Cup length and the Lusternik–Schnirelmann category #
For a commutative ring R, the cup length TauCeti.lsCupLength X R : ℕ∞ of a space X is the
supremum of the lengths n of the nonzero cup products a₁ ⌣ ⋯ ⌣ aₙ of classes aᵢ of positive
degree in the singular cohomology H*(X; R), the supremum of the empty collection being 0.
The cohomology and the cup product are the singular ones of
TauCeti.AlgebraicTopology.Cohomology.Cup.Basic, for the coefficient category ModuleCat R, the
chain and cochain coefficients R = 𝟙_ (ModuleCat R), and the unitors of ModuleCat R as the
diagonal of the chain coefficients and the multiplication of the cochain values.
The main result is the classical lower bound for the Lusternik–Schnirelmann category: on a nonempty
space, lsCupLength X R + 1 ≤ lsCategoryCount X univ, equivalently
lsCupLength X R ≤ normalizedLSCategory X, which also holds on the empty space. If U₁, …, Uₙ
are open sets whose inclusions into X are nullhomotopic, a class aᵢ of positive degree
restricts to zero on Uᵢ, because the positive-degree cohomology of a point vanishes, so the long
exact sequence of the pair lifts it to H*(X, Uᵢ). The relative cup product of open sets lifts
a₁ ⌣ ⋯ ⌣ aₙ to H*(X, U₁ ∪ ⋯ ∪ Uₙ), which vanishes when the Uᵢ cover X.
Main definitions #
TauCeti.IsPositiveDegreeCupProduct X R n x: the classxofH*(X; R)is a cup producta₁ ⌣ (a₂ ⌣ ⋯ ⌣ aₙ)ofn ≥ 1classes of positive degree.TauCeti.lsCupLength X R: the cup length ofXwith coefficients inR.
Main results #
TauCeti.natCast_le_lsCupLength_iff: forn ≥ 1,n ≤ lsCupLength X Rexactly when some product ofnpositive-degree classes is nonzero.TauCeti.lsCupLength_eq_zero_iff: the cup length vanishes exactly when every class of positive degree vanishes.TauCeti.IsPositiveDegreeCupProduct.exists_singularCohomologyπ_eq: a product ofnclasses of positive degree comes from the cohomology ofXrelative to the union of anynopen sets with nullhomotopic inclusions.TauCeti.lsCupLength_add_one_le_lsCategoryCount_univandTauCeti.lsCupLength_le_normalizedLSCategory: the cup-length lower bound for the Lusternik–Schnirelmann category.TauCeti.lsCupLength_le_of_injectiveandContinuousMap.HomotopyEquiv.lsCupLength_eq: monotonicity along maps injective on cohomology, and homotopy invariance.
References #
- O. Cornea, G. Lupton, J. Oprea and D. Tanré, Lusternik–Schnirelmann Category, Mathematical Surveys and Monographs 103, AMS, 2003, Chapter 1, the cup-length lower bound.
- A. Hatcher, Algebraic Topology,
Section 3.2, the relative cup product
Hᵏ(X, A) × Hˡ(X, B) ⟶ Hᵏ⁺ˡ(X, A ∪ B)for openAandB.
IsPositiveDegreeCupProduct X R n x: the class x ∈ Hᵈ(X; R) is a cup product
a₁ ⌣ (a₂ ⌣ ⋯ ⌣ aₙ) of n classes aᵢ ∈ H^{pᵢ}(X; R) of positive degree pᵢ ≠ 0, nested to the
right, so that n ≥ 1 and d = p₁ + ⋯ + pₙ. Here H*(X; R) is singular cohomology with chain and
cochain coefficients R = 𝟙_ (ModuleCat R), and the cup product uses the unitors of ModuleCat R
as diagonal and multiplication.
