Excision for relative singular homology #
Let (X, B) be a topological pair and A ⊆ X a subset such that the interiors of A and B
cover X. The inclusion of pairs (A, A ∩ B) ⟶ (X, B) induces an isomorphism on relative
singular homology in every degree, with coefficients in any object of an abelian category with
coproducts (TopPair.isIso_singularHomologyMap_excisionMap). Equivalently, excising a set Z
whose closure lies in the interior of B does not change relative homology
(TopPair.isIso_singularHomologyMap_excisionMap_compl).
Both follow from a statement about an arbitrary map of topological pairs f : P ⟶ P' whose map
on ambient spaces is an embedding and whose subspace is the full preimage of the subspace of P':
if the ambient space of P' has an open cover each of whose members lies in the subspace of P'
or in the image of f, then f induces isomorphisms on relative singular homology
(TopPair.isIso_singularHomologyMap_of_open_cover).
Two intermediate results are available on their own. The relative small-chain theorem
(TopPair.isIso_homologyMap_restrictι_smallSingularSubcomplex) identifies the relative homology
of a topological pair with that of its singular pair restricted to the simplices subordinate to
any open cover of the ambient space. Under the hypotheses of
TopPair.isIso_singularHomologyMap_of_open_cover, the map f puts the simplices subordinate to
the pulled-back cover which do not lie in the subspace of P in bijection with the simplices
subordinate to the cover which do not lie in the subspace of P'
(TopPair.relativeSimplex_restrictMap_bijective); this is the form in which the hypotheses on f
and the cover enter, and it feeds the complementary-simplex criterion of simplicial excision.
A continuous map g : X ⟶ Y carrying A into A' and B into B' is a map of excision data:
it induces a map of pairs TopPair.interPairMap g hA hB : (A, A ∩ B) ⟶ (A', A' ∩ B'), functorial
in g, which together with TopPair.ofSubsetMap g hB : (X, B) ⟶ (Y, B') forms a commutative
square with the excision maps (TopPair.interPairMap_comp_excisionMap), so the excision
isomorphisms are natural in the data.
Compatibility with the connecting morphism of the pair is TopPair.singularHomologyδ_naturality
applied to the excision map.
References #
- A. Hatcher, Algebraic Topology, Section 2.1, Theorem 2.20 and Proposition 2.21.
- S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology, Chapter VII.
If the subspace of P is the full preimage of the subspace of P', then a map of pairs sends
small simplices not lying in the subspace of P to small simplices not lying in the subspace of
P'.
Complementary small simplices correspond. For a map of pairs which is an embedding on ambient spaces, whose subspace is the full preimage of the subspace of the target, and an open cover of the target each of whose members lies in the subspace or in the image, the simplices small for the pulled-back cover not lying in the subspace correspond bijectively to the simplices small for the cover not lying in the subspace of the target.
Excision for relative singular homology. Let f : P ⟶ P' be a map of topological pairs
which is an embedding on ambient spaces and whose subspace is the full preimage of the subspace of
P'. If the ambient space of P' has an open cover each of whose members lies in the subspace of
P' or in the image of f, then f induces isomorphisms on relative singular homology.
The topological pair (A, A ∩ B), with A ∩ B realised as the preimage of B in the
subspace A.
Equations
- TopPair.interPair A B = TopPair.ofSubset (Subtype.val ⁻¹' B)
Instances For
The inclusion of pairs (A, A ∩ B) ⟶ (X, B).
Equations
- TopPair.excisionMap A B = TopPair.ofHom (TopCat.ofHom { toFun := Subtype.val, continuous_toFun := ⋯ }) (TopCat.ofHom { toFun := B.restrictPreimage Subtype.val, continuous_toFun := ⋯ }) ⋯
Instances For
Excision. If the interiors of A and B cover X, the inclusion of pairs
(A, A ∩ B) ⟶ (X, B) induces isomorphisms on relative singular homology.
Excision of a set with closure in the interior of the subspace. If closure Z ⊆ interior B,
the inclusion of pairs (X ∖ Z, B ∖ Z) ⟶ (X, B) induces isomorphisms on relative singular
homology.
A map of excision data: a continuous map g : X ⟶ Y carrying A into A' and B into B'
induces a map of pairs (A, A ∩ B) ⟶ (A', A' ∩ B').
Equations
- TopPair.interPairMap g hA hB = TopPair.ofSubsetMap (TopCat.ofHom { toFun := Set.MapsTo.restrict (⇑(CategoryTheory.ConcreteCategory.hom g)) A A' hA, continuous_toFun := ⋯ }) ⋯
Instances For
The excision maps are natural in the excision data.
The excision maps are natural in the excision data.