def homology(self,
dim=None,
base_ring=ZZ,
subcomplex=None,
generators=False,
cohomology=False,
algorithm='pari',
verbose=False,
reduced=True,
**kwds):
r"""
The (reduced) homology of this cell complex.
:param dim: If None, then return the homology in every
dimension. If ``dim`` is an integer or list, return the
homology in the given dimensions. (Actually, if ``dim`` is
a list, return the homology in the range from ``min(dim)``
to ``max(dim)``.)
:type dim: integer or list of integers or None; optional,
default None
:param base_ring: commutative ring, must be ZZ or a field.
:type base_ring: optional, default ZZ
:param subcomplex: a subcomplex of this simplicial complex.
Compute homology relative to this subcomplex.
:type subcomplex: optional, default empty
:param generators: If ``True``, return generators for the homology
groups along with the groups. NOTE: Since :trac:`6100`, the result
may not be what you expect when not using CHomP since its return
is in terms of the chain complex.
:type generators: boolean; optional, default False
:param cohomology: If True, compute cohomology rather than homology.
:type cohomology: boolean; optional, default False
:param algorithm: The options are 'auto', 'dhsw', 'pari' or 'no_chomp'.
See below for a description of what they mean.
:type algorithm: string; optional, default 'pari'
:param verbose: If True, print some messages as the homology is
computed.
:type verbose: boolean; optional, default False
:param reduced: If ``True``, return the reduced homology.
:type reduced: boolean; optional, default ``True``
ALGORITHM:
If ``algorithm`` is set to 'auto', then use
CHomP if available. (CHomP is available at the web page
http://chomp.rutgers.edu/. It is also an optional package
for Sage.)
CHomP computes homology, not cohomology, and only works over
the integers or finite prime fields. Therefore if any of
these conditions fails, or if CHomP is not present, or if
``algorithm`` is set to 'no_chomp', go to plan B: if this complex
has a ``_homology`` method -- each simplicial complex has
this, for example -- then call that. Such a method implements
specialized algorithms for the particular type of cell
complex.
Otherwise, move on to plan C: compute the chain complex of
this complex and compute its homology groups. To do this: over a
field, just compute ranks and nullities, thus obtaining
dimensions of the homology groups as vector spaces. Over the
integers, compute Smith normal form of the boundary matrices
defining the chain complex according to the value of
``algorithm``. If ``algorithm`` is 'auto' or 'no_chomp', then
for each relatively small matrix, use the standard Sage
method, which calls the Pari package. For any large matrix,
reduce it using the Dumas, Heckenbach, Saunders, and Welker
elimination algorithm: see
:func:`sage.homology.matrix_utils.dhsw_snf` for details.
Finally, ``algorithm`` may also be 'pari' or 'dhsw', which
forces the named algorithm to be used regardless of the size
of the matrices and regardless of whether CHomP is available.
As of this writing, ``'pari'`` is the fastest standard option.
The optional CHomP package may be better still.
EXAMPLES::
sage: P = delta_complexes.RealProjectivePlane()
sage: P.homology()
{0: 0, 1: C2, 2: 0}
sage: P.homology(reduced=False)
{0: Z, 1: C2, 2: 0}
sage: P.homology(base_ring=GF(2))
{0: Vector space of dimension 0 over Finite Field of size 2,
1: Vector space of dimension 1 over Finite Field of size 2,
2: Vector space of dimension 1 over Finite Field of size 2}
sage: S7 = delta_complexes.Sphere(7)
sage: S7.homology(7)
Z
sage: cubical_complexes.KleinBottle().homology(1, base_ring=GF(2))
Vector space of dimension 2 over Finite Field of size 2
If CHomP is installed, Sage can compute generators of homology
groups::
sage: S2 = simplicial_complexes.Sphere(2)
sage: S2.homology(dim=2, generators=True, base_ring=GF(2)) # optional - CHomP
(Vector space of dimension 1 over Finite Field of size 2, [(0, 1, 2) + (0, 1, 3) + (0, 2, 3) + (1, 2, 3)])
When generators are computed, Sage returns a pair for each
dimension: the group and the list of generators. For
simplicial complexes, each generator is represented as a
linear combination of simplices, as above, and for cubical
complexes, each generator is a linear combination of cubes::
sage: S2_cub = cubical_complexes.Sphere(2)
sage: S2_cub.homology(dim=2, generators=True) # optional - CHomP
(Z, [-[[0,1] x [0,1] x [0,0]] + [[0,1] x [0,1] x [1,1]] - [[0,0] x [0,1] x [0,1]] - [[0,1] x [1,1] x [0,1]] + [[0,1] x [0,0] x [0,1]] + [[1,1] x [0,1] x [0,1]]])
"""
from sage.interfaces.chomp import have_chomp, homcubes, homsimpl
from sage.homology.cubical_complex import CubicalComplex
from sage.homology.simplicial_complex import SimplicialComplex
from sage.modules.all import VectorSpace
from sage.homology.homology_group import HomologyGroup
if dim is not None:
if isinstance(dim, (list, tuple, range)):
low = min(dim) - 1
high = max(dim) + 2
else:
low = dim - 1
high = dim + 2
dims = range(low, high)
else:
