Chain homotopies and chain contractions

Chain homotopies are standard constructions in homological algebra: given chain complexes \(C\) and \(D\) and chain maps \(f, g: C \to D\), say with differential of degree \(-1\), a chain homotopy \(H\) between \(f\) and \(g\) is a collection of maps \(H_n: C_n \to D_{n+1}\) satisfying

\[\partial_D H + H \partial_C = f - g.\]

The presence of a chain homotopy defines an equivalence relation (chain homotopic) on chain maps. If \(f\) and \(g\) are chain homotopic, then one can show that \(f\) and \(g\) induce the same map on homology.

Chain contractions are not as well known. The papers [MAR2009], [RMA2009], and [PR2015] provide some references. Given two chain complexes \(C\) and \(D\), a chain contraction is a chain homotopy \(H: C \to C\) for which there are chain maps \(\pi: C \to D\) (“projection”) and \(\iota: D \to C\) (“inclusion”) such that

  • \(H\) is a chain homotopy between \(1_C\) and \(\iota \pi\),
  • \(\pi \iota = 1_D\),
  • \(\pi H = 0\),
  • \(H \iota = 0\),
  • \(H H = 0\).

Such a chain homotopy provides a strong relation between the chain complexes \(C\) and \(D\); for example, their homology groups are isomorphic.

class sage.homology.chain_homotopy.ChainContraction(matrices, pi, iota)

Bases: sage.homology.chain_homotopy.ChainHomotopy

A chain contraction.

An algebraic gradient vector field \(H: C \to C\) (that is a chain homotopy satisfying \(H H = 0\)) for which there are chain maps \(\pi: C \to D\) (“projection”) and \(\iota: D \to C\) (“inclusion”) such that

  • \(H\) is a chain homotopy between \(1_C\) and \(\iota \pi\),
  • \(\pi \iota = 1_D\),
  • \(\pi H = 0\),
  • \(H \iota = 0\).

H is defined by a dictionary matrices of matrices.

INPUT:

  • matrices – dictionary of matrices, keyed by dimension
  • pi – a chain map \(C \to D\)
  • iota – a chain map \(D \to C\)

EXAMPLES:

sage: from sage.homology.chain_homotopy import ChainContraction
sage: C = ChainComplex({0: zero_matrix(ZZ, 1), 1: identity_matrix(ZZ, 1)})
sage: D = ChainComplex({0: matrix(ZZ, 0, 1)})

The chain complex \(C\) is chain homotopy equivalent to \(D\), which is just a copy of \(\ZZ\) in degree 0, and we construct a chain contraction:

sage: pi = Hom(C,D)({0: identity_matrix(ZZ, 1)})
sage: iota = Hom(D,C)({0: identity_matrix(ZZ, 1)})
sage: H = ChainContraction({0: zero_matrix(ZZ, 0, 1), 1: zero_matrix(ZZ, 1), 2: identity_matrix(ZZ, 1)}, pi, iota)
dual()

The chain contraction dual to this one.

This is useful when switching from homology to cohomology.

EXAMPLES:

sage: S2 = simplicial_complexes.Sphere(2)
sage: phi, M = S2.algebraic_topological_model(QQ)
sage: phi.iota()
Chain complex morphism:
  From: Chain complex with at most 3 nonzero terms over Rational Field
  To: Chain complex with at most 3 nonzero terms over Rational Field

Lifting the degree zero homology class gives a single vertex, but the degree zero cohomology class needs to be detected on every vertex, and vice versa for degree 2:

sage: phi.iota().in_degree(0)
[0]
[0]
[0]
[1]
sage: phi.dual().iota().in_degree(0)
[1]
[1]
[1]
[1]
sage: phi.iota().in_degree(2)
[-1]
[ 1]
[-1]
[ 1]
sage: phi.dual().iota().in_degree(2)
[0]
[0]
[0]
[1]
iota()

The chain map \(\iota\) associated to this chain contraction.

