Surjectivity of the relative Hurewicz map, and the isomorphism for highly connected spaces #
Let A be a path-connected subspace of a space X and a₀ ∈ A. Suppose that every point of X
is joined by a path to a point of A, and that the relative homotopy sets πₖ(X, A, a₀) are
trivial for 0 < k < n, where n ≥ 2. This file proves that the relative Hurewicz map
πₙ(X, A, a₀) → Hₙ(X, A; ℤ), [p] ↦ p_*[Iⁿ], is surjective
(TauCeti.RelHomotopyGroup.surjective_hurewicz).
This is half of the relative Hurewicz theorem in degree n. Unlike injectivity, it needs no
assumption on the fundamental group of A: under these hypotheses the relative Hurewicz map
identifies Hₙ(X, A; ℤ) with the quotient of πₙ(X, A, a₀) by the action of π₁(A, a₀)
(Hatcher, Theorem 4.37), which is trivial when A is simply connected.
The file then proves injectivity, and so the relative Hurewicz isomorphism in degree n, when
A and X are both (n - 1)-connected
(TauCeti.RelHomotopyGroup.bijective_hurewicz_of_subsingleton_homotopyGroup). In degree two
this is the full relative Hurewicz theorem: if A and X are simply connected, then
π₂(X, A, a₀), which is commutative (TauCeti.RelHomotopyGroup.mul_comm_of_subsingleton), is
isomorphic to H₂(X, A; ℤ). The proof of injectivity is a diagram chase. As πₙ₋₁(A, a₀) is
trivial, every relative class comes from πₙ(X, a₀); the relative Hurewicz map extends the
absolute one (TauCeti.RelHomotopyGroup.hurewicz_ofHomotopyGroup), and the absolute Hurewicz
theorem for A and X (HomotopyGroup.bijective_hurewicz), together with exactness of the
homology sequence of the pair at Hₙ(X; ℤ), shows that a class in the kernel comes from
πₙ(A, a₀), hence is trivial in πₙ(X, A, a₀). For n ≥ 3 the relative Hurewicz theorem only
needs A to be simply connected; that case is not covered here.
Proof #
The proof is Eilenberg's deformation of singular chains, as for the vanishing of Hₖ(X, A) below
degree n (TauCeti.isZero_singularHomology_ofSubset_of_relHomotopyGroup).
- By the compression lemma the singular cubes of
Xdeform coherently intoAbelow dimensionn(TauCeti.nonempty_subspaceCubeDeformation_of_relHomotopyGroup). On relative normalized cubical chains the end of the deformation is chain homotopic to the identity, and it carries everyn-cube to a cube whose faces, of dimensionn - 1, lie inA. So every class ofHₙ(X, A)in cubical homology is a combination of classes of singlen-cubescsending the boundary of the cube intoA(TauCeti.SubspaceCubeDeformation.homologyπ_mem_of_cube). - Such a cube is a map of pairs
c : (Iⁿ, ∂Iⁿ) ⟶ (X, A), and its class is the image underc_*of the class of the fundamental cycle of(Iⁿ, ∂Iⁿ)(TauCeti.homologyMap_cubeBoundaryPairHom_cubeBoundaryPairCycle). The comparison of relative cubical and simplicial chains is natural, so in singular homology every class ofHₙ(X, A; ℤ)is a combination of the imagesc_*(x)withx ∈ Hₙ(Iⁿ, ∂Iⁿ; ℤ) ≅ ℤ, that is, of integer multiples of the classesc_*[Iⁿ]. - A map of pairs out of the cube is homotopic, through maps of pairs, to a relative cube at some
point of
A(TauCeti.exists_homotopicWith_relGenLoop), and transporting that relative cube along a path inAtoa₀is another homotopy through maps of pairs (TauCeti.RelGenLoop.HomotopyAlong.homotopicWith). Hencec_*[Iⁿ]is the relative Hurewicz class of a relative cube ata₀(TauCeti.RelHomotopyGroup.exists_hurewicz_eq_singularHomologyMap), and everyc_*(x)is a relative Hurewicz image (TauCeti.RelHomotopyGroup.exists_hurewicz_apply_one_eq_singularHomologyMap). So the class of every such cube, read in singular homology, is a relative Hurewicz image (TauCeti.RelHomotopyGroup.exists_hurewicz_apply_one_eq_singularCubicalHomologyIso_inv).
As the relative Hurewicz map is a homomorphism in degrees at least two
(TauCeti.RelHomotopyGroup.hurewiczIntHom), its image is a subgroup, and it contains a set of
generators of Hₙ(X, A; ℤ).
