The complement of an embedded cube is acyclic #
If h : Iᵏ → Sⁿ is an embedding of a cube into a sphere, then the complement Sⁿ ∖ h(Iᵏ) has
the reduced singular homology of a point. This is the first half of Hatcher's Proposition 2B.1;
the second half, that the complement of an embedded k-sphere has the reduced homology of an
(n - k - 1)-sphere, follows from it by the same Mayer–Vietoris argument, and with it the
Jordan–Brouwer separation theorem: an embedded (n - 1)-sphere separates Sⁿ into exactly two
path components.
The proof works in any Hausdorff space Y in which the complement of every point has vanishing
reduced homology, by induction on k. Write Iᵏ⁺¹ = I × Iᵏ and, for s ⊆ I, let W s be the
complement of the image of the slab s × Iᵏ. By induction, W {t} is acyclic for every t. A
class α of W I therefore vanishes in W J for every short enough interval J around a point,
since a singular cycle has compact support (TauCeti.exists_singularHomologyMap_inclusion_eq_zero),
and if it vanishes in W [a, b] and W [b, c] then it vanishes in W [a, c], by the
Mayer–Vietoris sequence of W [a, c] = W [a, b] ∩ W [b, c] inside W {b}
(TopCat.eq_zero_of_comp_reducedSingularHomologyFunctor_map_inclusion). Hatcher concludes by
repeated bisection; here the set of s with α vanishing in W [0, s] is shown to be open and
closed in I, hence all of it.
Coefficients are a module over a ring, since the compactness step is an argument about elements.
Main results #
TauCeti.isZero_reducedSingularHomologyFunctor_compl_range_cube: in a Hausdorff space whose point complements are acyclic, the complement of an embedded cube is acyclic.TauCeti.isZero_reducedSingularHomologyFunctor_sphere_compl_range_cube: the complement of a cube embedded in the unit sphere of a real normed space is acyclic.TauCeti.isZero_reducedSingularHomologyFunctor_compl_range_closedBall: the same for an embedded closed disc, which is homeomorphic to a cube.
References #
- A. Hatcher, Algebraic Topology, Section 2.B, Proposition 2B.1(a).
The complement of an embedded cube is acyclic. Let Y be a Hausdorff space in which the
complement of every point has vanishing reduced homology. Then the complement of the image of any
embedding of a cube Iᵏ into Y has vanishing reduced homology, with coefficients in any module.
Since the cube is compact and Y is Hausdorff, a continuous injection of it is an embedding.
This is Hatcher, Algebraic Topology, Proposition 2B.1(a), for Y a sphere; the proof is by
induction on k, cutting the cube into slabs along its first coordinate.
The complement of a cube embedded in a sphere is acyclic (Hatcher, Algebraic Topology,
Proposition 2B.1(a)): for every embedding h of a cube Iᵏ into the unit sphere of a real normed
space, the reduced singular homology of the complement of its image vanishes in every degree.
The complement of an embedded disc is acyclic. Let Y be a Hausdorff space in which the
complement of every point has vanishing reduced homology. Then the complement of the image of any
continuous injection of the closed unit ball of a finite-dimensional real normed space into Y
has vanishing reduced homology, with coefficients in any module.