EXAMPLES:

sage: S2 = simplicial_complexes.Sphere(2)
sage: phi, M = S2.algebraic_topological_model(QQ)
sage: phi.iota()
Chain complex morphism:
  From: Chain complex with at most 3 nonzero terms over Rational Field
  To: Chain complex with at most 3 nonzero terms over Rational Field

Lifting the degree zero homology class gives a single vertex:

sage: phi.iota().in_degree(0)
[0]
[0]
[0]
[1]

Lifting the degree two homology class gives the signed sum of all of the 2-simplices:

sage: phi.iota().in_degree(2)
[-1]
[ 1]
[-1]
[ 1]
pi()

The chain map \(\pi\) associated to this chain contraction.

EXAMPLES:

sage: S2 = simplicial_complexes.Sphere(2)
sage: phi, M = S2.algebraic_topological_model(QQ)
sage: phi.pi()
Chain complex morphism:
  From: Chain complex with at most 3 nonzero terms over Rational Field
  To: Chain complex with at most 3 nonzero terms over Rational Field
sage: phi.pi().in_degree(0)  # Every vertex represents a homology class.
[1 1 1 1]
sage: phi.pi().in_degree(1)  # No homology in degree 1.
[]

The degree 2 homology generator is detected on a single simplex:

sage: phi.pi().in_degree(2)
[0 0 0 1]
class sage.homology.chain_homotopy.ChainHomotopy(matrices, f, g=None)

Bases: sage.categories.morphism.Morphism

A chain homotopy.

A chain homotopy \(H\) between chain maps \(f, g: C \to D\) is a sequence of maps \(H_n: C_n \to D_{n+1}\) (if the chain complexes are graded homologically) satisfying

\[\partial_D H + H \partial_C = f - g.\]

INPUT:

  • matrices – dictionary of matrices, keyed by dimension
  • f – chain map \(C \to D\)
  • g (optional) – chain map \(C \to D\)

The dictionary matrices defines H by specifying the matrix defining it in each degree: the entry \(m\) corresponding to key \(i\) gives the linear transformation \(C_i \to D_{i+1}\).

If \(f\) is specified but not \(g\), then \(g\) can be recovered from the defining formula. That is, if \(g\) is not specified, then it is defined to be \(f - \partial_D H - H \partial_C\).

Note that the degree of the differential on the chain complex \(C\) must agree with that for \(D\), and those degrees determine the “degree” of the chain homotopy map: if the degree of the differential is \(d\), then the chain homotopy consists of a sequence of maps \(C_n \to C_{n-d}\). The keys in the dictionary matrices specify the starting degrees.

EXAMPLES:

sage: from sage.homology.chain_homotopy import ChainHomotopy
sage: C = ChainComplex({0: identity_matrix(ZZ, 1)})
sage: D = ChainComplex({0: zero_matrix(ZZ, 1)})
sage: f = Hom(C,D)({0: identity_matrix(ZZ, 1), 1: zero_matrix(ZZ, 1)})
sage: g = Hom(C,D)({0: zero_matrix(ZZ, 1), 1: zero_matrix(ZZ, 1)})
sage: H = ChainHomotopy({0: zero_matrix(ZZ, 0, 1), 1: identity_matrix(ZZ, 1)}, f, g)

Note that the maps \(f\) and \(g\) are stored in the attributes H._f and H._g:

sage: H._f
Chain complex morphism:
  From: Chain complex with at most 2 nonzero terms over Integer Ring
  To: Chain complex with at most 2 nonzero terms over Integer Ring
sage: H._f.in_degree(0)
[1]
sage: H._g.in_degree(0)
[0]

A non-example:

sage: H = ChainHomotopy({0: zero_matrix(ZZ, 0, 1), 1: zero_matrix(ZZ, 1)}, f, g)
Traceback (most recent call last):
...
ValueError: the data do not define a valid chain homotopy
dual()

Dual chain homotopy to this one.

That is, if this one is a chain homotopy between chain maps \(f, g: C \to D\), then its dual is a chain homotopy between the dual of \(f\) and the dual of \(g\), from \(D^*\) to \(C^*\). It is represented in each degree by the transpose of the corresponding matrix.

