def __call__(self, f):
"""
`f` is a dictionary of matrices in the basis of the chain complex.
EXAMPLES::
sage: S = simplicial_complexes.Sphere(5)
sage: H = Hom(S,S)
sage: i = H.identity()
sage: C = S.chain_complex()
sage: G = Hom(C,C)
sage: x = i.associated_chain_complex_morphism()
sage: f = x._matrix_dictionary
sage: y = G(f)
sage: x == y
True
"""
return ChainComplexMorphism(f, self.domain(), self.codomain())
def associated_chain_complex_morphism(self,
base_ring=ZZ,
augmented=False,
cochain=False):
"""
Returns the associated chain complex morphism of self.
EXAMPLES::
sage: S = simplicial_complexes.Sphere(1)
sage: T = simplicial_complexes.Sphere(2)
sage: H = Hom(S,T)
sage: f = {0:0,1:1,2:2}
sage: x = H(f)
sage: x
Simplicial complex morphism {0: 0, 1: 1, 2: 2} from Simplicial complex with vertex set (0, 1, 2) and facets {(1, 2), (0, 2), (0, 1)} to Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 2, 3), (0, 1, 2), (1, 2, 3), (0, 1, 3)}
sage: a = x.associated_chain_complex_morphism()
sage: a
Chain complex morphism from Chain complex with at most 2 nonzero terms over Integer Ring to Chain complex with at most 3 nonzero terms over Integer Ring
sage: a._matrix_dictionary
{0: [0 0 0]
[0 1 0]
[0 0 1]
[1 0 0],
1: [0 0 0]
[0 1 0]
[0 0 0]
[1 0 0]
[0 0 0]
[0 0 1],
2: []}
sage: x.associated_chain_complex_morphism(augmented=True)
Chain complex morphism from Chain complex with at most 3 nonzero terms over Integer Ring to Chain complex with at most 4 nonzero terms over Integer Ring
sage: x.associated_chain_complex_morphism(cochain=True)
Chain complex morphism from Chain complex with at most 3 nonzero terms over Integer Ring to Chain complex with at most 2 nonzero terms over Integer Ring
sage: x.associated_chain_complex_morphism(augmented=True,cochain=True)
Chain complex morphism from Chain complex with at most 4 nonzero terms over Integer Ring to Chain complex with at most 3 nonzero terms over Integer Ring
sage: x.associated_chain_complex_morphism(base_ring=GF(11))
Chain complex morphism from Chain complex with at most 2 nonzero terms over Finite Field of size 11 to Chain complex with at most 3 nonzero terms over Finite Field of size 11
Some simplicial maps which reverse the orientation of a few simplices::
sage: g = {0:1, 1:2, 2:0}
sage: H(g).associated_chain_complex_morphism()._matrix_dictionary
{0: [0 0 0]
[1 0 0]
[0 1 0]
[0 0 1], 1: [ 0 0 0]
[-1 0 0]
[ 0 0 0]
[ 0 0 1]
[ 0 0 0]
[ 0 -1 0], 2: []}
sage: X = SimplicialComplex(1, [[0, 1]])
sage: Hom(X,X)({0:1, 1:0}).associated_chain_complex_morphism()._matrix_dictionary
{0: [0 1]
[1 0], 1: [-1]}
"""
max_dim = max(self._domain.dimension(), self._codomain.dimension())
min_dim = min(self._domain.dimension(), self._codomain.dimension())
matrices = {}
if augmented is True:
m = matrix.Matrix(base_ring, 1, 1, 1)
if not cochain:
matrices[-1] = m
else:
matrices[-1] = m.transpose()
for dim in range(min_dim + 1):
# X_faces = list(self._domain.faces()[dim])
# Y_faces = list(self._codomain.faces()[dim])
X_faces = list(self._domain.n_cells(dim))
Y_faces = list(self._codomain.n_cells(dim))
num_faces_X = len(X_faces)
num_faces_Y = len(Y_faces)
mval = [0 for i in range(num_faces_X * num_faces_Y)]
for i in X_faces:
y, oriented = self(i, orientation=True)
if y.dimension() < dim:
pass
else:
mval[X_faces.index(i) +
(Y_faces.index(y) * num_faces_X)] = oriented
m = matrix.Matrix(base_ring,
num_faces_Y,
num_faces_X,
mval,
sparse=True)
if not cochain:
matrices[dim] = m
else:
matrices[dim] = m.transpose()
for dim in range(min_dim + 1, max_dim + 1):
try:
l1 = len(self._codomain.n_cells(dim))
except KeyError:
l1 = 0
try:
l2 = len(self._domain.n_cells(dim))
except KeyError:
l2 = 0
m = matrix.zero_matrix(base_ring, l1, l2, sparse=True)
if not cochain:
matrices[dim] = m
else:
matrices[dim] = m.transpose()
if not cochain:
return ChainComplexMorphism(matrices,\
self._domain.chain_complex(base_ring=base_ring,augmented=augmented,cochain=cochain),\
self._codomain.chain_complex(base_ring=base_ring,augmented=augmented,cochain=cochain))
else:
return ChainComplexMorphism(matrices,\
self._codomain.chain_complex(base_ring=base_ring,augmented=augmented,cochain=cochain),\
self._domain.chain_complex(base_ring=base_ring,augmented=augmented,cochain=cochain))
def __init__(self, matrices, f, g=None):
r"""
Create a chain homotopy between the given chain maps
from a dictionary of matrices.