dims = None
# try to use CHomP if computing homology (not cohomology) and
# working over Z or F_p, p a prime.
if (algorithm == 'auto' and cohomology is False
and (base_ring == ZZ or
(base_ring.is_prime_field() and base_ring != QQ))):
# homcubes, homsimpl seems fastest if all of homology is computed.
H = None
if isinstance(self, CubicalComplex):
if have_chomp('homcubes'):
H = homcubes(self,
subcomplex,
base_ring=base_ring,
verbose=verbose,
generators=generators)
elif isinstance(self, SimplicialComplex):
if have_chomp('homsimpl'):
H = homsimpl(self,
subcomplex,
base_ring=base_ring,
verbose=verbose,
generators=generators)
# now pick off the requested dimensions
if H:
answer = {}
if not dims:
dims = range(self.dimension() + 1)
for d in dims:
answer[d] = H.get(d, HomologyGroup(0, base_ring))
if dim is not None:
if not isinstance(dim, (list, tuple, range)):
answer = answer.get(dim, HomologyGroup(0, base_ring))
return answer
# Derived classes can implement specialized algorithms using a
# _homology_ method. See SimplicialComplex for one example.
# Those may allow for other arguments, so we pass **kwds.
if hasattr(self, '_homology_'):
return self._homology_(dim,
subcomplex=subcomplex,
cohomology=cohomology,
base_ring=base_ring,
verbose=verbose,
algorithm=algorithm,
reduced=reduced,
**kwds)
C = self.chain_complex(cochain=cohomology,
augmented=reduced,
dimensions=dims,
subcomplex=subcomplex,
base_ring=base_ring,
verbose=verbose)
answer = C.homology(base_ring=base_ring,
generators=generators,
verbose=verbose,
algorithm=algorithm)
if dim is None:
dim = range(self.dimension() + 1)
zero = HomologyGroup(0, base_ring)
if isinstance(dim, (list, tuple, range)):
return dict([d, answer.get(d, zero)] for d in dim)
return answer.get(dim, zero)
def homology(self, dim=None, base_ring=ZZ, subcomplex=None,
generators=False, cohomology=False, algorithm='pari',
verbose=False, reduced=True, **kwds):
r"""
The (reduced) homology of this cell complex.
:param dim: If None, then return the homology in every
dimension. If ``dim`` is an integer or list, return the
homology in the given dimensions. (Actually, if ``dim`` is
a list, return the homology in the range from ``min(dim)``
to ``max(dim)``.)
:type dim: integer or list of integers or None; optional,
default None
:param base_ring: commutative ring, must be ZZ or a field.
:type base_ring: optional, default ZZ
:param subcomplex: a subcomplex of this simplicial complex.
Compute homology relative to this subcomplex.
:type subcomplex: optional, default empty
:param generators: If ``True``, return generators for the homology
groups along with the groups. NOTE: Since :trac:`6100`, the result
may not be what you expect when not using CHomP since its return
is in terms of the chain complex.
:type generators: boolean; optional, default False
:param cohomology: If True, compute cohomology rather than homology.
:type cohomology: boolean; optional, default False
:param algorithm: The options are 'auto', 'dhsw', 'pari' or 'no_chomp'.
See below for a description of what they mean.
:type algorithm: string; optional, default 'pari'
:param verbose: If True, print some messages as the homology is
computed.
:type verbose: boolean; optional, default False
:param reduced: If ``True``, return the reduced homology.
