delta2 = chain_map_mod(g_x_g3_Delta2)
g2_x_g2_Delta2 = chain_map_mod(g2_x_g2_Delta2)
g3_x_g_Delta2 = chain_map_mod(g3_x_g_Delta2)
print u"\n( g " + OTIMES + " g^3 ) " + DELTA + "^2 =", format_morphism(g_x_g3_Delta2)
print u"\n( g^2 " + OTIMES + " g^2 ) " + DELTA + "^2 =", format_morphism(g2_x_g2_Delta2)
print u"\n( g^3 " + OTIMES + " g ) " + DELTA + "^2 =", format_morphism(g3_x_g_Delta2)
# (g x g x g^2) Delta^3 ## (g x g^2 x g) Delta^3 ## (g^2 x g x g) Delta^3 #
g_x_g_x_g2_Delta3 = {}
g_x_g2_x_g_Delta3 = {}
g2_x_g_x_g_Delta3 = {}
for h, hxhxhs in delta3.iteritems():
g_x_g_x_g2_Delta3[h] = [(g[l], g[m]) + r_cp for (l, m, r) in hxhxhs for r_cp in g2[r]]
g_x_g2_x_g_Delta3[h] = [(g[l], ) + m_cp + (g[r], ) for (l, m, r) in hxhxhs for m_cp in g2[m]]
g2_x_g_x_g_Delta3[h] = [l_cp + (g[m], g[r]) for (l, m, r) in hxhxhs for l_cp in g2[l]]
g_x_g_x_g2_Delta3 = chain_map_mod(expand_map_all(g_x_g_x_g2_Delta3))
g_x_g2_x_g_Delta3 = chain_map_mod(expand_map_all(g_x_g2_x_g_Delta3))
g2_x_g_x_g_Delta3 = chain_map_mod(expand_map_all(g2_x_g_x_g_Delta3))
print u"\n( g " + OTIMES + " g " + OTIMES + " g^2 ) " + DELTA + "^3 =", format_morphism(g_x_g_x_g2_Delta3)
print u"\n( g " + OTIMES + " g^2 " + OTIMES + " g ) " + DELTA + "^3 =", format_morphism(g_x_g2_x_g_Delta3)
print u"\n( g^2 " + OTIMES + " g " + OTIMES + " g ) " + DELTA + "^3 =", format_morphism(g2_x_g_x_g_Delta3)
# phi_2
phi_2 = reduce(add_maps_mod_2, [delta_c4_g, id_x_Delta_c3_g2, Delta_c3_x_id_g2,
id_x_id_x_Delta_g3, id_x_Delta_x_id_g3, Delta_x_id_x_id_g3,
g_x_g3_Delta2, g3_x_g_Delta2, g2_x_g2_Delta2,
g_x_g_x_g2_Delta3, g_x_g2_x_g_Delta3, g2_x_g_x_g_Delta3], {})
print "\n" + PHI + u"_2 =", format_morphism(phi_2)
# factor phi_2
factored_phi_2 = {h: factorize_cycles(chain, C) for h, chain in phi_2.iteritems() if chain}
print "\n" + PHI + u"_2 (factored) =", format_morphism(factored_phi_2)
delta4 = {h: [f_tensor(cxcxcxc) for cxcxcxc in cycles] for h, cycles in factored_phi_2.iteritems()}
# flatten delta3 and remove any empty elements (mod 2)
delta4 = chain_map_mod(expand_map_all(delta4))
print "\n" + DELTA + u"_4 =", format_morphism(delta4)
# however, we still need all keys to be available
for h in g.iterkeys():
if h not in delta4:
delta4[h] = []
"""
Check coassociativity of Delta^4
"""
print "\nChecking coassociativity...\n"
# (Delta^2 x 1 x 1 x 1) Delta^4 ## (1 x Delta^2 x 1 x 1) Delta^4 #
# (1 x 1 x Delta^2 x 1) Delta^4 ## (1 x 1 x 1 x Delta^2) Delta^4 #
delta2_x_id_x_id_x_id_Delta4 = {}
id_x_delta2_x_id_x_id_Delta4 = {}
id_x_id_x_delta2_x_id_Delta4 = {}
id_x_id_x_id_x_delta2_Delta4 = {}