EXAMPLES:
If ``g`` is not specified, it is set equal to
`f - (H \partial + \partial H)`. ::
sage: from sage.homology.chain_homotopy import ChainHomotopy
sage: C = ChainComplex({1: matrix(ZZ, 1, 2, (1,0)), 2: matrix(ZZ, 2, 1, (0, 2))}, degree_of_differential=-1)
sage: D = ChainComplex({2: matrix(ZZ, 1, 1, (6,))}, degree_of_differential=-1)
sage: f_d = {1: matrix(ZZ, 1, 2, (0,3)), 2: identity_matrix(ZZ, 1)}
sage: f = Hom(C,D)(f_d)
sage: H_d = {0: identity_matrix(ZZ, 1), 1: matrix(ZZ, 1, 2, (2,2))}
sage: H = ChainHomotopy(H_d, f)
sage: H._g.in_degree(0)
[]
sage: H._g.in_degree(1)
[-13 -9]
sage: H._g.in_degree(2)
[-3]
TESTS:
Try to construct a chain homotopy in which the maps do not
have matching domains and codomains::
sage: g = Hom(C,C)({}) # the zero chain map
sage: H = ChainHomotopy(H_d, f, g)
Traceback (most recent call last):
...
ValueError: the chain maps are not compatible
"""
domain = f.domain()
codomain = f.codomain()
deg = domain.degree_of_differential()
# Check that the chain complexes are compatible. This should
# never arise, because first there should be errors in
# constructing the chain maps. But just in case...
if domain.degree_of_differential() != codomain.degree_of_differential(
):
raise ValueError('the chain complexes are not compatible')
if g is not None:
# Check that the chain maps are compatible.
if not (domain == g.domain() and codomain == g.codomain()):
raise ValueError('the chain maps are not compatible')
# Check that the data define a chain homotopy.
for i in domain.differential():
if i in matrices and i + deg in matrices:
if not (codomain.differential(i - deg) * matrices[i] +
matrices[i + deg] * domain.differential(i)
== f.in_degree(i) - g.in_degree(i)):
raise ValueError(
'the data do not define a valid chain homotopy')
elif i in matrices:
if not (codomain.differential(i - deg) * matrices[i]
== f.in_degree(i) - g.in_degree(i)):
raise ValueError(
'the data do not define a valid chain homotopy')
elif i + deg in matrices:
if not (matrices[i + deg] * domain.differential(i)
== f.in_degree(i) - g.in_degree(i)):
raise ValueError(
'the data do not define a valid chain homotopy')
else:
# Define g.
g_data = {}
for i in domain.differential():
if i in matrices and i + deg in matrices:
g_data[i] = f.in_degree(
i) - matrices[i + deg] * domain.differential(
i) - codomain.differential(i - deg) * matrices[i]
elif i in matrices:
g_data[i] = f.in_degree(
i) - codomain.differential(i - deg) * matrices[i]
elif i + deg in matrices:
g_data[i] = f.in_degree(
i) - matrices[i + deg] * domain.differential(i)
g = ChainComplexMorphism(g_data, domain, codomain)
self._matrix_dictionary = {}
for i in matrices:
m = matrices[i]
# Use immutable matrices because they're hashable.
m.set_immutable()
self._matrix_dictionary[i] = m
self._f = f
self._g = g
Morphism.__init__(self, Hom(domain, codomain))
def __init__(self, matrices, pi, iota):
r"""
Create a chain contraction from the given data.