:type reduced: boolean; optional, default ``True``
ALGORITHM:
If ``algorithm`` is set to 'auto', then use
CHomP if available. (CHomP is available at the web page
http://chomp.rutgers.edu/. It is also an optional package
for Sage.)
CHomP computes homology, not cohomology, and only works over
the integers or finite prime fields. Therefore if any of
these conditions fails, or if CHomP is not present, or if
``algorithm`` is set to 'no_chomp', go to plan B: if this complex
has a ``_homology`` method -- each simplicial complex has
this, for example -- then call that. Such a method implements
specialized algorithms for the particular type of cell
complex.
Otherwise, move on to plan C: compute the chain complex of
this complex and compute its homology groups. To do this: over a
field, just compute ranks and nullities, thus obtaining
dimensions of the homology groups as vector spaces. Over the
integers, compute Smith normal form of the boundary matrices
defining the chain complex according to the value of
``algorithm``. If ``algorithm`` is 'auto' or 'no_chomp', then
for each relatively small matrix, use the standard Sage
method, which calls the Pari package. For any large matrix,
reduce it using the Dumas, Heckenbach, Saunders, and Welker
elimination algorithm: see
:func:`sage.homology.matrix_utils.dhsw_snf` for details.
Finally, ``algorithm`` may also be 'pari' or 'dhsw', which
forces the named algorithm to be used regardless of the size
of the matrices and regardless of whether CHomP is available.
As of this writing, ``'pari'`` is the fastest standard option.
The optional CHomP package may be better still.
EXAMPLES::
sage: P = delta_complexes.RealProjectivePlane()
sage: P.homology()
{0: 0, 1: C2, 2: 0}
sage: P.homology(reduced=False)
{0: Z, 1: C2, 2: 0}
sage: P.homology(base_ring=GF(2))
{0: Vector space of dimension 0 over Finite Field of size 2,
1: Vector space of dimension 1 over Finite Field of size 2,
2: Vector space of dimension 1 over Finite Field of size 2}
sage: S7 = delta_complexes.Sphere(7)
sage: S7.homology(7)
Z
sage: cubical_complexes.KleinBottle().homology(1, base_ring=GF(2))
Vector space of dimension 2 over Finite Field of size 2
If CHomP is installed, Sage can compute generators of homology
groups::
sage: S2 = simplicial_complexes.Sphere(2)
sage: S2.homology(dim=2, generators=True, base_ring=GF(2)) # optional - CHomP
(Vector space of dimension 1 over Finite Field of size 2, [(0, 1, 2) + (0, 1, 3) + (0, 2, 3) + (1, 2, 3)])
When generators are computed, Sage returns a pair for each
dimension: the group and the list of generators. For
simplicial complexes, each generator is represented as a
linear combination of simplices, as above, and for cubical
complexes, each generator is a linear combination of cubes::
sage: S2_cub = cubical_complexes.Sphere(2)
sage: S2_cub.homology(dim=2, generators=True) # optional - CHomP
(Z, [-[[0,1] x [0,1] x [0,0]] + [[0,1] x [0,1] x [1,1]] - [[0,0] x [0,1] x [0,1]] - [[0,1] x [1,1] x [0,1]] + [[0,1] x [0,0] x [0,1]] + [[1,1] x [0,1] x [0,1]]])
"""
from sage.interfaces.chomp import have_chomp, homcubes, homsimpl
from sage.homology.cubical_complex import CubicalComplex
from sage.homology.simplicial_complex import SimplicialComplex
from sage.modules.all import VectorSpace
from sage.homology.homology_group import HomologyGroup
if dim is not None:
if isinstance(dim, (list, tuple, range)):
low = min(dim) - 1
high = max(dim) + 2
else:
low = dim - 1
high = dim + 2
dims = range(low, high)
else:
dims = None
# try to use CHomP if computing homology (not cohomology) and
# working over Z or F_p, p a prime.
if (algorithm == 'auto' and cohomology is False
and (base_ring == ZZ or (base_ring.is_prime_field()
and base_ring != QQ))):