- single
{X : Type u}
[TopologicalSpace X]
{R : Type u}
[CommRing R]
{p : ℕ}
(hp : p ≠ 0)
(a :
↑(TopCat.singularCohomology (CategoryTheory.MonoidalCategoryStruct.tensorUnit (ModuleCat R)) R
(CategoryTheory.MonoidalCategoryStruct.tensorUnit (ModuleCat R)) (↧X) p))
: IsPositiveDegreeCupProduct X R 1 a
A single class of positive degree is a product of length one.
- cup
{X : Type u}
[TopologicalSpace X]
{R : Type u}
[CommRing R]
{n p q d : ℕ}
(hp : p ≠ 0)
(h : p + q = d)
(a :
↑(TopCat.singularCohomology (CategoryTheory.MonoidalCategoryStruct.tensorUnit (ModuleCat R)) R
(CategoryTheory.MonoidalCategoryStruct.tensorUnit (ModuleCat R)) (↧X) p))
{y :
↑(TopCat.singularCohomology (CategoryTheory.MonoidalCategoryStruct.tensorUnit (ModuleCat R)) R
(CategoryTheory.MonoidalCategoryStruct.tensorUnit (ModuleCat R)) (↧X) q)}
(hy : IsPositiveDegreeCupProduct X R n y)
: IsPositiveDegreeCupProduct X R (n + 1)
((((↧X).singularCup R
(CategoryTheory.MonoidalCategoryStruct.leftUnitor
(CategoryTheory.MonoidalCategoryStruct.tensorUnit (ModuleCat R))).inv
(CategoryTheory.MonoidalCategoryStruct.leftUnitor
(CategoryTheory.MonoidalCategoryStruct.tensorUnit (ModuleCat R))).hom
p q d h)
a)
y)
A class of positive degree cupped with a product of length
nis a product of lengthn + 1.
Instances For
The cup length of a space X with coefficients in a commutative ring R: the supremum of
the lengths n of the nonzero cup products a₁ ⌣ ⋯ ⌣ aₙ of classes of positive degree in
H*(X; R) (TauCeti.IsPositiveDegreeCupProduct). It is 0 when every class of positive degree
vanishes, and ⊤ when there are nonzero such products of every length.
Instances For
A product of positive-degree classes has at least one factor.
A product of positive-degree classes has positive degree.
The products of length one are exactly the classes of positive degree.
If a nonzero class is a product of n positive-degree classes, then for every 1 ≤ m ≤ n some
nonzero class is a product of m positive-degree classes: drop leading factors, since a cup product
with a zero factor vanishes.
Naturality: pulling back a product of positive-degree classes along a continuous map gives a product of positive-degree classes of the same length.
The defining property of the cup length: it bounds the length of every nonzero product of positive-degree classes.
The cup length is at most m exactly when every nonzero product of positive-degree classes
has length at most m.
Finite-product witnesses: for n ≥ 1, the cup length is at least n exactly when some
product of n classes of positive degree is nonzero.
The cup length vanishes exactly when every class of positive degree in H*(X; R) vanishes.
Monotonicity: if a continuous map f : X → Y induces injective maps on cohomology in every
positive degree, then the cup length of Y is at most that of X.
Homotopy invariance of the cup length.
The empty space has cup length zero.
The lower bound for the Lusternik–Schnirelmann category #
Products of positive-degree classes vanish on categorical opens. A product of n classes
of positive degree comes from the cohomology of X relative to the union U₁ ∪ ⋯ ∪ Uₙ of any n
open sets whose inclusions into X are nullhomotopic.
The cup-length lower bound for the Lusternik–Schnirelmann category: on a nonempty space
X, a cover by r open sets with nullhomotopic inclusions forces every product of r classes of
positive degree to vanish, so lsCupLength X R + 1 ≤ lsCategoryCount X univ.
The cup-length lower bound for the normalized Lusternik–Schnirelmann category:
lsCupLength X R ≤ normalizedLSCategory X.
A contractible space has cup length zero.