EXAMPLES::
sage: from sage.homology.chain_homotopy import ChainContraction
sage: C = ChainComplex({0: zero_matrix(ZZ, 1), 1: identity_matrix(ZZ, 1)})
sage: D = ChainComplex({0: matrix(ZZ, 0, 1)})
The chain complex `C` is chain homotopy equivalent to `D`,
which is just a copy of `\ZZ` in degree 0, and we try
construct a chain contraction, but get the map `\iota` wrong::
sage: pi = Hom(C,D)({0: identity_matrix(ZZ, 1)})
sage: iota = Hom(D,C)({0: zero_matrix(ZZ, 1)})
sage: H = ChainContraction({0: zero_matrix(ZZ, 0, 1), 1: zero_matrix(ZZ, 1), 2: identity_matrix(ZZ, 1)}, pi, iota)
Traceback (most recent call last):
...
ValueError: the composite 'pi iota' is not the identity
Another bad `\iota`::
sage: iota = pi # wrong domain, codomain
sage: H = ChainContraction({0: zero_matrix(ZZ, 0, 1), 1: zero_matrix(ZZ, 1), 2: identity_matrix(ZZ, 1)}, pi, iota)
Traceback (most recent call last):
...
ValueError: the chain maps are not composable
`\iota` is okay, but wrong data defining `H`::
sage: iota = Hom(D,C)({0: identity_matrix(ZZ, 1)})
sage: H = ChainContraction({0: zero_matrix(ZZ, 0, 1), 1: identity_matrix(ZZ, 1), 2: identity_matrix(ZZ, 1)}, pi, iota)
Traceback (most recent call last):
...
ValueError: not an algebraic gradient vector field
"""
from sage.matrix.constructor import identity_matrix
from .chain_complex_morphism import ChainComplexMorphism
if not (pi.domain() == iota.codomain()
and pi.codomain() == iota.domain()):
raise ValueError('the chain maps are not composable')
C = pi.domain()
D = pi.codomain()
base_ring = C.base_ring()
# Check that the composite 'pi iota' is 1.
for i in D.nonzero_degrees():
if pi.in_degree(i) * iota.in_degree(i) != identity_matrix(
base_ring, D.free_module_rank(i)):
raise ValueError("the composite 'pi iota' is not the identity")
# Construct the chain map 'id_C'.
id_C_dict = {}
for i in C.nonzero_degrees():
id_C_dict[i] = identity_matrix(base_ring, C.free_module_rank(i))
id_C = ChainComplexMorphism(id_C_dict, C, C)
# Now check that
# - `H` is a chain homotopy between `id_C` and `\iota \pi`
# - `HH = 0`
ChainHomotopy.__init__(self, matrices, id_C, iota * pi)
if not self.is_algebraic_gradient_vector_field():
raise ValueError('not an algebraic gradient vector field')
# Check that `\pi H = 0`:
deg = C.degree_of_differential()
for i in matrices:
if pi.in_degree(i - deg) * matrices[i] != 0:
raise ValueError(
'the data do not define a valid chain contraction: pi H != 0'
)
# Check that `H \iota = 0`:
for i in iota._matrix_dictionary:
if i in matrices:
if matrices[i] * iota.in_degree(i) != 0:
raise ValueError(
'the data do not define a valid chain contraction: H iota != 0'
)
self._pi = pi
self._iota = iota
def associated_chain_complex_morphism(self,base_ring=ZZ,augmented=False,cochain=False):
"""
Returns the associated chain complex morphism of ``self``.