EXAMPLES:

sage: from sage.homology.chain_homotopy import ChainHomotopy
sage: C = ChainComplex({1: matrix(ZZ, 0, 2)}) # one nonzero term in degree 1
sage: D = ChainComplex({0: matrix(ZZ, 0, 1)}) # one nonzero term in degree 0
sage: f = Hom(C, D)({})
sage: H = ChainHomotopy({1: matrix(ZZ, 1, 2, (3,1))}, f, f)
sage: H.in_degree(1)
[3 1]
sage: H.dual().in_degree(0)
[3]
[1]
in_degree(n)

The matrix representing this chain homotopy in degree n.

INPUT:

  • n – degree

EXAMPLES:

sage: from sage.homology.chain_homotopy import ChainHomotopy
sage: C = ChainComplex({1: matrix(ZZ, 0, 2)}) # one nonzero term in degree 1
sage: D = ChainComplex({0: matrix(ZZ, 0, 1)}) # one nonzero term in degree 0
sage: f = Hom(C, D)({})
sage: H = ChainHomotopy({1: matrix(ZZ, 1, 2, (3,1))}, f, f)
sage: H.in_degree(1)
[3 1]

This returns an appropriately sized zero matrix if the chain homotopy is not defined in degree n:

sage: H.in_degree(-3)
[]
is_algebraic_gradient_vector_field()

An algebraic gradient vector field is a linear map \(H: C \to C\) such that \(H H = 0\).

(Some authors also require that \(H \partial H = H\), whereas some make this part of the definition of “homology gradient vector field. We have made the second choice.) See Molina-Abril and Réal [MAR2009] and Réal and Molina-Abril [RMA2009] for this and related terminology.

See also is_homology_gradient_vector_field().

EXAMPLES:

sage: from sage.homology.chain_homotopy import ChainHomotopy
sage: C = ChainComplex({0: zero_matrix(ZZ, 1), 1: identity_matrix(ZZ, 1)})

The chain complex \(C\) is chain homotopy equivalent to a copy of \(\ZZ\) in degree 0. Two chain maps \(C \to C\) will be chain homotopic as long as they agree in degree 0.

sage: f = Hom(C,C)({0: identity_matrix(ZZ, 1), 1: matrix(ZZ, 1, 1, [3]), 2: matrix(ZZ, 1, 1, [3])})
sage: g = Hom(C,C)({0: identity_matrix(ZZ, 1), 1: matrix(ZZ, 1, 1, [2]), 2: matrix(ZZ, 1, 1, [2])})
sage: H = ChainHomotopy({0: zero_matrix(ZZ, 0, 1), 1: zero_matrix(ZZ, 1), 2: identity_matrix(ZZ, 1)}, f, g)
sage: H.is_algebraic_gradient_vector_field()
True

A chain homotopy which is not an algebraic gradient vector field:

sage: H = ChainHomotopy({0: zero_matrix(ZZ, 0, 1), 1: identity_matrix(ZZ, 1), 2: identity_matrix(ZZ, 1)}, f, g)
sage: H.is_algebraic_gradient_vector_field()
False
is_homology_gradient_vector_field()

A homology gradient vector field is an algebraic gradient vector field \(H: C \to C\) (i.e., a chain homotopy satisfying \(H H = 0\)) such that \(\partial H \partial = \partial\) and \(H \partial H = H\).

See Molina-Abril and Réal [MAR2009] and Réal and Molina-Abril [RMA2009] for this and related terminology.

See also is_algebraic_gradient_vector_field().

EXAMPLES:

sage: from sage.homology.chain_homotopy import ChainHomotopy
sage: C = ChainComplex({0: zero_matrix(ZZ, 1), 1: identity_matrix(ZZ, 1)})

sage: f = Hom(C,C)({0: identity_matrix(ZZ, 1), 1: matrix(ZZ, 1, 1, [3]), 2: matrix(ZZ, 1, 1, [3])})
sage: g = Hom(C,C)({0: identity_matrix(ZZ, 1), 1: matrix(ZZ, 1, 1, [2]), 2: matrix(ZZ, 1, 1, [2])})
sage: H = ChainHomotopy({0: zero_matrix(ZZ, 0, 1), 1: zero_matrix(ZZ, 1), 2: identity_matrix(ZZ, 1)}, f, g)
sage: H.is_homology_gradient_vector_field()
True