Evaluating cohomology classes on fundamental classes #
Let N be a compact topological n-manifold without boundary with an orientation o
(TauCeti.HomologicalOrientation N E n), and let f : N → X be a continuous map to a space X.
The fundamental class [N] ∈ Hₙ(N; ℤ) pushes forward to a class f_*[N] ∈ Hₙ(X; ℤ), and a
cohomology class c ∈ Hⁿ(X; M) with coefficients in an abelian group M can be evaluated on it
through the Kronecker map, giving ⟨c, f_*[N]⟩ ∈ M. For n = 2 and an embedded closed oriented
surface f : S → X, this is the evaluation of a degree-two class on the surface, as in the
Euler-class inequalities for foliations of 3-manifolds, where c is the Euler class of the
tangent plane field.
The pairing is natural: evaluating c on the image of [N] under f is evaluating the pull-back
f^* c on [N] itself, and for a further map g : X → Y, evaluating c on (g ∘ f)_*[N] is
evaluating g^* c on f_*[N]. The class f_*[N], and so the pairing, depends only on the
homotopy class of f, and reversing the orientation of N negates both. For a map f : N → P to
a second compact preconnected oriented n-manifold, the pairing scales by the degree of f:
⟨f^* c, [N]⟩ = deg f • ⟨c, [P]⟩.
The pairing is non-degenerate on the manifold itself. If N is moreover connected and M is an
injective abelian group, such as ℚ or ℚ / ℤ, then c ↦ ⟨c, [N]⟩ is a bijection
Hⁿ(N; M) → M: the Kronecker map is an isomorphism Hⁿ(N; M) ≃ Hom(Hₙ(N; ℤ), M) for injective
coefficients (TopCat.singularKroneckerEquiv), and Hₙ(N; ℤ) is freely generated by [N]
(TauCeti.HomologicalOrientation.fundamentalClassEquiv).
Main definitions #
TauCeti.HomologicalOrientation.evalFundamentalClass: theℤ-linear evaluationc ↦ ⟨c, f_*[N]⟩ofHⁿ(X; M)on the imageTauCeti.HomologicalOrientation.mapFundamentalClassof the fundamental class under a continuous mapf : N → X.
Main results #
TauCeti.HomologicalOrientation.evalFundamentalClass_id_singularCohomologyMap:⟨f^* c, [N]⟩ = ⟨c, f_*[N]⟩.TauCeti.HomologicalOrientation.evalFundamentalClass_comp:⟨c, (g ∘ f)_*[N]⟩ = ⟨g^* c, f_*[N]⟩.TauCeti.HomologicalOrientation.evalFundamentalClass_eq_of_homotopic: homotopic maps give the same pairing.TauCeti.HomologicalOrientation.evalFundamentalClass_neg: reversing the orientation negates the pairing.TauCeti.HomologicalOrientation.evalFundamentalClass_id_singularCohomologyMap_eq_degree_smul: for a mapf : N → Pof compact orientedn-manifolds withPpreconnected,⟨f^* c, [N]⟩ = deg f • ⟨c, [P]⟩, and more generallyTauCeti.HomologicalOrientation.evalFundamentalClass_comp_eq_degree_smul:⟨c, (g ∘ f)_*[N]⟩ = deg f • ⟨c, g_*[P]⟩.TauCeti.HomologicalOrientation.evalFundamentalClass_id_bijective: for a compact connected oriented manifold and injective coefficients, evaluation on the fundamental class is a bijectionHⁿ(N; M) → M.
Implementation notes #
The manifold N and the target space X live in Type, as for
TauCeti.HomologicalOrientation.fundamentalClass, and homology has the integral coefficients
ModuleCat.of ℤ (ULift ℤ) used there.
References #
- A. Hatcher, Algebraic Topology, Section 3.1 (the Kronecker pairing and the universal coefficient theorem) and Section 3.3 (fundamental classes).
- D. Calegari, Foliations and the Geometry of 3-Manifolds, Oxford (2007), Chapter 6 (the Euler class of a taut foliation evaluated on surfaces).
The evaluation of a cohomology class on the image of a fundamental class: for a compact
oriented n-manifold N and a continuous map f : N → X, the ℤ-linear map
Hⁿ(X; M) → M, c ↦ ⟨c, f_*[N]⟩, the Kronecker pairing of c with
TauCeti.HomologicalOrientation.mapFundamentalClass.
Instances For
The evaluation on f_*[N] is the Kronecker map applied to f_*[N].
Functoriality of the pairing, ⟨c, (g ∘ f)_*[N]⟩ = ⟨g^* c, f_*[N]⟩.
Naturality of the pairing, ⟨f^* c, [N]⟩ = ⟨c, f_*[N]⟩: evaluating the pull-back of a
class on the fundamental class is evaluating the class itself on the image of the fundamental
class.
Homotopy invariance of the pairing: homotopic maps give the same evaluation.
Reversing the orientation negates the pairing.
Evaluation on the fundamental class is an isomorphism for injective coefficients. For a
compact connected oriented n-manifold N and an injective abelian group M, such as ℚ, the
map Hⁿ(N; M) → M, c ↦ ⟨c, [N]⟩, is bijective.
The degree formula for the pairing: for a map f : N → P of compact oriented
n-manifolds with P preconnected and any continuous g : P → X,
⟨c, (g ∘ f)_*[N]⟩ = deg f • ⟨c, g_*[P]⟩.
The degree formula for characteristic numbers: for a map f : N → P of compact oriented
n-manifolds with P preconnected, evaluating the pull-back of c ∈ Hⁿ(P; M) on [N] is the
degree of f times evaluating c on [P]: ⟨f^* c, [N]⟩ = deg f • ⟨c, [P]⟩.