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 a cell complex.

INPUT:

base_ring – must be a field
cell_complex – the cell complex whose homology we are computing
category – (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

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 a cell 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 field GF(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 ring
deg_codomain – the degree of the codomain in the cohomology ring
side – (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 \(n\) for deg_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() and cohomology_ring() for the class of cell complexes.

INPUT:

base_ring – must be a field
cell_complex – the cell complex whose homology we are computing
cohomology – (default: False) if True, return the cohomology as a module
category – (optional) a subcategory of modules with basis

EXAMPLES:

Homology classes are denoted by h_{d,i} where d is the degree of the homology class and i is their index in the list of basis elements in that degree. Cohomology classes are denoted h^{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 at other.

INPUT:

other – an element of the dual space; if self is an element of cohomology in dimension \(n\), then other 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 see algebraic_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. If self 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 of cell 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 field GF(2)
cell_complex – the cell complex whose homology we are computing
category – (optional) a subcategory of modules with basis

This does not include the cohomology argument present for HomologyVectorSpaceWithBasis: use CohomologyRing_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

class Element#: Bases: Element

sage.homology.homology_vector_space_with_basis.is_GF2(R)#

Return True iff R 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 integer
i_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]]