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# -*- coding: utf-8 -*- 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
.. MATH::
\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. """
######################################################################## # Copyright (C) 2015 John H. Palmieri <palmieri@math.washington.edu> # # Distributed under the terms of the GNU General Public License (GPL) # as published by the Free Software Foundation; either version 2 of # the License, or (at your option) any later version. # # http://www.gnu.org/licenses/ ########################################################################
# In a perfect world, this would inherit from something like # "TwoMorphism" rather than "Morphism"... r""" 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
.. MATH::
\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 """ r""" Create a chain homotopy between the given chain maps from a dictionary of matrices.
EXAMPLES:
If ``g`` is not specified, it is set equal to `f - (H \partial + \partial H)`. ::
sage: from sage.homology.chain_homotopy import ChainHomotopy sage: C = ChainComplex({1: matrix(ZZ, 1, 2, (1,0)), 2: matrix(ZZ, 2, 1, (0, 2))}, degree_of_differential=-1) sage: D = ChainComplex({2: matrix(ZZ, 1, 1, (6,))}, degree_of_differential=-1) sage: f_d = {1: matrix(ZZ, 1, 2, (0,3)), 2: identity_matrix(ZZ, 1)} sage: f = Hom(C,D)(f_d) sage: H_d = {0: identity_matrix(ZZ, 1), 1: matrix(ZZ, 1, 2, (2,2))} sage: H = ChainHomotopy(H_d, f) sage: H._g.in_degree(0) [] sage: H._g.in_degree(1) [-13 -9] sage: H._g.in_degree(2) [-3]
TESTS:
Try to construct a chain homotopy in which the maps do not have matching domains and codomains::
sage: g = Hom(C,C)({}) # the zero chain map sage: H = ChainHomotopy(H_d, f, g) Traceback (most recent call last): ... ValueError: the chain maps are not compatible """ # Check that the chain complexes are compatible. This should # never arise, because first there should be errors in # constructing the chain maps. But just in case... raise ValueError('the chain complexes are not compatible') # Check that the chain maps are compatible. g.codomain()): # Check that the data define a chain homotopy. raise ValueError('the data do not define a valid chain homotopy') raise ValueError('the data do not define a valid chain homotopy') else: # Define g. # Use immutable matrices because they're hashable.
r""" 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 :meth:`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 """ return False
r""" 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 :meth:`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 """ return False return False return False
""" 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) [] """
r""" 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] """
""" TESTS::
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: hash(H) # random 314159265358979 """
""" String representation
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: ChainHomotopy({0: zero_matrix(ZZ, 0, 1), 1: identity_matrix(ZZ, 1)}, f, g) Chain homotopy between: 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 and 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 """
r""" 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) """ r""" Create a chain contraction from the given data.
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 try construct a chain contraction, but get the map `\iota` wrong::
sage: pi = Hom(C,D)({0: identity_matrix(ZZ, 1)}) sage: iota = Hom(D,C)({0: zero_matrix(ZZ, 1)}) sage: H = ChainContraction({0: zero_matrix(ZZ, 0, 1), 1: zero_matrix(ZZ, 1), 2: identity_matrix(ZZ, 1)}, pi, iota) Traceback (most recent call last): ... ValueError: the composite 'pi iota' is not the identity
Another bad `\iota`::
sage: iota = pi # wrong domain, codomain sage: H = ChainContraction({0: zero_matrix(ZZ, 0, 1), 1: zero_matrix(ZZ, 1), 2: identity_matrix(ZZ, 1)}, pi, iota) Traceback (most recent call last): ... ValueError: the chain maps are not composable
`\iota` is okay, but wrong data defining `H`::
sage: iota = Hom(D,C)({0: identity_matrix(ZZ, 1)}) sage: H = ChainContraction({0: zero_matrix(ZZ, 0, 1), 1: identity_matrix(ZZ, 1), 2: identity_matrix(ZZ, 1)}, pi, iota) Traceback (most recent call last): ... ValueError: not an algebraic gradient vector field """
and pi.codomain() == iota.domain()):
# Check that the composite 'pi iota' is 1.
# Construct the chain map 'id_C'.
# Now check that # - `H` is a chain homotopy between `id_C` and `\iota \pi` # - `HH = 0` # Check that `\pi H = 0`: raise ValueError('the data do not define a valid chain contraction: pi H != 0') # Check that `H \iota = 0`: raise ValueError('the data do not define a valid chain contraction: H iota != 0')
r""" 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] """
r""" 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] """
""" 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] """
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