Morphisms of simplicial complexes¶
AUTHORS:
- Benjamin Antieau <d.ben.antieau@gmail.com> (2009.06)
- Travis Scrimshaw (2012-08-18): Made all simplicial complexes immutable to work with the homset cache.
This module implements morphisms of simplicial complexes. The input is given by a dictionary on the vertex set of a simplicial complex. The initialization checks that faces are sent to faces.
There is also the capability to create the fiber product of two morphisms with the same codomain.
EXAMPLES:
sage: S = SimplicialComplex([[0,2],[1,5],[3,4]], is_mutable=False)
sage: H = Hom(S,S.product(S, is_mutable=False))
sage: H.diagonal_morphism()
Simplicial complex morphism:
From: Simplicial complex with vertex set (0, 1, 2, 3, 4, 5) and facets {(3, 4), (1, 5), (0, 2)}
To: Simplicial complex with 36 vertices and 18 facets
Defn: [0, 1, 2, 3, 4, 5] --> ['L0R0', 'L1R1', 'L2R2', 'L3R3', 'L4R4', 'L5R5']
sage: S = SimplicialComplex([[0,2],[1,5],[3,4]], is_mutable=False)
sage: T = SimplicialComplex([[0,2],[1,3]], is_mutable=False)
sage: f = {0:0,1:1,2:2,3:1,4:3,5:3}
sage: H = Hom(S,T)
sage: x = H(f)
sage: x.image()
Simplicial complex with vertex set (0, 1, 2, 3) and facets {(1, 3), (0, 2)}
sage: x.is_surjective()
True
sage: x.is_injective()
False
sage: x.is_identity()
False
sage: S = simplicial_complexes.Sphere(2)
sage: H = Hom(S,S)
sage: i = H.identity()
sage: i.image()
Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 2, 3), (0, 1, 2), (1, 2, 3), (0, 1, 3)}
sage: i.is_surjective()
True
sage: i.is_injective()
True
sage: i.is_identity()
True
sage: S = simplicial_complexes.Sphere(2)
sage: H = Hom(S,S)
sage: i = H.identity()
sage: j = i.fiber_product(i)
sage: j
Simplicial complex morphism:
From: Simplicial complex with 4 vertices and 4 facets
To: Minimal triangulation of the 2-sphere
Defn: L1R1 |--> 1
L3R3 |--> 3
L2R2 |--> 2
L0R0 |--> 0
sage: S = simplicial_complexes.Sphere(2)
sage: T = S.product(SimplicialComplex([[0,1]]), rename_vertices = False, is_mutable=False)
sage: H = Hom(T,S)
sage: T
Simplicial complex with 8 vertices and 12 facets
sage: T.vertices()
((0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1), (3, 0), (3, 1))
sage: f = {(0, 0): 0, (0, 1): 0, (1, 0): 1, (1, 1): 1, (2, 0): 2, (2, 1): 2, (3, 0): 3, (3, 1): 3}
sage: x = H(f)
sage: U = simplicial_complexes.Sphere(1)
sage: G = Hom(U,S)
sage: U
Minimal triangulation of the 1-sphere
sage: g = {0:0,1:1,2:2}
sage: y = G(g)
sage: z = y.fiber_product(x)
sage: z # this is the mapping path space
Simplicial complex morphism:
From: Simplicial complex with 6 vertices and 6 facets
To: Minimal triangulation of the 2-sphere
Defn: ['L2R(2, 0)', 'L2R(2, 1)', 'L0R(0, 0)', 'L0R(0, 1)', 'L1R(1, 0)', 'L1R(1, 1)'] --> [2, 2, 0, 0, 1, 1]
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class
sage.homology.simplicial_complex_morphism.
SimplicialComplexMorphism
(f, X, Y)¶ Bases:
sage.categories.morphism.Morphism
An element of this class is a morphism of simplicial complexes.