# homcubes, homsimpl seems fastest if all of homology is computed.
H = None
if isinstance(self, CubicalComplex):
if have_chomp('homcubes'):
H = homcubes(self, subcomplex, base_ring=base_ring,
verbose=verbose, generators=generators)
elif isinstance(self, SimplicialComplex):
if have_chomp('homsimpl'):
H = homsimpl(self, subcomplex, base_ring=base_ring,
verbose=verbose, generators=generators)
# now pick off the requested dimensions
if H:
answer = {}
if not dims:
dims = range(self.dimension() + 1)
for d in dims:
answer[d] = H.get(d, HomologyGroup(0, base_ring))
if dim is not None:
if not isinstance(dim, (list, tuple, range)):
answer = answer.get(dim, HomologyGroup(0, base_ring))
return answer
# Derived classes can implement specialized algorithms using a
# _homology_ method. See SimplicialComplex for one example.
# Those may allow for other arguments, so we pass **kwds.
if hasattr(self, '_homology_'):
return self._homology_(dim, subcomplex=subcomplex,
cohomology=cohomology, base_ring=base_ring,
verbose=verbose, algorithm=algorithm,
reduced=reduced, **kwds)
C = self.chain_complex(cochain=cohomology, augmented=reduced,
dimensions=dims, subcomplex=subcomplex,
base_ring=base_ring, verbose=verbose)
answer = C.homology(base_ring=base_ring, generators=generators,
verbose=verbose, algorithm=algorithm)
if dim is None:
dim = range(self.dimension() + 1)
zero = HomologyGroup(0, base_ring)
if isinstance(dim, (list, tuple, range)):
return dict([d, answer.get(d, zero)] for d in dim)
return answer.get(dim, zero)
def homology(self, dim=None, **kwds):
r"""
The reduced homology of this cell complex.
:param dim: If None, then return the homology in every
dimension. If ``dim`` is an integer or list, return the
homology in the given dimensions. (Actually, if ``dim`` is
a list, return the homology in the range from ``min(dim)``
to ``max(dim)``.)
:type dim: integer or list of integers or None; optional,
default None
:param base_ring: commutative ring, must be ZZ or a field.
:type base_ring: optional, default ZZ
:param subcomplex: a subcomplex of this simplicial complex.
Compute homology relative to this subcomplex.
:type subcomplex: optional, default empty
:param generators: If True, return generators for the homology
groups along with the groups. NOTE: this is only implemented
if the CHomP package is available.
:type generators: boolean; optional, default False
:param cohomology: If True, compute cohomology rather than homology.
:type cohomology: boolean; optional, default False
:param algorithm: The options are 'auto', 'dhsw', 'pari' or 'no_chomp'.
See below for a description of what they mean.
:type algorithm: string; optional, default 'auto'
:param verbose: If True, print some messages as the homology is
computed.
:type verbose: boolean; optional, default False
.. note::
The keyword arguments to this function get passed on to
:meth:``chain_complex`` and its homology.
ALGORITHM:
If ``algorithm`` is set to 'auto' (the default), then use
CHomP if available. (CHomP is available at the web page
http://chomp.rutgers.edu/. It is also an experimental package
for Sage.)
CHomP computes homology, not cohomology, and only works over
the integers or finite prime fields. Therefore if any of
these conditions fails, or if CHomP is not present, or if
``algorithm`` is set to 'no_chomp', go to plan B: if ``self``
has a ``_homology`` method -- each simplicial complex has
this, for example -- then call that. Such a method implements
specialized algorithms for the particular type of cell
complex.
Otherwise, move on to plan C: compute the chain complex of
``self`` and compute its homology groups. To do this: over a
field, just compute ranks and nullities, thus obtaining
dimensions of the homology groups as vector spaces. Over the
integers, compute Smith normal form of the boundary matrices
defining the chain complex according to the value of
``algorithm``. If ``algorithm`` is 'auto' or 'no_chomp', then
for each relatively small matrix, use the standard Sage
method, which calls the Pari package. For any large matrix,
reduce it using the Dumas, Heckenbach, Saunders, and Welker
elimination algorithm: see
:func:`sage.homology.chain_complex.dhsw_snf` for details.
Finally, ``algorithm`` may also be 'pari' or 'dhsw', which
forces the named algorithm to be used regardless of the size
of the matrices and regardless of whether CHomP is available.
As of this writing, CHomP is by far the fastest option,
followed by the 'auto' or 'no_chomp' setting of using the
Dumas, Heckenbach, Saunders, and Welker elimination algorithm
for large matrices and Pari for small ones.
