The Künneth short exact sequence for singular homology #
Let X and Y be topological spaces and M and N modules over a commutative ring k.
Suppose that the cycles Z_q(Y; N) of the singular chain complex C(Y; N) and the quotients
C_q(Y; N) / Z_q(Y; N) are projective. Over a principal ideal domain this holds for every
projective N, for instance N = k, since the singular chain modules C_q(Y; N) = ⨁ N are then
projective (SSet.projective_chainComplex_X). More generally, both hypotheses are found by
instance search when every k-module has projective dimension at most one
(HomologicalComplex.projective_cycles_of_hasProjectiveDimensionLE_one and
HomologicalComplex.projective_cokernel_iCycles_of_hasProjectiveDimensionLE_one). Then the
Künneth sequence
0 ⟶ ⨁_{p + q = m + 1} Hₚ(X; M) ⊗ H_q(Y; N) ⟶ Hₘ₊₁(X × Y; M ⊗ N)
⟶ ⨁_{p + q = m} Tor₁(Hₚ(X; M), H_q(Y; N)) ⟶ 0
is exact (TopCat.singularHomologyKunnethShortComplex_shortExact). The first map is the Künneth
map TopCat.singularHomologyKunneth, assembled from the homology cross products. The second map
TopCat.singularHomologyToTor is the map induced by the Alexander–Whitney map
C(X × Y; M ⊗ N) ⟶ C(X; M) ⊗ C(Y; N) followed by the algebraic map
HomologicalComplex.homologyToTor of the Künneth sequence of chain complexes
(HomologicalComplex.homologyKunnethShortComplex_shortExact). The two sequences are isomorphic
through the Eilenberg–Zilber homotopy equivalence TopCat.eilenbergZilberHomotopyEquiv, whose
inverse is the shuffle map through which the Künneth map is defined.
Over a hereditary commutative ring k, expressed by requiring every k-module to have
projective dimension at most one, taking M = N = k and the unitor k ⊗ k ≅ k gives the sequence
0 ⟶ ⨁_{p + q = m + 1} Hₚ(X; k) ⊗ H_q(Y; k) ⟶ Hₘ₊₁(X × Y; k)
⟶ ⨁_{p + q = m} Tor₁(Hₚ(X; k), H_q(Y; k)) ⟶ 0
(TopCat.singularHomologyKunnethUnitShortComplex_shortExact).
The map TopCat.singularHomologyToTor is natural in both spaces and in both coefficient modules
(TopCat.singularHomologyToTor_naturality and
TopCat.singularHomologyToTor_coefficient_naturality), as is the Künneth map, so the Künneth
sequence is natural: continuous maps and maps of coefficient modules induce morphisms of Künneth
short complexes (TopCat.singularHomologyKunnethShortComplexMap and
TopCat.singularHomologyKunnethShortComplexCoefficientMap).
If the cycles of C(X; M) split off its terms as well, for instance for projective M and N
over a hereditary commutative ring, the Künneth map is a split monomorphism
(TopCat.isSplitMono_singularHomologyKunneth) and the sequence splits
(TopCat.singularHomologyKunnethSplitting, and TopCat.singularHomologyKunnethUnitSplitting for
coefficients in a hereditary commutative ring). The splitting is built from chosen splittings of
the cycles of the singular chain complexes and need not be natural in X and Y; that no natural
splitting exists is not proved here.
Through the splitting, a natural number e that kills the positive-degree homology of X and of
Y, with coefficients in a hereditary commutative ring, also kills that of X × Y
(TopCat.nsmul_singularHomology_tensor_eq_zero). Every tensor product summand has a factor of
positive degree, and so does every Tor₁ summand except Tor₁(H₀(X; k), H_q(Y; k)), which
vanishes because H₀(X; k) is a free k-module. The same number e works for the product;
without the splitting the extension would only show that e² kills it.
Main definitions and results #
TopCat.singularHomologyToTor: the mapHₘ₊₁(X × Y; M ⊗ N) ⟶ ⨁_{p + q = m} Tor₁(Hₚ(X; M), H_q(Y; N)).TopCat.singularHomologyToTor_naturalityandTopCat.singularHomologyToTor_coefficient_naturality: its naturality.TopCat.singularHomologyKunnethShortComplexandTopCat.singularHomologyKunnethShortComplex_shortExact: the Künneth short exact sequence.TopCat.mono_singularHomologyKunneth: the Künneth map is a monomorphism in every degree.TopCat.singularHomologyKunnethUnitShortComplexandTopCat.singularHomologyKunnethUnitShortComplex_shortExact: its form with coefficients in a hereditary commutative ring.TopCat.singularHomologyKunnethShortComplexMapandTopCat.singularHomologyKunnethShortComplexCoefficientMap: the morphisms of Künneth short complexes induced by a pair of continuous maps and by a pair of maps of coefficient modules.TopCat.singularHomologyKunnethSplittingandTopCat.singularHomologyKunnethUnitSplitting: the splitting of the Künneth sequence.TopCat.nsmul_singularHomology_tensor_eq_zero: a number killing the positive-degree homology ofXandYkills that ofX × Y.
References #
- A. Hatcher, Algebraic Topology, Section 3.B, Theorem 3B.6.
- C. Weibel, An Introduction to Homological Algebra, Theorem 3.6.3, the Künneth formula.
The map Hₘ₊₁(X × Y; M ⊗ N) ⟶ ⨁_{p + q = m} Tor₁(Hₚ(X; M), H_q(Y; N)) of the Künneth short
exact sequence (TopCat.singularHomologyKunnethShortComplex_shortExact): the map induced by the
Alexander–Whitney map, followed by the map HomologicalComplex.homologyToTor of the Künneth
sequence of the singular chain complexes C(X; M) and C(Y; N).