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associated_chain_complex_morphism
(base_ring=Integer Ring, augmented=False, cochain=False)¶ Returns the associated chain complex morphism of
self
.EXAMPLES:
sage: S = simplicial_complexes.Sphere(1) sage: T = simplicial_complexes.Sphere(2) sage: H = Hom(S,T) sage: f = {0:0,1:1,2:2} sage: x = H(f) sage: x Simplicial complex morphism: From: Minimal triangulation of the 1-sphere To: Minimal triangulation of the 2-sphere Defn: 0 |--> 0 1 |--> 1 2 |--> 2 sage: a = x.associated_chain_complex_morphism() sage: a Chain complex morphism: From: Chain complex with at most 2 nonzero terms over Integer Ring To: Chain complex with at most 3 nonzero terms over Integer Ring sage: a._matrix_dictionary {0: [1 0 0] [0 1 0] [0 0 1] [0 0 0], 1: [1 0 0] [0 1 0] [0 0 0] [0 0 1] [0 0 0] [0 0 0], 2: []} sage: x.associated_chain_complex_morphism(augmented=True) Chain complex morphism: From: Chain complex with at most 3 nonzero terms over Integer Ring To: Chain complex with at most 4 nonzero terms over Integer Ring sage: x.associated_chain_complex_morphism(cochain=True) Chain complex morphism: From: Chain complex with at most 3 nonzero terms over Integer Ring To: Chain complex with at most 2 nonzero terms over Integer Ring sage: x.associated_chain_complex_morphism(augmented=True,cochain=True) Chain complex morphism: From: Chain complex with at most 4 nonzero terms over Integer Ring To: Chain complex with at most 3 nonzero terms over Integer Ring sage: x.associated_chain_complex_morphism(base_ring=GF(11)) Chain complex morphism: From: Chain complex with at most 2 nonzero terms over Finite Field of size 11 To: Chain complex with at most 3 nonzero terms over Finite Field of size 11
Some simplicial maps which reverse the orientation of a few simplices:
sage: g = {0:1, 1:2, 2:0} sage: H(g).associated_chain_complex_morphism()._matrix_dictionary {0: [0 0 1] [1 0 0] [0 1 0] [0 0 0], 1: [ 0 -1 0] [ 0 0 -1] [ 0 0 0] [ 1 0 0] [ 0 0 0] [ 0 0 0], 2: []} sage: X = SimplicialComplex([[0, 1]], is_mutable=False) sage: Hom(X,X)({0:1, 1:0}).associated_chain_complex_morphism()._matrix_dictionary {0: [0 1] [1 0], 1: [-1]}
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fiber_product
(other, rename_vertices=True)¶ Fiber product of
self
andother
. Both morphisms should have the same codomain. The method returns a morphism of simplicial complexes, which is the morphism from the space of the fiber product to the codomain.EXAMPLES:
sage: S = SimplicialComplex([[0,1],[1,2]], is_mutable=False) sage: T = SimplicialComplex([[0,2],[1]], is_mutable=False) sage: U = SimplicialComplex([[0,1],[2]], is_mutable=False) sage: H = Hom(S,U) sage: G = Hom(T,U) sage: f = {0:0,1:1,2:0} sage: g = {0:0,1:1,2:1} sage: x = H(f) sage: y = G(g) sage: z = x.fiber_product(y) sage: z Simplicial complex morphism: From: Simplicial complex with 4 vertices and facets {('L2R0',), ('L1R1',), ('L0R0', 'L1R2')} To: Simplicial complex with vertex set (0, 1, 2) and facets {(2,), (0, 1)} Defn: L1R2 |--> 1 L1R1 |--> 1 L2R0 |--> 0 L0R0 |--> 0
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image
()¶ Computes the image simplicial complex of \(f\).
EXAMPLES:
sage: S = SimplicialComplex([[0,1],[2,3]], is_mutable=False) sage: T = SimplicialComplex([[0,1]], is_mutable=False) sage: f = {0:0,1:1,2:0,3:1} sage: H = Hom(S,T) sage: x = H(f) sage: x.image() Simplicial complex with vertex set (0, 1) and facets {(0, 1)} sage: S = SimplicialComplex(is_mutable=False) sage: H = Hom(S,S) sage: i = H.identity() sage: i.image() Simplicial complex with vertex set () and facets {()} sage: i.is_surjective() True sage: S = SimplicialComplex([[0,1]], is_mutable=False) sage: T = SimplicialComplex([[0,1], [0,2]], is_mutable=False) sage: f = {0:0,1:1} sage: g = {0:0,1:1} sage: k = {0:0,1:2} sage: H = Hom(S,T) sage: x = H(f) sage: y = H(g) sage: z = H(k) sage: x == y True sage: x == z False sage: x.image() Simplicial complex with vertex set (0, 1) and facets {(0, 1)} sage: y.image() Simplicial complex with vertex set (0, 1) and facets {(0, 1)} sage: z.image() Simplicial complex with vertex set (0, 2) and facets {(0, 2)}
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induced_homology_morphism
(base_ring=None, cohomology=False)¶ The map in (co)homology induced by this map
INPUT:
base_ring
– must be a field (optional, defaultQQ
)cohomology
– boolean (optional, defaultFalse
). IfTrue
, the map induced in cohomology rather than homology.
