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 $π : C \to D$ (“projection”) and $ι : D \to C$ (“inclusion”) such that

$H$ is a chain homotopy between $1_{C}$ and $ι π$ ,
$π ι = 1_{D}$ ,
$π H = 0$ ,
$H ι = 0$ ,
$H H = 0$ .

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 $π : C \to D$ (“projection”) and $ι : D \to C$ (“inclusion”) such that

$H$ is a chain homotopy between $1_{C}$ and $ι π$ ,
$π ι = 1_{D}$ ,
$π H = 0$ ,
$H ι = 0$ .

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 $Z$ 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 $ι$ 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 $π$ 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.

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 $Z$ 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.

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