Generic cell complexes#
AUTHORS:
John H. Palmieri (2009-08)
This module defines a class of abstract finite cell complexes. This
is meant as a base class from which other classes (like
SimplicialComplex
,
CubicalComplex
, and
DeltaComplex
) should derive. As
such, most of its properties are not implemented. It is meant for use
by developers producing new classes, not casual users.
Note
Keywords for chain_complex()
,
homology()
, etc.: any keywords given to
the homology()
method get passed on to
the chain_complex()
method and also to
the constructor for chain complexes in
sage.homology.chain_complex.ChainComplex_class
,
as well as its associated
homology()
method.
This means that those keywords should have consistent meaning in
all of those situations. It also means that it is easy to
implement new keywords: for example, if you implement a new
keyword for the
sage.homology.chain_complex.ChainComplex_class.homology()
method,
then it will be automatically accessible through the
homology()
method for cell complexes –
just make sure it gets documented.
- class sage.topology.cell_complex.GenericCellComplex#
Bases:
sage.structure.sage_object.SageObject
Class of abstract cell complexes.
This is meant to be used by developers to produce new classes, not by casual users. Classes which derive from this are
SimplicialComplex
,DeltaComplex
, andCubicalComplex
.Most of the methods here are not implemented, but probably should be implemented in a derived class. Most of the other methods call a non-implemented one; their docstrings contain examples from derived classes in which the various methods have been defined. For example,
homology()
callschain_complex()
; the classDeltaComplex
implementschain_complex()
, and so thehomology()
method here is illustrated with examples involving \(\Delta\)-complexes.EXAMPLES:
It’s hard to give informative examples of the base class, since essentially nothing is implemented.
sage: from sage.topology.cell_complex import GenericCellComplex sage: A = GenericCellComplex()
- alexander_whitney(cell, dim_left)#
The decomposition of
cell
in this complex into left and right factors, suitable for computing cup products. This should provide a cellular approximation for the diagonal map \(K \to K \times K\).This method is not implemented for generic cell complexes, but must be implemented for any derived class to make cup products work in
self.cohomology_ring()
.INPUT:
cell
– a cell in this complexdim_left
– the dimension of the left-hand factors in the decomposition
OUTPUT: a list containing triples
(c, left, right)
.left
andright
should be cells in this complex, andc
an integer. In the cellular approximation of the diagonal map, the chain represented bycell
should get sent to the sum of terms \(c (left \otimes right)\) in the tensor product \(C(K) \otimes C(K)\) of the chain complex for this complex with itself.This gets used in the method
product_on_basis()
for the class of cohomology rings.For simplicial and cubical complexes, the decomposition can be done at the level of individual cells: see
alexander_whitney()
andalexander_whitney()
. Then the method for simplicial complexes just calls the method for individual simplices, and similarly for cubical complexes. For \(\Delta\)-complexes and simplicial sets, the method is instead defined at the level of the cell complex.EXAMPLES:
sage: from sage.topology.cell_complex import GenericCellComplex sage: A = GenericCellComplex() sage: A.alexander_whitney(None, 2) Traceback (most recent call last): ... NotImplementedError: <abstract method alexander_whitney at ...>
- algebraic_topological_model(base_ring=Rational Field)#
Algebraic topological model for this cell complex with coefficients in
base_ring
.The term “algebraic topological model” is defined by Pilarczyk and Réal [PR2015].
This is not implemented for generic cell complexes. For any classes deriving from this one, when this method is implemented, it should essentially just call either
algebraic_topological_model()
oralgebraic_topological_model_delta_complex()
.EXAMPLES:
sage: from sage.topology.cell_complex import GenericCellComplex sage: A = GenericCellComplex() sage: A.algebraic_topological_model(QQ) Traceback (most recent call last): ... NotImplementedError
- betti(dim=None, subcomplex=None)#
The Betti numbers of this simplicial complex as a dictionary (or a single Betti number, if only one dimension is given): the ith Betti number is the rank of the ith homology group.
