Homology and cohomology with a basis#
This module provides homology and cohomology vector spaces suitable for computing cup products and cohomology operations.
REFERENCES:
AUTHORS:
John H. Palmieri, Travis Scrimshaw (2015-09)
- class sage.homology.homology_vector_space_with_basis.CohomologyRing(base_ring, cell_complex, category=None)#
Bases:
HomologyVectorSpaceWithBasis
The cohomology ring.
Note
This is not intended to be created directly by the user, but instead via the
cohomology ring
of acell complex
.INPUT:
base_ring
– must be a fieldcell_complex
– the cell complex whose homology we are computingcategory
– (optional) a subcategory of modules with basis
EXAMPLES:
sage: CP2 = simplicial_complexes.ComplexProjectivePlane() sage: H = CP2.cohomology_ring(QQ) sage: H.basis(2) Finite family {(2, 0): h^{2,0}} sage: x = H.basis(2)[2,0]
The product structure is the cup product:
sage: x.cup_product(x) -h^{4,0} sage: x * x -h^{4,0}
- class Element#
Bases:
Element
- cup_product(other)#
Return the cup product of this element and
other
.Algorithm: see González-Díaz and Réal [GDR2003], p. 88. Given two cohomology classes, lift them to cocycle representatives via the chain contraction for this complex, using
to_cycle()
. In the sum of their dimensions, look at all of the homology classes \(\gamma\): lift each of those to a cycle representative, apply the Alexander-Whitney diagonal map to each cell in the cycle, evaluate the two cocycles on these factors, and multiply. The result is the value of the cup product cocycle on this homology class. After this has been done for all homology classes, since homology and cohomology are dual, one can tell which cohomology class corresponds to the cup product.See also
EXAMPLES:
sage: RP3 = simplicial_complexes.RealProjectiveSpace(3) sage: H = RP3.cohomology_ring(GF(2)) sage: c = H.basis()[1,0] sage: c.cup_product(c) h^{2,0} sage: c * c * c h^{3,0}
We can also take powers:
sage: RP2 = simplicial_complexes.RealProjectivePlane() sage: a = RP2.cohomology_ring(GF(2)).basis()[1,0] sage: a**0 h^{0,0} sage: a**1 h^{1,0} sage: a**2 h^{2,0} sage: a**3 0
A non-connected example:
sage: K = cubical_complexes.Torus().disjoint_union(cubical_complexes.Sphere(2)) sage: a,b = K.cohomology_ring(QQ).basis(2) sage: a**0 h^{0,0} + h^{0,1}
- one()#
The multiplicative identity element.
EXAMPLES:
sage: H = simplicial_complexes.Torus().cohomology_ring(QQ) sage: H.one() h^{0,0} sage: all(H.one() * x == x == x * H.one() for x in H.basis()) True
- product_on_basis(li, ri)#
The cup product of the basis elements indexed by
li
andri
in this cohomology ring.INPUT:
li
,ri
– index of a cohomology class
See also
CohomologyRing.Element.cup_product()
– the documentation for this method describes the algorithm.EXAMPLES:
sage: RP3 = simplicial_complexes.RealProjectiveSpace(3) sage: H = RP3.cohomology_ring(GF(2)) sage: c = H.basis()[1,0] sage: c.cup_product(c).cup_product(c) # indirect doctest h^{3,0} sage: T = simplicial_complexes.Torus() sage: x,y = T.cohomology_ring(QQ).basis(1) sage: x.cup_product(y) -h^{2,0} sage: x.cup_product(x) 0 sage: one = T.cohomology_ring(QQ).basis()[0,0] sage: x.cup_product(one) h^{1,0} sage: one.cup_product(y) == y True sage: one.cup_product(one) h^{0,0} sage: x.cup_product(y) + y.cup_product(x) 0
