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)

Bases: sage.homology.homology_vector_space_with_basis.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

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}

There are mod 2 cohomology operations defined, also, for simplicial complexes and simplicial sets:

sage: Hmod2 = CP2.cohomology_ring(GF(2))
sage: y = Hmod2.basis(2)[2,0]
sage: y.Sq(2)
h^{4,0}

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
class Element

Bases: sage.homology.homology_vector_space_with_basis.HomologyVectorSpaceWithBasis.Element

Sq(i)

Return the result of applying \(Sq^i\) to this element.

INPUT:

  • i – nonnegative integer

Warning

This is only implemented for simplicial complexes.

This cohomology operation is only defined in characteristic 2.

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: 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)  # long time
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)
sage: z_ss = RP4_ss.cohomology_ring(GF(2)).basis()[3,0]
sage: z_ss.Sq(1)
h^{4,0}
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.

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 and ri 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)
sage: x = RP5.cohomology_ring(GF(2)).basis()[1,0]
sage: x**4
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.HomologyVectorSpaceWithBasis(base_ring, cell_complex, cohomology=False, category=None)

Bases: sage.combinat.free_module.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: 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: 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: sage.modules.with_basis.indexed_element.IndexedFreeModuleElement

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
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]]