- Parameters
dim (integer or list of integers or
None
; optional, defaultNone
) – IfNone
, then return every Betti number, as a dictionary with keys the non-negative integers. Ifdim
is an integer or list, return the Betti number for each given dimension. (Actually, ifdim
is a list, return the Betti numbers, as a dictionary, in the range frommin(dim)
tomax(dim)
. Ifdim
is a number, return the Betti number in that dimension.)subcomplex (optional, default
None
) – a subcomplex of this cell complex. Compute the Betti numbers of the homology relative to this subcomplex.
EXAMPLES:
Build the two-sphere as a three-fold join of a two-point space with itself:
sage: S = SimplicialComplex([[0], [1]]) sage: (S*S*S).betti() {0: 1, 1: 0, 2: 1} sage: (S*S*S).betti([1,2]) {1: 0, 2: 1} sage: (S*S*S).betti(2) 1
Or build the two-sphere as a \(\Delta\)-complex:
sage: S2 = delta_complexes.Sphere(2) sage: S2.betti([1,2]) {1: 0, 2: 1}
Or as a cubical complex:
sage: S2c = cubical_complexes.Sphere(2) sage: S2c.betti(2) 1
- cells(subcomplex=None)#
The cells of this cell complex, in the form of a dictionary: the keys are integers, representing dimension, and the value associated to an integer \(d\) is the set of \(d\)-cells. If the optional argument
subcomplex
is present, then return only the cells which are not in the subcomplex.- Parameters
subcomplex (optional, default None) – a subcomplex of this cell complex. Return the cells which are not in this subcomplex.
This is not implemented in general; it should be implemented in any derived class. When implementing, see the warning in the
dimension()
method.This method is used by various other methods, such as
n_cells()
andf_vector()
.EXAMPLES:
sage: from sage.topology.cell_complex import GenericCellComplex sage: A = GenericCellComplex() sage: A.cells() Traceback (most recent call last): ... NotImplementedError: <abstract method cells at ...>
- chain_complex(subcomplex=None, augmented=False, verbose=False, check=True, dimensions=None, base_ring='ZZ', cochain=False)#
This is not implemented for general cell complexes.
Some keywords to possibly implement in a derived class:
subcomplex
– a subcomplex: compute the relative chain complexaugmented
– a bool: whether to return the augmented complexverbose
– a bool: whether to print informational messages as the chain complex is being computedcheck
– a bool: whether to check that the each composite of two consecutive differentials is zerodimensions
– ifNone
, compute the chain complex in alldimensions. If a list or tuple of integers, compute the chain complex in those dimensions, setting the chain groups in all other dimensions to zero.
Definitely implement the following:
base_ring
– commutative ring (optional, default ZZ)cochain
– a bool: whether to return the cochain complex
EXAMPLES:
sage: from sage.topology.cell_complex import GenericCellComplex sage: A = GenericCellComplex() sage: A.chain_complex() Traceback (most recent call last): ... NotImplementedError: <abstract method chain_complex at ...>
- cohomology(dim=None, base_ring=Integer Ring, subcomplex=None, generators=False, algorithm='pari', verbose=False, reduced=True)#
The reduced cohomology of this cell complex.