This also works with cubical complexes:
sage: T = cubical_complexes.Torus() sage: x,y = T.cohomology_ring(QQ).basis(1) sage: x.cup_product(y) h^{2,0} sage: x.cup_product(x) 0
\(\Delta\)-complexes:
sage: T_d = delta_complexes.Torus() sage: a,b = T_d.cohomology_ring(QQ).basis(1) sage: a.cup_product(b) h^{2,0} sage: b.cup_product(a) -h^{2,0} sage: RP2 = delta_complexes.RealProjectivePlane() sage: w = RP2.cohomology_ring(GF(2)).basis()[1,0] sage: w.cup_product(w) h^{2,0}
and simplicial sets:
sage: from sage.topology.simplicial_set_examples import RealProjectiveSpace sage: RP5 = RealProjectiveSpace(5) # needs sage.groups sage: x = RP5.cohomology_ring(GF(2)).basis()[1,0] # needs sage.groups sage: x**4 # needs sage.groups h^{4,0}
A non-connected example:
sage: K = cubical_complexes.Torus().disjoint_union(cubical_complexes.Torus()) sage: a,b,c,d = K.cohomology_ring(QQ).basis(1) sage: x,y = K.cohomology_ring(QQ).basis(0) sage: a.cup_product(x) == a True sage: a.cup_product(y) 0
- class sage.homology.homology_vector_space_with_basis.CohomologyRing_mod2(base_ring, cell_complex)#
Bases:
CohomologyRing
The mod 2 cohomology ring.
Based on
CohomologyRing
, with Steenrod operations included.Note
This is not intended to be created directly by the user, but instead via the
cohomology ring
of acell complex
.Todo
Implement Steenrod operations on (co)homology at odd primes, and thereby implement this class over \(\GF{p}\) for any \(p\).
INPUT:
base_ring
– must be the fieldGF(2)
cell_complex
– the cell complex whose homology we are computing
EXAMPLES:
Mod 2 cohomology operations are defined on both the left and the right:
sage: CP2 = simplicial_complexes.ComplexProjectivePlane() sage: Hmod2 = CP2.cohomology_ring(GF(2)) sage: y = Hmod2.basis(2)[2,0] sage: y.Sq(2) h^{4,0} sage: # needs sage.groups sage: Y = simplicial_sets.RealProjectiveSpace(6).suspension() sage: H_Y = Y.cohomology_ring(GF(2)) sage: b = H_Y.basis()[2,0] sage: b.Sq(1) h^{3,0} sage: b.Sq(2) 0 sage: c = H_Y.basis()[4,0] sage: c.Sq(1) h^{5,0} sage: c.Sq(2) h^{6,0} sage: c.Sq(3) h^{7,0} sage: c.Sq(4) 0
Cohomology can be viewed as a left module over the Steenrod algebra, and also as a right module:
sage: # needs sage.groups sage: RP4 = simplicial_sets.RealProjectiveSpace(4) sage: H = RP4.cohomology_ring(GF(2)) sage: x = H.basis()[1,0] sage: Sq(0,1) * x h^{4,0} sage: Sq(3) * x 0 sage: x * Sq(3) h^{4,0}
- class Element#
Bases:
Element
- Sq(i)#
Return the result of applying \(Sq^i\) to this element.
INPUT:
i
– nonnegative integer
Warning
The main implementation is only for simplicial complexes and simplicial sets; cubical complexes are converted to simplicial complexes first. Note that this converted complex may be large and so computations may be slow. There is no implementation for \(\Delta\)-complexes.
This cohomology operation is only defined in characteristic 2. Odd primary Steenrod operations are not implemented.
Algorithm: see González-Díaz and Réal [GDR1999], Corollary 3.2.