EXAMPLES:
sage: S = simplicial_complexes.Sphere(1) sage: T = S.product(S, is_mutable=False) sage: H = Hom(S,T) sage: diag = H.diagonal_morphism() sage: h = diag.induced_homology_morphism(QQ) sage: h Graded vector space morphism: From: Homology module of Minimal triangulation of the 1-sphere over Rational Field To: Homology module of Simplicial complex with 9 vertices and 18 facets over Rational Field Defn: induced by: Simplicial complex morphism: From: Minimal triangulation of the 1-sphere To: Simplicial complex with 9 vertices and 18 facets Defn: 0 |--> L0R0 1 |--> L1R1 2 |--> L2R2
We can view the matrix form for the homomorphism:
sage: h.to_matrix(0) # in degree 0 [1] sage: h.to_matrix(1) # in degree 1 [1] [0] sage: h.to_matrix() # the entire homomorphism [1|0] [-+-] [0|1] [0|0] [-+-] [0|0]
We can evaluate it on (co)homology classes:
sage: coh = diag.induced_homology_morphism(QQ, cohomology=True) sage: coh.to_matrix(1) [1 0] sage: x,y = list(T.cohomology_ring(QQ).basis(1)) sage: coh(x) h^{1,0} sage: coh(2*x+3*y) 2*h^{1,0}
Note that the complexes must be immutable for this to work. Many, but not all, complexes are immutable when constructed:
sage: S.is_immutable() True sage: S.barycentric_subdivision().is_immutable() False sage: S2 = S.suspension() sage: S2.is_immutable() False sage: h = Hom(S,S2)({0: 0, 1:1, 2:2}).induced_homology_morphism() Traceback (most recent call last): ... ValueError: the domain and codomain complexes must be immutable sage: S2.set_immutable(); S2.is_immutable() True sage: h = Hom(S,S2)({0: 0, 1:1, 2:2}).induced_homology_morphism()
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is_identity
()¶ If
self
is an identity morphism, returnsTrue
. Otherwise,False
.EXAMPLES:
sage: T = simplicial_complexes.Sphere(1) sage: G = Hom(T,T) sage: T Minimal triangulation of the 1-sphere sage: j = G({0:0,1:1,2:2}) sage: j.is_identity() True sage: S = simplicial_complexes.Sphere(2) sage: T = simplicial_complexes.Sphere(3) sage: H = Hom(S,T) sage: f = {0:0,1:1,2:2,3:3} sage: x = H(f) sage: x Simplicial complex morphism: From: Minimal triangulation of the 2-sphere To: Minimal triangulation of the 3-sphere Defn: 0 |--> 0 1 |--> 1 2 |--> 2 3 |--> 3 sage: x.is_identity() False
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is_injective
()¶ Returns
True
if and only ifself
is injective.EXAMPLES:
sage: S = simplicial_complexes.Sphere(1) sage: T = simplicial_complexes.Sphere(2) sage: U = simplicial_complexes.Sphere(3) sage: H = Hom(T,S) sage: G = Hom(T,U) sage: f = {0:0,1:1,2:0,3:1} sage: x = H(f) sage: g = {0:0,1:1,2:2,3:3} sage: y = G(g) sage: x.is_injective() False sage: y.is_injective() True
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is_surjective
()¶ Returns
True
if and only ifself
is surjective.EXAMPLES:
sage: S = SimplicialComplex([(0,1,2)], is_mutable=False) sage: S Simplicial complex with vertex set (0, 1, 2) and facets {(0, 1, 2)} sage: T = SimplicialComplex([(0,1)], is_mutable=False) sage: T Simplicial complex with vertex set (0, 1) and facets {(0, 1)} sage: H = Hom(S,T) sage: x = H({0:0,1:1,2:1}) sage: x.is_surjective() True sage: S = SimplicialComplex([[0,1],[2,3]], is_mutable=False) sage: T = SimplicialComplex([[0,1]], is_mutable=False) sage: f = {0:0,1:1,2:0,3:1} sage: H = Hom(S,T) sage: x = H(f) sage: x.is_surjective() True
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mapping_torus
()¶ The mapping torus of a simplicial complex endomorphism
The mapping torus is the simplicial complex formed by taking the product of the domain of
self
with a \(4\) point interval \([I_0, I_1, I_2, I_3]\) and identifying vertices of the form \((I_0, v)\) with \((I_3, w)\) where \(w\) is the image of \(v\) under the given morphism.See Wikipedia article Mapping torus
EXAMPLES:
sage: C = simplicial_complexes.Sphere(1) # Circle sage: T = Hom(C,C).identity().mapping_torus() ; T # Torus Simplicial complex with 9 vertices and 18 facets sage: T.homology() == simplicial_complexes.Torus().homology() True sage: f = Hom(C,C)({0:0,1:2,2:1}) sage: K = f.mapping_torus() ; K # Klein Bottle Simplicial complex with 9 vertices and 18 facets sage: K.homology() == simplicial_complexes.KleinBottle().homology() True
TESTS:
sage: g = Hom(simplicial_complexes.Simplex([1]),C)({1:0}) sage: g.mapping_torus() Traceback (most recent call last): ... ValueError: self must have the same domain and codomain.
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sage.homology.simplicial_complex_morphism.
is_SimplicialComplexMorphism
(x)¶ Returns
True
if and only ifx
is a morphism of simplicial complexes.EXAMPLES:
sage: from sage.homology.simplicial_complex_morphism import is_SimplicialComplexMorphism sage: S = SimplicialComplex([[0,1],[3,4]], is_mutable=False) sage: H = Hom(S,S) sage: f = {0:0,1:1,3:3,4:4} sage: x = H(f) sage: is_SimplicialComplexMorphism(x) True