EXAMPLES::
sage: P = delta_complexes.RealProjectivePlane()
sage: P.homology()
{0: 0, 1: C2, 2: 0}
sage: P.homology(base_ring=GF(2))
{0: Vector space of dimension 0 over Finite Field of size 2,
1: Vector space of dimension 1 over Finite Field of size 2,
2: Vector space of dimension 1 over Finite Field of size 2}
sage: S7 = delta_complexes.Sphere(7)
sage: S7.homology(7)
Z
sage: cubical_complexes.KleinBottle().homology(1, base_ring=GF(2))
Vector space of dimension 2 over Finite Field of size 2
If CHomP is installed, Sage can compute generators of homology
groups::
sage: S2 = simplicial_complexes.Sphere(2)
sage: S2.homology(dim=2, generators=True, base_ring=GF(2)) # optional - CHomP
(Vector space of dimension 1 over Finite Field of size 2, [(0, 1, 2) + (0, 1, 3) + (0, 2, 3) + (1, 2, 3)])
When generators are computed, Sage returns a pair for each
dimension: the group and the list of generators. For
simplicial complexes, each generator is represented as a
linear combination of simplices, as above, and for cubical
complexes, each generator is a linear combination of cubes::
sage: S2_cub = cubical_complexes.Sphere(2)
sage: S2_cub.homology(dim=2, generators=True) # optional - CHomP
(Z, [-[[0,1] x [0,1] x [0,0]] + [[0,1] x [0,1] x [1,1]] - [[0,0] x [0,1] x [0,1]] - [[0,1] x [1,1] x [0,1]] + [[0,1] x [0,0] x [0,1]] + [[1,1] x [0,1] x [0,1]]])
"""
from sage.interfaces.chomp import have_chomp, homcubes, homsimpl
from sage.homology.cubical_complex import CubicalComplex
from sage.homology.simplicial_complex import SimplicialComplex
from sage.modules.all import VectorSpace
from sage.homology.chain_complex import HomologyGroup
base_ring = kwds.get('base_ring', ZZ)
cohomology = kwds.get('cohomology', False)
subcomplex = kwds.get('subcomplex', None)
verbose = kwds.get('verbose', False)
algorithm = kwds.get('algorithm', 'auto')
if dim is not None:
if isinstance(dim, (list, tuple)):
low = min(dim) - 1
high = max(dim) + 2
else:
low = dim - 1
high = dim + 2
dims = range(low, high)
else:
dims = None
# try to use CHomP if computing homology (not cohomology) and
# working over Z or F_p, p a prime.
if (algorithm == 'auto' and cohomology is False
and (base_ring == ZZ or (base_ring.is_prime_field()
and base_ring != QQ))):
# homcubes, homsimpl seems fastest if all of homology is computed.
H = None
if isinstance(self, CubicalComplex):
if have_chomp('homcubes'):
if 'subcomplex' in kwds:
del kwds['subcomplex']
H = homcubes(self, subcomplex, **kwds)
elif isinstance(self, SimplicialComplex):
if have_chomp('homsimpl'):
if 'subcomplex' in kwds:
del kwds['subcomplex']
H = homsimpl(self, subcomplex, **kwds)
# now pick off the requested dimensions
if H:
answer = {}
if not dims:
for d in range(self.dimension() + 1):
if base_ring == ZZ:
answer[d] = H.get(d, HomologyGroup(0))
else:
answer[d] = H.get(d, VectorSpace(base_ring, 0))
else:
for d in dims:
if base_ring == ZZ:
answer[d] = H.get(d, HomologyGroup(0))
else:
answer[d] = H.get(d, VectorSpace(base_ring, 0))
if dim is not None:
if not isinstance(dim, (list, tuple)):
if base_ring == ZZ:
answer = answer.get(dim, HomologyGroup(0))
else:
answer = answer.get(dim, VectorSpace(base_ring, 0))