EXAMPLES:
sage: RP2 = simplicial_complexes.RealProjectiveSpace(2) sage: x = RP2.cohomology_ring(GF(2)).basis()[1,0] sage: x.Sq(1) h^{2,0} sage: K = RP2.suspension() sage: K.set_immutable() sage: y = K.cohomology_ring(GF(2)).basis()[2,0] sage: y.Sq(1) h^{3,0} sage: # long time sage: # needs sage.groups sage: RP4 = simplicial_complexes.RealProjectiveSpace(4) sage: H = RP4.cohomology_ring(GF(2)) sage: x = H.basis()[1,0] sage: y = H.basis()[2,0] sage: z = H.basis()[3,0] sage: x.Sq(1) == y True sage: z.Sq(1) h^{4,0}
This calculation is much faster with simplicial sets (on one machine, 20 seconds with a simplicial complex, 4 ms with a simplicial set).
sage: RP4_ss = simplicial_sets.RealProjectiveSpace(4) # needs sage.groups sage: z_ss = RP4_ss.cohomology_ring(GF(2)).basis()[3,0] # needs sage.groups sage: z_ss.Sq(1) # needs sage.groups h^{4,0}
- steenrod_module_map(deg_domain, deg_codomain, side='left')#
Return a component of the module structure map \(A \otimes H \to H\), where \(H\) is this cohomology ring and \(A\) is the Steenrod algebra.
INPUT:
deg_domain
– the degree of the domain in the cohomology ringdeg_codomain
– the degree of the codomain in the cohomology ringside
– (default'left'
) whether we are computing the action as a left module action or a right module
We will write this with respect to the left action; for the right action, just switch all of the the tensors. Writing \(m\) for
deg_domain
and \(n\) fordeg_codomain
, this returns \(A^{n-m} \otimes H^{m} \to H^{n}\), one single component of the map making \(H\) into an \(A\)-module.Warning
This is only implemented in characteristic two. The main implementation is only for simplicial complexes and simplicial sets; cubical complexes are converted to simplicial complexes first. Note that this converted complex may be large and so computations may be slow. There is no implementation for \(\Delta\)-complexes.
ALGORITHM:
Use the Milnor basis for the truncated Steenrod algebra \(A\), and for cohomology, use the basis with which it is equipped. For each pair of basis elements \(a\) and \(h\), compute the product \(a \otimes h\), and use this to assemble a matrix defining the action map via multiplication on the appropriate side. That is, if
side
is'left'
, return a matrix suitable for multiplication on the left, etc.EXAMPLES:
sage: # needs sage.groups sage: RP4 = simplicial_sets.RealProjectiveSpace(4) sage: H = RP4.cohomology_ring(GF(2)) sage: H.steenrod_module_map(1, 2) [1] sage: H.steenrod_module_map(1, 3) [0] sage: H.steenrod_module_map(1, 4, 'left') [1 0] sage: H.steenrod_module_map(1, 4, 'right') [1] [1]
Products of projective spaces:
sage: RP3 = simplicial_sets.RealProjectiveSpace(3) sage: K = RP3.product(RP3) sage: H = K.cohomology_ring(GF(2)) sage: H Cohomology ring of RP^3 x RP^3 over Finite Field of size 2
There is one column for each element \(a \otimes b\), where \(a\) is a basis element for the Steenrod algebra and \(b\) is a basis element for the cohomology algebra. There is one row for each basis element of the cohomology algebra. Unfortunately, the chosen basis for this truncated polynomial algebra is not the monomial basis:
sage: x1, x2 = H.basis(1) sage: x1 * x1 h^{2,0} + h^{2,1} sage: x2 * x2 h^{2,2} sage: x1 * x2 h^{2,0} sage: H.steenrod_module_map(1, 2) [1 0] [1 0] [0 1] sage: H.steenrod_module_map(1, 3, 'left') [0 0] [0 0] [0 0] [0 0] sage: H.steenrod_module_map(1, 3, 'right') [0 0 0 0] [0 0 0 0] sage: H.steenrod_module_map(2, 3) [0 0 0] [1 1 0] [0 0 0] [0 0 0]
- class sage.homology.homology_vector_space_with_basis.HomologyVectorSpaceWithBasis(base_ring, cell_complex, cohomology=False, category=None)#
Bases:
CombinatorialFreeModule
Homology (or cohomology) vector space.
This provides enough structure to allow the computation of cup products and cohomology operations. See the class
CohomologyRing
(which derives from this) for examples.It also requires field coefficients (hence the “VectorSpace” in the name of the class).