Main results #
TauCeti.RelHomotopyGroup.exists_hurewicz_eq_singularHomologyMap: whenAis path connected, the image of the generator ofHₙ(Iⁿ, ∂Iⁿ)under a cube sending the boundary of the cube intoAis a relative Hurewicz class at any point ofA.TauCeti.RelHomotopyGroup.surjective_hurewicz: the relative Hurewicz mapπₙ₊₂(X, A, a₀) → Hₙ₊₂(X, A; ℤ)is surjective whenAis path connected and the pair is(n + 1)-connected.TauCeti.RelHomotopyGroup.injective_hurewicz_of_subsingleton_homotopyGroup,TauCeti.RelHomotopyGroup.bijective_hurewicz_of_subsingleton_homotopyGroup: the relative Hurewicz mapπₙ₊₂(X, A, a₀) → Hₙ₊₂(X, A; ℤ)is injective, and bijective, whenAandXare(n + 1)-connected.TauCeti.RelHomotopyGroup.hurewiczAddEquivOfSubsingletonHomotopyGroup: the resulting additive isomorphismπₙ₊₂(X, A, a₀) ≃ Hₙ₊₂(X, A; ℤ).
References #
- S. Eilenberg, Singular homology theory, Ann. of Math. (2) 45 (1944), 407–447, for the deformation of singular chains of a highly connected pair.
- A. Hatcher, Algebraic Topology, Section 4.2, Theorem 4.32 (the relative Hurewicz theorem) and Theorem 4.37.
Cubes with boundary in the subspace #
The class of a cube with boundary in a path-connected subspace is a relative Hurewicz
class. If B is path connected and b ∈ B, then for every singular (n + 1)-cube c sending
the boundary of the cube into B, the image under c : (Iⁿ⁺¹, ∂Iⁿ⁺¹) ⟶ (Y, B) of the generator
of Hₙ₊₁(Iⁿ⁺¹, ∂Iⁿ⁺¹; R) is the relative Hurewicz class of a relative cube at b.
Surjectivity of the relative Hurewicz map #
Over ℤ, if B is path connected and b ∈ B, then for every singular (n + 2)-cube c
sending the boundary of the cube into B, the image of every class of Hₙ₊₂(Iⁿ⁺², ∂Iⁿ⁺²; ℤ) under
c : (Iⁿ⁺², ∂Iⁿ⁺²) ⟶ (Y, B) is the relative Hurewicz image of a class of πₙ₊₂(Y, B, b).
Over ℤ, if B is path connected and b ∈ B, then the class of every singular (n + 2)-cube
c sending the boundary of the cube into B, read in relative singular homology
Hₙ₊₂(Y, B; ℤ), is the relative Hurewicz image of a class of πₙ₊₂(Y, B, b).
The relative Hurewicz map is surjective. Let B be a path-connected subset of Y and
b ∈ B. If every point of Y is joined by a path to a point of B, and the relative homotopy set
π_{k+1}(Y, B, b) is trivial for every k + 1 < n + 2, then the relative Hurewicz map
πₙ₊₂(Y, B, b) → Hₙ₊₂(Y, B; ℤ), [p] ↦ p_*[Iⁿ⁺²], is surjective.
The relative Hurewicz isomorphism for highly connected pairs #
Injectivity of the relative Hurewicz map for highly connected spaces. If πₖ(A, a₀) and
πₖ(X, a₀) are trivial for every k < n + 2, then the relative Hurewicz map
πₙ₊₂(X, A, a₀) → Hₙ₊₂(X, A; ℤ), [p] ↦ p_*[Iⁿ⁺²], is injective. In degree two, that is for
n = 0, the hypotheses say that A and X are simply connected.
The relative Hurewicz isomorphism for highly connected spaces. Let B be a subspace of
Y and b ∈ B. If πₖ(B, b) and πₖ(Y, b) are trivial for every k < n + 2, then the
relative Hurewicz map πₙ₊₂(Y, B, b) → Hₙ₊₂(Y, B; ℤ), [p] ↦ p_*[Iⁿ⁺²], is bijective. In degree
two, that is for n = 0, the hypotheses say that B and Y are simply connected.
The relative Hurewicz isomorphism πₙ₊₂(Y, B, b) ≃ Hₙ₊₂(Y, B; ℤ), [p] ↦ p_*[Iⁿ⁺²],
when πₖ(B, b) and πₖ(Y, b) are trivial for every k < n + 2. In degree two, where B and Y
are simply connected, this identifies π₂(Y, B, b), which is then commutative
(TauCeti.RelHomotopyGroup.mul_comm_of_subsingleton), with H₂(Y, B; ℤ).