The long exact sequence and excision for cubical singular chains #
The short exact sequence 0 ⟶ N(A) ⟶ N(X) ⟶ N(X, A) ⟶ 0 of normalized cubical chains of a
topological pair gives the long exact sequence of the pair in cubical homology, with
connecting morphism TopPair.cubicalHomologyδ. The comparison maps form a morphism of short
exact sequences from the simplicial one, so this connecting morphism corresponds to the singular
one under the comparison isomorphisms (TopPair.singularCubicalHomologyIso_hom_cubicalHomologyδ).
Conversely, a relative class c ∈ Hₘ₊₁(X, A) lifts to a normalized cubical chain of X with a
prescribed boundary: any cycle z of A representing the image of c under the connecting
morphism is the boundary of a chain of X which is a relative cycle representing c
(TopPair.exists_boundary_eq_of_cubicalHomologyδ_eq), and two such chains differ by a boundary
plus a cycle of A (TopPair.exists_sub_eq_boundary_add_map). The class c may equally be given
in relative singular homology (TopPair.exists_boundary_eq_of_singularHomologyδ_eq). Applied to
a compact oriented manifold M with boundary, with c = [M, ∂M] and z a fundamental cycle of
∂M (so that the connecting morphism sends c to the class of z), this gives a fundamental
chain of M relative to its boundary whose boundary is exactly z.
The simplest relative cycles are single cubes. A singular cube c of X sending the boundary of
the cube into A has its faces in A, and the signed sum of these faces is a cycle of A
(TopPair.cubicalBoundaryCycleOfCube) whose image in X is the boundary of c. The connecting
morphism sends the relative class of c to the class of this cycle
(TopPair.cubicalHomologyδ_homologyπ_cubicalCycleOfCube), and it is an isomorphism in positive
degrees when X is contractible (TopPair.isIso_cubicalHomologyδ_of_contractibleSpace).
Excision is transported from relative singular homology: a map of pairs inducing isomorphisms
on relative singular homology induces isomorphisms on relative cubical homology
(TopPair.isIso_homologyMap_cubicalChainComplexMap_of_isIso), which applies to the excision maps
of TauCeti.AlgebraicTopology.Singular.Excision.
Main definitions #
TopPair.cubicalHomologyδ: the connecting morphismHₙ(X, A) ⟶ Hₘ(A)of cubical homology.
Main results #
TopPair.cubicalHomology_exact_subspace,TopPair.cubicalHomology_exact_space,TopPair.cubicalHomology_exact_relative: exactness of the long exact sequence.TopPair.singularCubicalHomologyIso_hom_cubicalHomologyδ: compatibility of the connecting morphisms with the comparison.TopPair.exists_boundary_eq_of_cubicalHomologyδ_eq,TopPair.exists_boundary_eq_of_singularHomologyδ_eq: relative classes lift to chains with a prescribed boundary.TopPair.exists_sub_eq_boundary_add_map: such chains are unique up to boundaries and cycles of the subspace.TopPair.cubicalHomologyδ_homologyπ_cubicalCycleOfCube: the connecting morphism sends the class of a relative cube to the class of its boundary.TopPair.isIso_cubicalHomologyδ_of_contractibleSpace: with contractible ambient space, the connecting morphismHₘ₊₂(X, A) ⟶ Hₘ₊₁(A)is an isomorphism.TopPair.isIso_homologyMap_cubicalChainComplexMap_excisionMap,TopPair.isIso_homologyMap_cubicalChainComplexMap_excisionMap_compl: excision.
References #
- S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology, Princeton, 1952, Chapters I–III and VII.
- W. S. Massey, Singular Homology Theory, GTM 70, Springer, 1980, Chapter VII.
The long exact sequence of a pair #
The connecting morphism followed by the map induced by the inclusion of the subspace is zero.
The map to relative homology followed by the connecting morphism is zero.
Exactness at the cubical homology of the subspace.
Exactness at the cubical homology of the ambient space.
Exactness at the relative cubical homology.
The connecting morphisms of cubical and singular homology correspond under the comparison isomorphisms.
Chains with a prescribed boundary #
Relative cubical classes lift to chains with a prescribed boundary. Let c be a class of
Hₘ₊₁(X, A) and z an m-cycle of A representing the image of c under the connecting
morphism. Then some normalized cubical (m + 1)-chain s of X has boundary exactly the image
of z in X, and is a relative cycle representing c.
Applied to a compact manifold M with boundary, c = [M, ∂M] and a fundamental cycle z of
∂M, this produces a fundamental chain of M relative to its boundary whose boundary is z.
Chains with a prescribed boundary are unique up to boundaries and cycles of the subspace.
Let s, s' be normalized cubical (m + 1)-chains of X with the same boundary which are relative
cycles representing the same class of Hₘ₊₁(X, A). Then s - s' is the sum of a boundary and
the image of a cycle of A.
Singular relative classes lift to cubical chains with a prescribed boundary. Let μ be a
class of the relative singular homology Hₘ₊₁(X, A) and z a normalized cubical m-cycle of
A representing, under the comparison of cubical and singular homology, the image of μ under
the singular connecting morphism. Then some normalized cubical (m + 1)-chain s of X has
boundary exactly the image of z in X, and is a relative cycle whose cubical class corresponds
to μ.
The hypothesis on z is a statement about singular homology alone once z is fixed, so this is
the form in which singular fundamental classes rel boundary produce cubical fundamental chains
with a prescribed boundary.
The connecting morphism of a relative cube #
The connecting morphism of a relative cube is its boundary. For a singular cube c of X
sending the boundary of the cube into A, the connecting morphism Hₘ₊₁(X, A) ⟶ Hₘ(A) sends the
class of c to the class of its boundary, read in A.
If the ambient space of a pair is contractible, the connecting morphism
Hₘ₊₂(X, A) ⟶ Hₘ₊₁(A) of cubical homology is an isomorphism: its neighbours Hₘ₊₂(X) and
Hₘ₊₁(X) in the long exact sequence vanish.
Excision #
A map of pairs inducing isomorphisms on relative singular homology induces isomorphisms on relative cubical homology.
Excision for cubical homology. If the interiors of A and B cover X, the inclusion of
pairs (A, A ∩ B) ⟶ (X, B) induces isomorphisms on relative cubical homology.