Note
This is not intended to be created directly by the user, but instead via the methods
homology_with_basis()
andcohomology_ring()
for the class ofcell complexes
.INPUT:
base_ring
– must be a fieldcell_complex
– the cell complex whose homology we are computingcohomology
– (default:False
) ifTrue
, return the cohomology as a modulecategory
– (optional) a subcategory of modules with basis
EXAMPLES:
Homology classes are denoted by
h_{d,i}
whered
is the degree of the homology class andi
is their index in the list of basis elements in that degree. Cohomology classes are denotedh^{1,0}
:sage: RP2 = cubical_complexes.RealProjectivePlane() sage: RP2.homology_with_basis(GF(2)) Homology module of Cubical complex with 21 vertices and 81 cubes over Finite Field of size 2 sage: RP2.cohomology_ring(GF(2)) Cohomology ring of Cubical complex with 21 vertices and 81 cubes over Finite Field of size 2 sage: simplicial_complexes.Torus().homology_with_basis(QQ) Homology module of Minimal triangulation of the torus over Rational Field
To access a basis element, use its degree and index (0 or 1 in the 1st cohomology group of a torus):
sage: H = simplicial_complexes.Torus().cohomology_ring(QQ) sage: H.basis(1) Finite family {(1, 0): h^{1,0}, (1, 1): h^{1,1}} sage: x = H.basis()[1,0]; x h^{1,0} sage: y = H.basis()[1,1]; y h^{1,1} sage: 2*x-3*y 2*h^{1,0} - 3*h^{1,1}
You can compute cup products of cohomology classes:
sage: x.cup_product(y) -h^{2,0} sage: y.cup_product(x) h^{2,0} sage: x.cup_product(x) 0
This works with simplicial, cubical, and \(\Delta\)-complexes, and also simplicial sets:
sage: Torus_c = cubical_complexes.Torus() sage: H = Torus_c.cohomology_ring(GF(2)) sage: x,y = H.basis(1) sage: x.cup_product(x) 0 sage: x.cup_product(y) h^{2,0} sage: y.cup_product(y) 0 sage: Klein_d = delta_complexes.KleinBottle() sage: H = Klein_d.cohomology_ring(GF(2)) sage: u,v = sorted(H.basis(1)) sage: u.cup_product(u) h^{2,0} sage: u.cup_product(v) 0 sage: v.cup_product(v) h^{2,0}
An isomorphism between the rings for the cubical model and the \(\Delta\)-complex model can be obtained by sending \(x\) to \(u+v\), \(y\) to \(v\).
sage: # needs sage.groups sage: X = simplicial_sets.RealProjectiveSpace(6) sage: H_X = X.cohomology_ring(GF(2)) sage: a = H_X.basis()[1,0] sage: a**6 h^{6,0} sage: a**7 0
All products of positive-dimensional elements in a suspension should be zero:
sage: # needs sage.groups sage: Y = X.suspension() sage: H_Y = Y.cohomology_ring(GF(2)) sage: b = H_Y.basis()[2,0] sage: b**2 0 sage: B = sorted(H_Y.basis())[1:] sage: B [h^{2,0}, h^{3,0}, h^{4,0}, h^{5,0}, h^{6,0}, h^{7,0}] sage: import itertools sage: [a*b for (a,b) in itertools.combinations(B, 2)] [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
The basis elements in the simplicial complex case have been chosen differently; apply the change of basis \(x \mapsto a + b\), \(y \mapsto b\) to see the same product structure.