The arguments are the same as for the
homology()
method, except thathomology()
accepts acohomology
key word, while this function does not:cohomology
is automatically true here. Indeed, this function just callshomology()
withcohomology
set toTrue
.- Parameters
dim –
base_ring –
subcomplex –
algorithm –
verbose –
reduced –
EXAMPLES:
sage: circle = SimplicialComplex([[0,1], [1,2], [0, 2]]) sage: circle.cohomology(0) 0 sage: circle.cohomology(1) Z sage: P2 = SimplicialComplex([[0,1,2], [0,2,3], [0,1,5], [0,4,5], [0,3,4], [1,2,4], [1,3,4], [1,3,5], [2,3,5], [2,4,5]]) # projective plane sage: P2.cohomology(2) C2 sage: P2.cohomology(2, base_ring=GF(2)) Vector space of dimension 1 over Finite Field of size 2 sage: P2.cohomology(2, base_ring=GF(3)) Vector space of dimension 0 over Finite Field of size 3 sage: cubical_complexes.KleinBottle().cohomology(2) C2
Relative cohomology:
sage: T = SimplicialComplex([[0,1]]) sage: U = SimplicialComplex([[0], [1]]) sage: T.cohomology(1, subcomplex=U) Z
A \(\Delta\)-complex example:
sage: s5 = delta_complexes.Sphere(5) sage: s5.cohomology(base_ring=GF(7))[5] Vector space of dimension 1 over Finite Field of size 7
- cohomology_ring(base_ring=Rational Field)#
Return the unreduced cohomology with coefficients in
base_ring
with a chosen basis.This is implemented for simplicial, cubical, and \(\Delta\)-complexes, not for arbitrary generic cell complexes. The resulting elements are suitable for computing cup products. For simplicial complexes, they should be suitable for computing cohomology operations; so far, only mod 2 cohomology operations have been implemented.
INPUT:
base_ring
– coefficient ring (optional, defaultQQ
); must be a field
The basis elements in dimension
dim
are named ‘h^{dim,i}’ where \(i\) ranges between 0 and \(r-1\), if \(r\) is the rank of the cohomology group.Note
For all but the smallest complexes, this is likely to be slower than
cohomology()
(with field coefficients), possibly by several orders of magnitude. This and its companionhomology_with_basis()
carry extra information which allows computation of cup products, for example, but because of speed issues, you may only wish to use these if you need that extra information.EXAMPLES:
sage: K = simplicial_complexes.KleinBottle() sage: H = K.cohomology_ring(QQ); H Cohomology ring of Minimal triangulation of the Klein bottle over Rational Field sage: sorted(H.basis(), key=str) [h^{0,0}, h^{1,0}] sage: H = K.cohomology_ring(GF(2)); H Cohomology ring of Minimal triangulation of the Klein bottle over Finite Field of size 2 sage: sorted(H.basis(), key=str) [h^{0,0}, h^{1,0}, h^{1,1}, h^{2,0}] sage: X = delta_complexes.SurfaceOfGenus(2) sage: H = X.cohomology_ring(QQ); H Cohomology ring of Delta complex with 3 vertices and 29 simplices over Rational Field sage: sorted(H.basis(1), key=str) [h^{1,0}, h^{1,1}, h^{1,2}, h^{1,3}] sage: H = simplicial_complexes.Torus().cohomology_ring(QQ); H Cohomology ring of Minimal triangulation of the torus over Rational Field sage: x = H.basis()[1,0]; x h^{1,0} sage: y = H.basis()[1,1]; y h^{1,1}
You can compute cup products of cohomology classes:
sage: x.cup_product(y) -h^{2,0} sage: x * y # alternate notation -h^{2,0} sage: y.cup_product(x) h^{2,0} sage: x.cup_product(x) 0
Cohomology operations:
sage: RP2 = simplicial_complexes.RealProjectivePlane() sage: K = RP2.suspension() sage: K.set_immutable() sage: y = K.cohomology_ring(GF(2)).basis()[2,0]; y h^{2,0} sage: y.Sq(1) h^{3,0}
To compute the cohomology ring, the complex must be “immutable”. This is only relevant for simplicial complexes, and most simplicial complexes are immutable, but certain constructions make them mutable. The suspension is one example, and this is the reason for calling
K.set_immutable()
above. Another example:sage: S1 = simplicial_complexes.Sphere(1) sage: T = S1.product(S1) sage: T.is_immutable() False sage: T.cohomology_ring() Traceback (most recent call last): ... ValueError: this simplicial complex must be immutable; call set_immutable() sage: T.set_immutable() sage: T.cohomology_ring() Cohomology ring of Simplicial complex with 9 vertices and 18 facets over Rational Field
- dimension()#
The dimension of this cell complex: the maximum dimension of its cells.