for h, hxhxhxhs in delta4.iteritems():
delta2_x_id_x_id_x_id_Delta4[h] = [cp + (h2, h3, h4) for (h1, h2, h3, h4) in hxhxhxhs for cp in delta2[h1]]
id_x_delta2_x_id_x_id_Delta4[h] = [(h1, ) + cp + (h3, h4) for (h1, h2, h3, h4) in hxhxhxhs for cp in delta2[h2]]
id_x_id_x_delta2_x_id_Delta4[h] = [(h1, h2) + cp + (h4, ) for (h1, h2, h3, h4) in hxhxhxhs for cp in delta2[h3]]
id_x_id_x_id_x_delta2_Delta4[h] = [(h1, h2, h3) + cp for (h1, h2, h3, h4) in hxhxhxhs for cp in delta2[h4]]
delta2_x_id_x_id_x_id_Delta4 = chain_map_mod(expand_map_all(delta2_x_id_x_id_x_id_Delta4))
id_x_delta2_x_id_x_id_Delta4 = chain_map_mod(expand_map_all(id_x_delta2_x_id_x_id_Delta4))
id_x_id_x_delta2_x_id_Delta4 = chain_map_mod(expand_map_all(id_x_id_x_delta2_x_id_Delta4))
id_x_id_x_id_x_delta2_Delta4 = chain_map_mod(expand_map_all(id_x_id_x_id_x_delta2_Delta4))
print u"\n( " + DELTA + "^2 " + OTIMES + " 1 " + OTIMES + " 1 " + OTIMES + " 1 ) " + DELTA + "_4 =", format_morphism(delta2_x_id_x_id_x_id_Delta4)
print u"\n( 1 " + OTIMES, DELTA + "^2 " + OTIMES + " 1 " + OTIMES + " 1 ) " + DELTA + "_4 =", format_morphism(id_x_delta2_x_id_x_id_Delta4)
print u"\n( 1 " + OTIMES + " 1 " + OTIMES, DELTA + "^2 " + OTIMES + " 1 ) " + DELTA + "_4 =", format_morphism(id_x_id_x_delta2_x_id_Delta4)
print u"\n( 1 " + OTIMES + " 1 " + OTIMES + " 1 " + OTIMES, DELTA + "^2 ) " + DELTA + "_4 =", format_morphism(id_x_id_x_id_x_delta2_Delta4)
# (Delta^3 x 1 x 1) Delta^3 ## (1 x Delta^3 x 1) Delta^3 ## (1 x 1 x Delta^3) Delta^3 #
delta3_x_id_x_id_Delta3 = {}
id_x_delta3_x_id_Delta3 = {}
id_x_id_x_delta3_Delta3 = {}
for h, hxhxhs in delta3.iteritems():
delta3_x_id_x_id_Delta3[h] = [l_cp + (m, r) for (l, m, r) in hxhxhs for l_cp in delta3[l]]
id_x_delta3_x_id_Delta3[h] = [(l, ) + m_cp + (r, ) for (l, m, r) in hxhxhs for m_cp in delta3[m]]
id_x_id_x_delta3_Delta3[h] = [(l, m) + r_cp for (l, m, r) in hxhxhs for r_cp in delta3[r]]
delta3_x_id_x_id_Delta3 = chain_map_mod(expand_map_all(delta3_x_id_x_id_Delta3))
id_x_delta3_x_id_Delta3 = chain_map_mod(expand_map_all(id_x_delta3_x_id_Delta3))
id_x_id_x_delta3_Delta3 = chain_map_mod(expand_map_all(id_x_id_x_delta3_Delta3))
print u"\n( " + DELTA + "^3 " + OTIMES + " 1 " + OTIMES + " 1 ) " + DELTA + "_3 =", format_morphism(delta3_x_id_x_id_Delta3)
print u"\n( 1 " + OTIMES, DELTA + "^3 " + OTIMES + " 1 ) " + DELTA + "_3 =", format_morphism(id_x_delta3_x_id_Delta3)
print u"\n( 1 " + OTIMES + " 1 " + OTIMES, DELTA + "^3 ) " + DELTA + "_3 =", format_morphism(id_x_id_x_delta3_Delta3)
# (Delta^4 x 1) Delta^2 ## (1 x Delta^4) Delta^2 #
delta4_x_id_Delta2 = {}