sage: Klein_s = simplicial_complexes.KleinBottle() sage: H = Klein_s.cohomology_ring(GF(2)) sage: a,b = H.basis(1) sage: a.cup_product(a) 0 sage: a.cup_product(b) h^{2,0} sage: (a+b).cup_product(a+b) h^{2,0} sage: b.cup_product(b) h^{2,0}
- class Element#
Bases:
IndexedFreeModuleElement
- eval(other)#
Evaluate
self
atother
.INPUT:
other
– an element of the dual space; ifself
is an element of cohomology in dimension \(n\), thenother
should be an element of homology in dimension \(n\), and vice versa
This just calls the
eval()
method on the representing chains and cochains.EXAMPLES:
sage: T = simplicial_complexes.Torus() sage: homology = T.homology_with_basis(QQ) sage: cohomology = T.cohomology_ring(QQ) sage: a1, a2 = homology.basis(1) sage: alpha1, alpha2 = cohomology.basis(1) sage: a1.to_cycle() (0, 3) - (0, 6) + (3, 6) sage: alpha1.to_cycle() -\chi_(1, 3) - \chi_(1, 4) - \chi_(2, 3) - \chi_(2, 4) - \chi_(2, 5) + \chi_(3, 6) sage: a1.eval(alpha1) 1 sage: alpha2.to_cycle() \chi_(1, 3) + \chi_(1, 4) + \chi_(1, 6) + \chi_(2, 4) - \chi_(4, 5) + \chi_(5, 6) sage: alpha2.eval(a1) 0 sage: (2 * alpha2).eval(a1 + a2) 2
- to_cycle()#
(Co)cycle representative of this homogeneous (co)homology class.
EXAMPLES:
sage: S2 = simplicial_complexes.Sphere(2) sage: H = S2.homology_with_basis(QQ) sage: h20 = H.basis()[2,0]; h20 h_{2,0} sage: h20.to_cycle() -(0, 1, 2) + (0, 1, 3) - (0, 2, 3) + (1, 2, 3)
Chains are written as linear combinations of simplices \(\sigma\). Cochains are written as linear combinations of characteristic functions \(\chi_{\sigma}\) for those simplices:
sage: S2.cohomology_ring(QQ).basis()[2,0].to_cycle() \chi_(1, 2, 3) sage: S2.cohomology_ring(QQ).basis()[0,0].to_cycle() \chi_(0,) + \chi_(1,) + \chi_(2,) + \chi_(3,)
- basis(d=None)#
Return (the degree
d
homogeneous component of) the basis of this graded vector space.INPUT:
d
– (optional) the degree
EXAMPLES:
sage: RP2 = simplicial_complexes.ProjectivePlane() sage: H = RP2.homology_with_basis(QQ) sage: H.basis() Finite family {(0, 0): h_{0,0}} sage: H.basis(0) Finite family {(0, 0): h_{0,0}} sage: H.basis(1) Finite family {} sage: H.basis(2) Finite family {}
- complex()#
The cell complex whose homology is being computed.
EXAMPLES:
sage: H = simplicial_complexes.Simplex(2).homology_with_basis(QQ) sage: H.complex() The 2-simplex
- contraction()#
The chain contraction associated to this homology computation.
That is, to work with chain representatives of homology classes, we need the chain complex \(C\) associated to the cell complex, the chain complex \(H\) of its homology (with trivial differential), chain maps \(\pi: C \to H\) and \(\iota: H \to C\), and a chain contraction \(\phi\) giving a chain homotopy between \(1_C\) and \(\iota \circ \pi\).
OUTPUT: \(\phi\)
See
ChainContraction
for information about chain contractions, and seealgebraic_topological_model()
for the construction of this particular chain contraction \(\phi\).EXAMPLES:
sage: H = simplicial_complexes.Simplex(2).homology_with_basis(QQ) sage: H.contraction() Chain homotopy between: Chain complex endomorphism of Chain complex with at most 3 nonzero terms over Rational Field and Chain complex endomorphism of Chain complex with at most 3 nonzero terms over Rational Field
From the chain contraction, one can also recover the maps \(\pi\) and \(\iota\):
sage: phi = H.contraction() sage: phi.pi() Chain complex morphism: From: Chain complex with at most 3 nonzero terms over Rational Field To: Chain complex with at most 1 nonzero terms over Rational Field sage: phi.iota() Chain complex morphism: From: Chain complex with at most 1 nonzero terms over Rational Field To: Chain complex with at most 3 nonzero terms over Rational Field
- degree_on_basis(i)#
Return the degree of the basis element indexed by
i
.EXAMPLES:
sage: H = simplicial_complexes.Torus().homology_with_basis(GF(7)) sage: H.degree_on_basis((2,0)) 2
- dual()#
Return the dual space.