Warning
If the
cells()
method callsdimension()
, then you’ll get an infinite loop. So either don’t usedimension()
or overridedimension()
.EXAMPLES:
sage: simplicial_complexes.RandomComplex(d=5, n=8).dimension() 5 sage: delta_complexes.Sphere(3).dimension() 3 sage: T = cubical_complexes.Torus() sage: T.product(T).dimension() 4
- disjoint_union(right)#
The disjoint union of this cell complex with another one.
- Parameters
right – the other cell complex (the right-hand factor)
Disjoint unions are not implemented for general cell complexes.
EXAMPLES:
sage: from sage.topology.cell_complex import GenericCellComplex sage: A = GenericCellComplex(); B = GenericCellComplex() sage: A.disjoint_union(B) Traceback (most recent call last): ... NotImplementedError: <abstract method disjoint_union at ...>
- euler_characteristic()#
The Euler characteristic of this cell complex: the alternating sum over \(n \geq 0\) of the number of \(n\)-cells.
EXAMPLES:
sage: simplicial_complexes.Simplex(5).euler_characteristic() 1 sage: delta_complexes.Sphere(6).euler_characteristic() 2 sage: cubical_complexes.KleinBottle().euler_characteristic() 0
- f_vector()#
The \(f\)-vector of this cell complex: a list whose \(n^{th}\) item is the number of \((n-1)\)-cells. Note that, like all lists in Sage, this is indexed starting at 0: the 0th element in this list is the number of \((-1)\)-cells (which is 1: the empty cell is the only \((-1)\)-cell).
EXAMPLES:
sage: simplicial_complexes.KleinBottle().f_vector() [1, 8, 24, 16] sage: delta_complexes.KleinBottle().f_vector() [1, 1, 3, 2] sage: cubical_complexes.KleinBottle().f_vector() [1, 42, 84, 42]
- face_poset()#
The face poset of this cell complex, the poset of nonempty cells, ordered by inclusion.
This uses the
cells()
method, and also assumes that for each cellf
, all off.faces()
,tuple(f)
, andf.dimension()
make sense. (If this is not the case in some derived class, as happens with \(\Delta\)-complexes, then override this method.)EXAMPLES:
sage: P = SimplicialComplex([[0, 1], [1,2], [2,3]]).face_poset(); P Finite poset containing 7 elements sage: sorted(P.list()) [(0,), (0, 1), (1,), (1, 2), (2,), (2, 3), (3,)] sage: S2 = cubical_complexes.Sphere(2) sage: S2.face_poset() Finite poset containing 26 elements
- graph()#
The 1-skeleton of this cell complex, as a graph.
This is not implemented for general cell complexes.
EXAMPLES:
sage: from sage.topology.cell_complex import GenericCellComplex sage: A = GenericCellComplex() sage: A.graph() Traceback (most recent call last): ... NotImplementedError
- homology(dim=None, base_ring=Integer Ring, subcomplex=None, generators=False, cohomology=False, algorithm='pari', verbose=False, reduced=True, **kwds)#
The (reduced) homology of this cell complex.
- Parameters
dim (integer or list of integers or None; optional, default None) – If None, then return the homology in every dimension. If
dim
is an integer or list, return the homology in the given dimensions. (Actually, ifdim
is a list, return the homology in the range frommin(dim)
tomax(dim)
.)base_ring (optional, default ZZ) – commutative ring, must be ZZ or a field.
subcomplex (optional, default empty) – a subcomplex of this simplicial complex. Compute homology relative to this subcomplex.
generators (boolean; optional, default False) – If
True
, return generators for the homology groups along with the groups.cohomology (boolean; optional, default False) – If True, compute cohomology rather than homology.
algorithm (string; optional, default 'pari') – The options are ‘auto’, ‘dhsw’, or ‘pari’. See below for a description of what they mean.
verbose (boolean; optional, default False) – If True, print some messages as the homology is computed.
reduced (boolean; optional, default
True
) – IfTrue
, return the reduced homology.