id_x_delta4_Delta2 = {}
for h, hxhs in delta2.iteritems():
delta4_x_id_Delta2[h] = [l_cp + (r, ) for (l, r) in hxhs for l_cp in delta4[l]]
id_x_delta4_Delta2[h] = [(l, ) + r_cp for (l, r) in hxhs for r_cp in delta4[r]]
delta4_x_id_Delta2 = chain_map_mod(delta4_x_id_Delta2)
id_x_delta4_Delta2 = chain_map_mod(id_x_delta4_Delta2)
print u"\n( " + DELTA + "^4 " + OTIMES + " 1 ) " + DELTA + "_2 =", format_morphism(delta4_x_id_Delta2)
print u"\n( 1 " + OTIMES, DELTA + "^4 ) " + DELTA + "_2 =", format_morphism(id_x_delta4_Delta2)
# z3
z3 = reduce(add_maps_mod_2, [delta2_x_id_x_id_x_id_Delta4, id_x_delta2_x_id_x_id_Delta4,
id_x_id_x_delta2_x_id_Delta4, id_x_id_x_id_x_delta2_Delta4,
delta3_x_id_x_id_Delta3, id_x_delta3_x_id_Delta3, id_x_id_x_delta3_Delta3,
delta4_x_id_Delta2, id_x_delta4_Delta2], {})
print DELTA + "^4 is co-associative?", not any(z3.values())
if any(z3.values()):
print "\nz3 = " + NABLA + "(" + DELTA + "^5) =",
print format_morphism(z3)
print "\nz3 = " + NABLA + "(" + DELTA + "^5) (factored) =",
print format_morphism({cell: factorize(factorize_cycles(chain, C), C) for cell, chain in z3.iteritems()})
"""
Compute g^4
"""
# (g x g x g x g) Delta4
gxgxgxg_delta4 = {}
for h, chain in delta4.iteritems():
gxgxgxg_delta4[h] = [g_tensor(hxhxhxh) for hxhxhxh in chain]
gxgxgxg_delta4 = chain_map_mod(expand_map_all(gxgxgxg_delta4))
print u"\n(g " + OTIMES + " g " + OTIMES + " g " + OTIMES + " g)" + DELTA + "_4 =", format_morphism(gxgxgxg_delta4)
# nabla g^4
nabla_g4 = add_maps_mod_2(gxgxgxg_delta4, phi_2)
print u"\n(g " + OTIMES + " g " + OTIMES + " g " + OTIMES + " g)" + DELTA + "_4 + " + PHI + "_2 =", format_morphism(nabla_g4)
# g^4
g4 = {h: chain_integrate(chain, C) for h, chain in nabla_g4.iteritems()}
g4 = chain_map_mod(expand_map_all(g4))
print u"\ng^4 =", format_morphism(g4)
print e.strerror
argparser.print_help()
raise SystemExit
# load file
data = args.file.read()
args.file.close()
# parse file contents
result = CellChainParse.parse(data)
if not result:
raise SystemExit
# construct coalgebra
C = Coalgebra(result["groups"], result["differentials"], result["coproducts"])
"""
COMPUTE HOMOLOGY
"""
differential = {n: C.incidence_matrix(n, sparse=False) for n in range(1, C.topDimension() + 1)}
# create temporary file for CHomP
scratch = open(temp_mat, 'w+')
for n, entries in differential.iteritems():
print >> scratch, n - 1
incidences = [(l, r, v % 2) for (l, r), v in entries.iteritems()]
for entry in ["{} {} {}".format(l, r, v) for (l, r, v) in sorted(incidences, cmp=compare_incidences)]: