Local homology of a compact subset of a manifold #
Let M be a Hausdorff space modelled on a real normed space E of finite dimension n, that is,
a topological n-manifold without boundary, and let A ⊆ M be compact. This file proves that
the local homology Hₖ(M, M ∖ A) (TauCeti.localHomology) with coefficients in a module
vanishes above the dimension (TauCeti.isZero_localHomology_of_isCompact), and that in degree
n a class is determined by its restrictions Hₙ(M, M ∖ A) ⟶ Hₙ(M, M ∖ {x}) to the points of
A (TauCeti.eq_zero_of_forall_localHomologyRestriction_eq_zero and
TauCeti.localHomology_ext_of_isCompact). Conversely, a family of local classes
αₓ ∈ Hₙ(M, M ∖ {x}), x ∈ A, that is locally the family of restrictions of a single class comes
from a class of Hₙ(M, M ∖ A) (TauCeti.exists_localHomologyRestriction_eq_of_isCompact), so
from exactly one (TauCeti.existsUnique_localHomologyRestriction_eq_of_isCompact). For M
compact and A = M, applied to local generators that vary in this way, this is the class that
becomes a fundamental class. For compact M, A = M also shows that the singular homology Hₖ(M)
vanishes for k > n (TauCeti.isZero_singularHomology_of_compactSpace).
Main results #
TauCeti.isZero_localHomology_of_isCompact:Hₖ(M, M ∖ A) = 0for compactAandk > n.TauCeti.eq_zero_of_forall_localHomologyRestriction_eq_zero: a class ofHₙ(M, M ∖ A)whose restriction to every point ofAvanishes is zero.TauCeti.localHomology_ext_of_isCompact: two classes ofHₙ(M, M ∖ A)with the same restrictions to the points ofAare equal.TauCeti.exists_localHomologyRestriction_eq_of_isCompactandTauCeti.existsUnique_localHomologyRestriction_eq_of_isCompact: a family of local classes at the points ofAthat is locally the family of restrictions of a single class is the family of restrictions of exactly one class ofHₙ(M, M ∖ A).TauCeti.isZero_singularHomology_of_compactSpace:Hₖ(M) = 0for compactMandk > n.
References #
- A. Hatcher, Algebraic Topology, Section 3.3, Lemma 3.27 and its proof.
Local homology of a compact set vanishes above the dimension. For a compact subset A of
a Hausdorff space modelled on a real normed space of dimension n, Hₖ(M, M ∖ A) = 0 for
k > n. This is Hatcher's Lemma 3.27(b).
A local homology class of a compact set is detected at its points. For a compact subset
A of a Hausdorff space modelled on a real normed space of dimension n, a class of
Hₙ(M, M ∖ A) whose restriction to Hₙ(M, M ∖ {x}) vanishes for every x ∈ A is zero.
Uniqueness of local homology classes of a compact set. For a compact subset A of a
Hausdorff space modelled on a real normed space of dimension n, two classes of
Hₙ(M, M ∖ A) with the same restrictions to Hₙ(M, M ∖ {x}) for every x ∈ A are equal. This
is the uniqueness part of Hatcher's Lemma 3.27(a).
Existence of local homology classes of a compact set. Let A be a compact subset of a
Hausdorff space modelled on a real normed space of dimension n, and let α x ∈ Hₙ(M, M ∖ {x})
be local classes that are locally restrictions of a single class: every x ∈ A has a
neighbourhood U within A and a class of Hₙ(M, M ∖ U) restricting to α y at each point
y ∈ U ∩ A. Then some class of Hₙ(M, M ∖ A) restricts to α x at every x ∈ A. This is the
existence part of Hatcher's Lemma 3.27(a).
Local homology classes of a compact set are glued from local data. Let A be a compact
subset of a Hausdorff space modelled on a real normed space of dimension n, and let
α x ∈ Hₙ(M, M ∖ {x}) be local classes such that every x ∈ A has a neighbourhood U within A
and a class of Hₙ(M, M ∖ U) restricting to α y at each point y ∈ U ∩ A. Then exactly one
class of Hₙ(M, M ∖ A) restricts to α x at every x ∈ A. This is Hatcher's Lemma 3.27(a),
with the continuous section of the covering space M_R there replaced by its defining property:
locally, the family is the family of restrictions of a single class.
The homology of a compact manifold vanishes above the dimension. For a compact Hausdorff
space modelled on a real normed space of dimension n, Hₖ(M) = 0 for k > n: it is the local
homology Hₖ(M, M ∖ M).