return answer
# Derived classes can implement specialized algorithms using a
# _homology_ method. See SimplicialComplex for one example.
if hasattr(self, '_homology_'):
return self._homology_(dim, **kwds)
C = self.chain_complex(cochain=cohomology, augmented=True,
dimensions=dims, **kwds)
if 'subcomplex' in kwds:
del kwds['subcomplex']
answer = C.homology(**kwds)
if isinstance(answer, dict):
if cohomology:
too_big = self.dimension() + 1
if (not ((isinstance(dim, (list, tuple)) and too_big in dim)
or too_big == dim)
and too_big in answer):
del answer[too_big]
if -2 in answer:
del answer[-2]
if -1 in answer:
del answer[-1]
for d in range(self.dimension() + 1):
if d not in answer:
if base_ring == ZZ:
answer[d] = HomologyGroup(0)
else:
answer[d] = VectorSpace(base_ring, 0)
if dim is not None:
if isinstance(dim, (list, tuple)):
temp = {}
for n in dim:
temp[n] = answer[n]
answer = temp
else: # just a single dimension
if base_ring == ZZ:
answer = answer.get(dim, HomologyGroup(0))
else:
answer = answer.get(dim, VectorSpace(base_ring, 0))
return answer
def homology(self, dim=None, **kwds):
r"""
The reduced homology of this cell complex.
:param dim: If None, then return the homology in every
dimension. If ``dim`` is an integer or list, return the
homology in the given dimensions. (Actually, if ``dim`` is
a list, return the homology in the range from ``min(dim)``
to ``max(dim)``.)
:type dim: integer or list of integers or None; optional,
default None
:param base_ring: commutative ring, must be ZZ or a field.
:type base_ring: optional, default ZZ
:param subcomplex: a subcomplex of this simplicial complex.
Compute homology relative to this subcomplex.
:type subcomplex: optional, default empty
:param generators: If True, return generators for the homology
groups along with the groups. NOTE: this is only implemented
if the CHomP package is available.
:type generators: boolean; optional, default False
:param cohomology: If True, compute cohomology rather than homology.
:type cohomology: boolean; optional, default False
:param algorithm: The options are 'auto', 'dhsw', 'pari' or 'no_chomp'.
See below for a description of what they mean.
:type algorithm: string; optional, default 'auto'
:param verbose: If True, print some messages as the homology is
computed.
:type verbose: boolean; optional, default False
.. note::
The keyword arguments to this function get passed on to
:meth:``chain_complex`` and its homology.
ALGORITHM:
If ``algorithm`` is set to 'auto' (the default), then use
CHomP if available. (CHomP is available at the web page
http://chomp.rutgers.edu/. It is also an experimental package
for Sage.)
CHomP computes homology, not cohomology, and only works over
the integers or finite prime fields. Therefore if any of
these conditions fails, or if CHomP is not present, or if
``algorithm`` is set to 'no_chomp', go to plan B: if ``self``
has a ``_homology`` method -- each simplicial complex has
this, for example -- then call that. Such a method implements
specialized algorithms for the particular type of cell
complex.
Otherwise, move on to plan C: compute the chain complex of
``self`` and compute its homology groups. To do this: over a
field, just compute ranks and nullities, thus obtaining
dimensions of the homology groups as vector spaces. Over the
integers, compute Smith normal form of the boundary matrices
defining the chain complex according to the value of
``algorithm``. If ``algorithm`` is 'auto' or 'no_chomp', then
for each relatively small matrix, use the standard Sage
method, which calls the Pari package. For any large matrix,
reduce it using the Dumas, Heckenbach, Saunders, and Welker
elimination algorithm: see
:func:`sage.homology.chain_complex.dhsw_snf` for details.
Finally, ``algorithm`` may also be 'pari' or 'dhsw', which
forces the named algorithm to be used regardless of the size
of the matrices and regardless of whether CHomP is available.
As of this writing, CHomP is by far the fastest option,
followed by the 'auto' or 'no_chomp' setting of using the
Dumas, Heckenbach, Saunders, and Welker elimination algorithm
for large matrices and Pari for small ones.