ALGORITHM:
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 ofalgorithm
. Ifalgorithm
is'auto'
, 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 [DHSW2003]: seedhsw_snf()
for details.'no_chomp'
is a synonym for'auto'
, maintained for backward-compatibility.algorithm
may also be'pari'
or'dhsw'
, which forces the named algorithm to be used regardless of the size of the matrices.As of this writing,
'pari'
is the fastest standard option.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
Sage can compute generators of homology groups:
sage: S2 = simplicial_complexes.Sphere(2) sage: S2.homology(dim=2, generators=True, base_ring=GF(2)) [(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) [(Z, [0,0] x [0,1] x [0,1] - [0,1] x [0,0] x [0,1] + [0,1] x [0,1] x [0,0] - [0,1] x [0,1] x [1,1] + [0,1] x [1,1] x [0,1] - [1,1] x [0,1] x [0,1])]
Similarly for simpicial sets:
sage: S = simplicial_sets.Sphere(2) sage: S.homology(generators=True) {0: [], 1: 0, 2: [(Z, sigma_2)]}
- homology_with_basis(base_ring=Rational Field, cohomology=False)#
Return the unreduced homology of this complex with coefficients in
base_ring
with a chosen basis.This is implemented for simplicial, cubical, and \(\Delta\)-complexes, not for arbitrary generic cell complexes.
INPUT:
base_ring
– coefficient ring (optional, defaultQQ
); must be a fieldcohomology
– boolean (optional, defaultFalse
); ifTrue
, return cohomology instead of homology
Homology basis elements are named ‘h_{dim,i}’ where i ranges between 0 and \(r-1\), if \(r\) is the rank of the homology group. Cohomology basis elements are denoted \(h^{dim,i}\) instead.
See also
If
cohomology
isTrue
, this returns the cohomology as a graded module. For the ring structure, usecohomology_ring()
.EXAMPLES:
sage: K = simplicial_complexes.KleinBottle() sage: H = K.homology_with_basis(QQ); H Homology module of Minimal triangulation of the Klein bottle over Rational Field sage: sorted(H.basis(), key=str) [h_{0,0}, h_{1,0}] sage: H = K.homology_with_basis(GF(2)); H Homology module of Minimal triangulation of the Klein bottle over Finite Field of size 2 sage: sorted(H.basis(), key=str) [h_{0,0}, h_{1,0}, h_{1,1}, h_{2,0}]
The homology is constructed as a graded object, so for example, you can ask for the basis in a single degree:
sage: H.basis(1) Finite family {(1, 0): h_{1,0}, (1, 1): h_{1,1}} sage: S3 = delta_complexes.Sphere(3) sage: H = S3.homology_with_basis(QQ, cohomology=True) sage: list(H.basis(3)) [h^{3,0}]
- is_acyclic(base_ring=Integer Ring)#
True if the reduced homology with coefficients in
base_ring
of this cell complex is zero.INPUT:
base_ring
– optional, defaultZZ
. Compute homology with coefficients in this ring.
EXAMPLES:
sage: RP2 = simplicial_complexes.RealProjectivePlane() sage: RP2.is_acyclic() False sage: RP2.is_acyclic(QQ) True
This first computes the Euler characteristic: if it is not 1, the complex cannot be acyclic. So this should return
False
reasonably quickly on complexes with Euler characteristic not equal to 1:sage: K = cubical_complexes.KleinBottle() sage: C = cubical_complexes.Cube(2) sage: P = K.product(C) sage: P Cubical complex with 168 vertices and 1512 cubes sage: P.euler_characteristic() 0 sage: P.is_acyclic() False
- is_connected()#
True if this cell complex is connected.