EXAMPLES::
sage: S = simplicial_complexes.Sphere(1)
sage: T = simplicial_complexes.Sphere(2)
sage: H = Hom(S,T)
sage: f = {0:0,1:1,2:2}
sage: x = H(f)
sage: x
Simplicial complex morphism:
From: Minimal triangulation of the 1-sphere
To: Minimal triangulation of the 2-sphere
Defn: 0 |--> 0
1 |--> 1
2 |--> 2
sage: a = x.associated_chain_complex_morphism()
sage: a
Chain complex morphism:
From: Chain complex with at most 2 nonzero terms over Integer Ring
To: Chain complex with at most 3 nonzero terms over Integer Ring
sage: a._matrix_dictionary
{0: [1 0 0]
[0 1 0]
[0 0 1]
[0 0 0], 1: [1 0 0]
[0 1 0]
[0 0 0]
[0 0 1]
[0 0 0]
[0 0 0], 2: []}
sage: x.associated_chain_complex_morphism(augmented=True)
Chain complex morphism:
From: Chain complex with at most 3 nonzero terms over Integer Ring
To: Chain complex with at most 4 nonzero terms over Integer Ring
sage: x.associated_chain_complex_morphism(cochain=True)
Chain complex morphism:
From: Chain complex with at most 3 nonzero terms over Integer Ring
To: Chain complex with at most 2 nonzero terms over Integer Ring
sage: x.associated_chain_complex_morphism(augmented=True,cochain=True)
Chain complex morphism:
From: Chain complex with at most 4 nonzero terms over Integer Ring
To: Chain complex with at most 3 nonzero terms over Integer Ring
sage: x.associated_chain_complex_morphism(base_ring=GF(11))
Chain complex morphism:
From: Chain complex with at most 2 nonzero terms over Finite Field of size 11
To: Chain complex with at most 3 nonzero terms over Finite Field of size 11
Some simplicial maps which reverse the orientation of a few simplices::
sage: g = {0:1, 1:2, 2:0}
sage: H(g).associated_chain_complex_morphism()._matrix_dictionary
{0: [0 0 1]
[1 0 0]
[0 1 0]
[0 0 0], 1: [ 0 -1 0]
[ 0 0 -1]
[ 0 0 0]
[ 1 0 0]
[ 0 0 0]
[ 0 0 0], 2: []}
sage: X = SimplicialComplex([[0, 1]], is_mutable=False)
sage: Hom(X,X)({0:1, 1:0}).associated_chain_complex_morphism()._matrix_dictionary
{0: [0 1]
[1 0], 1: [-1]}
"""
max_dim = max(self.domain().dimension(),self.codomain().dimension())
min_dim = min(self.domain().dimension(),self.codomain().dimension())
matrices = {}
if augmented is True:
m = matrix(base_ring,1,1,1)
if not cochain:
matrices[-1] = m
else:
matrices[-1] = m.transpose()
for dim in range(min_dim+1):
X_faces = list(self.domain().n_cells(dim))
Y_faces = list(self.codomain().n_cells(dim))
num_faces_X = len(X_faces)
num_faces_Y = len(Y_faces)
mval = [0 for i in range(num_faces_X*num_faces_Y)]
for i in X_faces:
y, oriented = self(i, orientation=True)
if y.dimension() < dim:
pass
else:
mval[X_faces.index(i)+(Y_faces.index(y)*num_faces_X)] = oriented
m = matrix(base_ring,num_faces_Y,num_faces_X,mval,sparse=True)
if not cochain:
matrices[dim] = m
else:
matrices[dim] = m.transpose()
for dim in range(min_dim+1,max_dim+1):
try:
l1 = len(self.codomain().n_cells(dim))
except KeyError:
l1 = 0
try:
l2 = len(self.domain().n_cells(dim))
except KeyError:
l2 = 0
m = zero_matrix(base_ring,l1,l2,sparse=True)
if not cochain:
matrices[dim] = m
else:
matrices[dim] = m.transpose()
if not cochain:
return ChainComplexMorphism(matrices,\
self.domain().chain_complex(base_ring=base_ring,augmented=augmented,cochain=cochain),\
self.codomain().chain_complex(base_ring=base_ring,augmented=augmented,cochain=cochain))
else:
return ChainComplexMorphism(matrices,\
self.codomain().chain_complex(base_ring=base_ring,augmented=augmented,cochain=cochain),\
self.domain().chain_complex(base_ring=base_ring,augmented=augmented,cochain=cochain))