EXAMPLES::
sage: P = delta_complexes.RealProjectivePlane()
sage: P.homology()
{0: 0, 1: C2, 2: 0}
sage: P.homology(base_ring=GF(2))
{0: Vector space of dimension 0 over Finite Field of size 2,
1: Vector space of dimension 1 over Finite Field of size 2,
2: Vector space of dimension 1 over Finite Field of size 2}
sage: S7 = delta_complexes.Sphere(7)
sage: S7.homology(7)
Z
sage: cubical_complexes.KleinBottle().homology(1, base_ring=GF(2))
Vector space of dimension 2 over Finite Field of size 2
If CHomP is installed, Sage can compute generators of homology
groups::
sage: S2 = simplicial_complexes.Sphere(2)
sage: S2.homology(dim=2, generators=True, base_ring=GF(2)) # optional - CHomP
(Vector space of dimension 1 over Finite Field of size 2, [(0, 1, 2) + (0, 1, 3) + (0, 2, 3) + (1, 2, 3)])
When generators are computed, Sage returns a pair for each
dimension: the group and the list of generators. For
simplicial complexes, each generator is represented as a
linear combination of simplices, as above, and for cubical
complexes, each generator is a linear combination of cubes::
sage: S2_cub = cubical_complexes.Sphere(2)
sage: S2_cub.homology(dim=2, generators=True) # optional - CHomP
(Z, [-[[0,1] x [0,1] x [0,0]] + [[0,1] x [0,1] x [1,1]] - [[0,0] x [0,1] x [0,1]] - [[0,1] x [1,1] x [0,1]] + [[0,1] x [0,0] x [0,1]] + [[1,1] x [0,1] x [0,1]]])
"""
from sage.interfaces.chomp import have_chomp, homcubes, homsimpl
from sage.homology.cubical_complex import CubicalComplex
from sage.homology.simplicial_complex import SimplicialComplex
from sage.modules.all import VectorSpace
from sage.homology.chain_complex import HomologyGroup
base_ring = kwds.get('base_ring', ZZ)
cohomology = kwds.get('cohomology', False)
subcomplex = kwds.get('subcomplex', None)
verbose = kwds.get('verbose', False)
algorithm = kwds.get('algorithm', 'auto')
if dim is not None:
if isinstance(dim, (list, tuple)):
low = min(dim) - 1
high = max(dim) + 2
else:
low = dim - 1
high = dim + 2
dims = range(low, high)
else:
dims = None
# try to use CHomP if computing homology (not cohomology) and
# working over Z or F_p, p a prime.
if (algorithm == 'auto' and cohomology is False
and (base_ring == ZZ or (base_ring.is_prime_field()
and base_ring != QQ))):
# homcubes, homsimpl seems fastest if all of homology is computed.
H = None
if isinstance(self, CubicalComplex):
if have_chomp('homcubes'):
if 'subcomplex' in kwds:
del kwds['subcomplex']
H = homcubes(self, subcomplex, **kwds)
elif isinstance(self, SimplicialComplex):
if have_chomp('homsimpl'):
if 'subcomplex' in kwds:
del kwds['subcomplex']
H = homsimpl(self, subcomplex, **kwds)
# now pick off the requested dimensions
if H:
answer = {}
if not dims:
for d in range(self.dimension() + 1):
if base_ring == ZZ:
answer[d] = H.get(d, HomologyGroup(0))
else:
answer[d] = H.get(d, VectorSpace(base_ring, 0))
else:
for d in dims:
if base_ring == ZZ:
answer[d] = H.get(d, HomologyGroup(0))
else:
answer[d] = H.get(d, VectorSpace(base_ring, 0))
if dim is not None:
if not isinstance(dim, (list, tuple)):
if base_ring == ZZ:
answer = answer.get(dim, HomologyGroup(0))
else:
answer = answer.get(dim, VectorSpace(base_ring, 0))
return answer
# Derived classes can implement specialized algorithms using a
# _homology_ method. See SimplicialComplex for one example.
if hasattr(self, '_homology_'):
return self._homology_(dim, **kwds)
C = self.chain_complex(cochain=cohomology, augmented=True,
dimensions=dims, **kwds)
if 'subcomplex' in kwds:
del kwds['subcomplex']
answer = C.homology(**kwds)
if isinstance(answer, dict):
if cohomology:
too_big = self.dimension() + 1
if (not ((isinstance(dim, (list, tuple)) and too_big in dim)
or too_big == dim)
and too_big in answer):
del answer[too_big]
if -2 in answer:
del answer[-2]
if -1 in answer:
del answer[-1]
for d in range(self.dimension() + 1):
if d not in answer:
if base_ring == ZZ:
answer[d] = HomologyGroup(0)
else:
answer[d] = VectorSpace(base_ring, 0)
if dim is not None:
if isinstance(dim, (list, tuple)):
temp = {}
for n in dim:
temp[n] = answer[n]
answer = temp
else: # just a single dimension
if base_ring == ZZ:
answer = answer.get(dim, HomologyGroup(0))
else:
answer = answer.get(dim, VectorSpace(base_ring, 0))
return answer