EXAMPLES:
sage: V = SimplicialComplex([[0,1,2],[3]]) sage: V Simplicial complex with vertex set (0, 1, 2, 3) and facets {(3,), (0, 1, 2)} sage: V.is_connected() False sage: X = SimplicialComplex([[0,1,2]]) sage: X.is_connected() True sage: U = simplicial_complexes.ChessboardComplex(3,3) sage: U.is_connected() True sage: W = simplicial_complexes.Sphere(3) sage: W.is_connected() True sage: S = SimplicialComplex([[0,1],[2,3]]) sage: S.is_connected() False sage: cubical_complexes.Sphere(0).is_connected() False sage: cubical_complexes.Sphere(2).is_connected() True
- join(right)#
The join of this cell complex with another one.
- Parameters
right – the other cell complex (the right-hand factor)
Joins are not implemented for general cell complexes. They may be implemented in some derived classes (like simplicial complexes).
EXAMPLES:
sage: from sage.topology.cell_complex import GenericCellComplex sage: A = GenericCellComplex(); B = GenericCellComplex() sage: A.join(B) Traceback (most recent call last): ... NotImplementedError: <abstract method join at ...>
- n_cells(n, subcomplex=None)#
List of cells of dimension
n
of this cell complex. If the optional argumentsubcomplex
is present, then return then
-dimensional cells which are not in the subcomplex.- Parameters
n (non-negative integer) – the dimension
subcomplex (optional, default
None
) – a subcomplex of this cell complex. Return the cells which are not in this subcomplex.
Note
The resulting list need not be sorted. If you want a sorted list of \(n\)-cells, use
_n_cells_sorted()
.EXAMPLES:
sage: delta_complexes.Torus().n_cells(1) [(0, 0), (0, 0), (0, 0)] sage: cubical_complexes.Cube(1).n_cells(0) [[1,1], [0,0]]
- n_chains(n, base_ring=Integer Ring, cochains=False)#
Return the free module of chains in degree
n
overbase_ring
.INPUT:
n
– integerbase_ring
– ring (optional, default \(\ZZ\))cochains
– boolean (optional, defaultFalse
); ifTrue
, return cochains instead
The only difference between chains and cochains is notation. In a simplicial complex, for example, a simplex
(0,1,2)
is written as “(0,1,2)” in the group of chains but as “chi_(0,1,2)” in the group of cochains.EXAMPLES:
sage: S2 = simplicial_complexes.Sphere(2) sage: S2.n_chains(1, QQ) Free module generated by {(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)} over Rational Field sage: list(simplicial_complexes.Sphere(2).n_chains(1, QQ, cochains=False).basis()) [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)] sage: list(simplicial_complexes.Sphere(2).n_chains(1, QQ, cochains=True).basis()) [\chi_(0, 1), \chi_(0, 2), \chi_(0, 3), \chi_(1, 2), \chi_(1, 3), \chi_(2, 3)]
- n_skeleton(n)#
The \(n\)-skeleton of this cell complex: the cell complex obtained by discarding all of the simplices in dimensions larger than \(n\).
- Parameters
n – non-negative integer
This is not implemented for general cell complexes.
EXAMPLES:
sage: from sage.topology.cell_complex import GenericCellComplex sage: A = GenericCellComplex() sage: A.n_skeleton(3) Traceback (most recent call last): ... NotImplementedError: <abstract method n_skeleton at ...>
- product(right, rename_vertices=True)#
The (Cartesian) product of this cell complex with another one.
Products are not implemented for general cell complexes. They may be implemented in some derived classes (like simplicial complexes).
EXAMPLES:
sage: from sage.topology.cell_complex import GenericCellComplex sage: A = GenericCellComplex(); B = GenericCellComplex() sage: A.product(B) Traceback (most recent call last): ... NotImplementedError: <abstract method product at ...>
- wedge(right)#
The wedge (one-point union) of this cell complex with another one.
- Parameters
right – the other cell complex (the right-hand factor)
Wedges are not implemented for general cell complexes.
EXAMPLES:
sage: from sage.topology.cell_complex import GenericCellComplex sage: A = GenericCellComplex(); B = GenericCellComplex() sage: A.wedge(B) Traceback (most recent call last): ... NotImplementedError: <abstract method wedge at ...>