If
self
is homology, return the cohomology ring. Ifself
is cohomology, return the homology as a vector space.EXAMPLES:
sage: T = simplicial_complexes.Torus() sage: hom = T.homology_with_basis(GF(2)) sage: coh = T.cohomology_ring(GF(2)) sage: hom.dual() is coh True sage: coh.dual() is hom True
- class sage.homology.homology_vector_space_with_basis.HomologyVectorSpaceWithBasis_mod2(base_ring, cell_complex, category=None)#
Bases:
HomologyVectorSpaceWithBasis
Homology vector space mod 2.
Based on
HomologyVectorSpaceWithBasis
, with Steenrod operations included.Note
This is not intended to be created directly by the user, but instead via the method
homology_with_basis()
for the class ofcell complexes
.Todo
Implement Steenrod operations on (co)homology at odd primes, and thereby implement this class over \(\GF{p}\) for any \(p\).
INPUT:
base_ring
– must be the fieldGF(2)
cell_complex
– the cell complex whose homology we are computingcategory
– (optional) a subcategory of modules with basis
This does not include the
cohomology
argument present forHomologyVectorSpaceWithBasis
: useCohomologyRing_mod2
for cohomology.EXAMPLES:
Mod 2 cohomology operations are defined on both the left and the right:
sage: # needs sage.groups sage: RP4 = simplicial_sets.RealProjectiveSpace(5) sage: H = RP4.homology_with_basis(GF(2)) sage: x4 = H.basis()[4,0] sage: x4 * Sq(1) h_{3,0} sage: Sq(1) * x4 h_{3,0} sage: Sq(2) * x4 h_{2,0} sage: Sq(3) * x4 h_{1,0} sage: Sq(0,1) * x4 h_{1,0} sage: x4 * Sq(0,1) h_{1,0} sage: Sq(3) * x4 h_{1,0} sage: x4 * Sq(3) 0
- sage.homology.homology_vector_space_with_basis.is_GF2(R)#
Return
True
iffR
is isomorphic to the field \(\GF{2}\).EXAMPLES:
sage: from sage.homology.homology_vector_space_with_basis import is_GF2 sage: is_GF2(GF(2)) True sage: is_GF2(GF(2, impl='ntl')) True sage: is_GF2(GF(3)) False
- sage.homology.homology_vector_space_with_basis.sum_indices(k, i_k_plus_one, S_k_plus_one)#
This is a recursive function for computing the indices for the nested sums in González-Díaz and Réal [GDR1999], Corollary 3.2.
In the paper, given indices \(i_n\), \(i_{n-1}\), …, \(i_{k+1}\), given \(k\), and given \(S(k+1)\), the number \(S(k)\) is defined to be
\[S(k) = -S(k+1) + floor(k/2) + floor((k+1)/2) + i_{k+1},\]and \(i_k\) ranges from \(S(k)\) to \(i_{k+1}-1\). There are two special cases: if \(k=0\), then \(i_0 = S(0)\). Also, the initial case of \(S(k)\) is \(S(n)\), which is set in the method
Sq()
before calling this function. For this function, given \(k\), \(i_{k+1}\), and \(S(k+1)\), return a list consisting of the allowable possible indices \([i_k, i_{k-1}, ..., i_1, i_0]\) given by the above formula.INPUT:
k
– non-negative integeri_k_plus_one
– the positive integer \(i_{k+1}\)S_k_plus_one
– the integer \(S(k+1)\)
EXAMPLES:
sage: from sage.homology.homology_vector_space_with_basis import sum_indices sage: sum_indices(1, 3, 3) [[1, 0], [2, 1]] sage: sum_indices(0, 4, 2) [[2]]