def cohomology_2_rel_boundary(t, field, as_matrix_with_column_vectors = False):
"""
computes H^2(t,boundary of t; field)
i.e. t is a manifold with boundary (cusps are cut), coefficients are in field
>>> from algebra.field_p import field_p
>>> from fractions import Fraction
>>> Fraction.one = staticmethod(lambda : Fraction(1,1))
>>> Fraction.zero = staticmethod(lambda : Fraction(0,1))
>>> def is_zero(m):
... for r in m.values:
... for c in r:
... if not c == 0:
... return False
... return True
>>> from manifold3D.triangulation import *
>>> t = read_triangulation_from_file("lib/triangulations/m003.trig")
>>> p = d2(t) * d3(t)
>>> is_zero(p)
True
H_1(m003)=H^2(m003,boundary) = Z + Z/5,
dim(H^2(m003,boundary;Q)) = 1
>>> len(cohomology_2_rel_boundary(t, Fraction))
1
>>> t = read_triangulation_from_file("lib/triangulations/m206.trig")
>>> is_zero(d2(t) * d3(t))
True
>>> field = field_p(5)
H_1(m206)=H^2(m206,boundary) = Z + Z/5,
dim(H^2(m206,boundary;Z/5)) = 2
>>> len(cohomology_2_rel_boundary(t, field))
2
>>> t = read_triangulation_from_file("lib/triangulations/m053.trig")
>>> is_zero(d2(t) * d3(t))
True
>>> len(cohomology_2_rel_boundary(t, Fraction))
1
>>> cohomology_2_rel_boundary(t, Fraction, True).no_columns()
1
"""
assert isinstance(t,triangulation)
m_d3 = matrix(d3(t), field).transpose()
m_d2 = matrix(d2(t), field).transpose()
H = homology_generators(m_d2, m_d3, as_matrix_with_column_vectors)
return H
def d2(t):
"""
the differential C_2 -> C_1 of chains with Z coefficients as a matrix
"""
faces = t.get_face_classes()
edges = t.get_edge_classes()
def boundary(face,an_edge_class):
res = 0
for an_edge in an_edge_class:
if face.tet1()==an_edge.tet():
if not (face.face1()==an_edge.vert_0() or
face.face1()==an_edge.vert_1()):
if an_edge.vert_0()>face.face1():
vert0 = an_edge.vert_0()-1
else:
vert0 = an_edge.vert_0()
if an_edge.vert_1()>face.face1():
vert1 = an_edge.vert_1()-1
else:
vert1 = an_edge.vert_1()
if (vert0,vert1) in [(0,1),(1,2),(2,0)]:
res += + (-1) ** face.face1()
else:
assert (vert0,vert1) in [(1,0),(2,1),(0,2)]
res += - (-1) ** face.face1()
return res
return matrix([[boundary(i,j) for i in faces] for j in edges],int)
def d3(t):
"""
the differential C_3 -> C_2 of chains with Z coefficients as a matrix
"""
tets = t.tet_list
faces = t.get_face_classes()
def boundary(tet,a_face_class):
res = 0
if tet.index==a_face_class.tet1():
res += +1
#res += (-1)**a_face_class.face1()
if tet.index==a_face_class.tet2():
res += a_face_class.orient()
#res += ((-1)**a_face_class.face2()) * a_face_class.orient()
return res
return matrix([[boundary(i,j) for i in tets] for j in faces],int)
return m