Instances For
The map to ⨁ Tor₁(Hₚ(X; M), H_q(Y; N)) is the map induced by the Alexander–Whitney map
followed by the algebraic map of the Künneth sequence of the singular chain complexes.
The Künneth map followed by the map to ⨁ Tor₁(Hₚ(X; M), H_q(Y; N)) vanishes.
The Künneth map followed by the map to ⨁ Tor₁(Hₚ(X; M), H_q(Y; N)) vanishes.
Naturality of the Künneth short exact sequence in both spaces: the map
TopCat.singularHomologyToTor to ⨁_{p + q = m} Tor₁(Hₚ(X; M), H_q(Y; N)) commutes with the maps
induced by continuous maps f : X ⟶ X' and g : Y ⟶ Y'. Together with the naturality of the
Künneth map (TopCat.singularHomologyKunneth_naturality), this makes the Künneth short exact
sequence natural in X and Y.
Naturality of the Künneth short exact sequence in both spaces: the map
TopCat.singularHomologyToTor to ⨁_{p + q = m} Tor₁(Hₚ(X; M), H_q(Y; N)) commutes with the maps
induced by continuous maps f : X ⟶ X' and g : Y ⟶ Y'. Together with the naturality of the
Künneth map (TopCat.singularHomologyKunneth_naturality), this makes the Künneth short exact
sequence natural in X and Y.
Naturality of the Künneth short exact sequence in both coefficient modules: the map
TopCat.singularHomologyToTor commutes with the maps induced by module maps φ : M ⟶ M' and
ψ : N ⟶ N'.
Naturality of the Künneth short exact sequence in both coefficient modules: the map
TopCat.singularHomologyToTor commutes with the maps induced by module maps φ : M ⟶ M' and
ψ : N ⟶ N'.
The Künneth short complex
⨁_{p + q = m + 1} Hₚ(X; M) ⊗ H_q(Y; N) ⟶ Hₘ₊₁(X × Y; M ⊗ N)
⟶ ⨁_{p + q = m} Tor₁(Hₚ(X; M), H_q(Y; N))
formed by the Künneth map and TopCat.singularHomologyToTor.
Instances For
The Künneth short exact sequence for singular homology. Let M and N be modules over a
commutative ring such that the cycles Z_q(Y; N) and the quotients C_q(Y; N) / Z_q(Y; N) are
projective, for instance any projective N over a principal ideal domain. Then
0 ⟶ ⨁_{p + q = m + 1} Hₚ(X; M) ⊗ H_q(Y; N) ⟶ Hₘ₊₁(X × Y; M ⊗ N)
⟶ ⨁_{p + q = m} Tor₁(Hₚ(X; M), H_q(Y; N)) ⟶ 0
is exact, the first map being the Künneth map and the second TopCat.singularHomologyToTor.
The Künneth map is a monomorphism when the cycles Z_q(Y; N) and the quotients
C_q(Y; N) / Z_q(Y; N) are projective, for instance for any projective N over a principal ideal
domain.
The Künneth short exact sequence for singular homology splits when, in addition, the
cycles of the singular chain complex C(X; M) split off its terms; for instance for projective
modules M and N over a principal ideal domain. The splitting comes from a retraction of the
Künneth map (TopCat.isSplitMono_singularHomologyKunneth) built from chosen splittings of the
cycles, so it need not be natural in X and Y.
Instances For
The retraction of the Künneth splitting is the chosen retraction of the Künneth map.
The morphism of Künneth short complexes induced by continuous maps f : X ⟶ X' and
g : Y ⟶ Y': the maps f_* ⊗ g_* on ⨁ Hₚ(X; M) ⊗ H_q(Y; N), (f × g)_* on
Hₘ₊₁(X × Y; M ⊗ N) and Tor₁(f_*, g_*) on ⨁ Tor₁(Hₚ(X; M), H_q(Y; N)). Together with
TopCat.singularHomologyKunnethShortComplex_shortExact this makes the Künneth short exact sequence
natural in both spaces.
Instances For
On ⨁ Hₚ(X; M) ⊗ H_q(Y; N), the morphism of Künneth short complexes is f_* ⊗ g_*.
On Hₘ₊₁(X × Y; M ⊗ N), the morphism of Künneth short complexes is (f × g)_*.
On ⨁ Tor₁(Hₚ(X; M), H_q(Y; N)), the morphism of Künneth short complexes is
Tor₁(f_*, g_*).
The morphism of Künneth short complexes induced by linear maps φ : M ⟶ M' and ψ : N ⟶ N'
of coefficient modules: the maps φ_* ⊗ ψ_* on ⨁ Hₚ(X; M) ⊗ H_q(Y; N), the map induced by
φ ⊗ ψ on Hₘ₊₁(X × Y; M ⊗ N) and Tor₁(φ_*, ψ_*) on ⨁ Tor₁(Hₚ(X; M), H_q(Y; N)). Together
with TopCat.singularHomologyKunnethShortComplex_shortExact this makes the Künneth short exact
sequence natural in both coefficient modules.
Instances For
On ⨁ Hₚ(X; M) ⊗ H_q(Y; N), the morphism of Künneth short complexes is φ_* ⊗ ψ_*.
On Hₘ₊₁(X × Y; M ⊗ N), the morphism of Künneth short complexes is the map induced by
φ ⊗ ψ.
On ⨁ Tor₁(Hₚ(X; M), H_q(Y; N)), the morphism of Künneth short complexes is
Tor₁(φ_*, ψ_*).
The Künneth short complex with coefficients in the ring:
⨁_{p + q = m + 1} Hₚ(X; k) ⊗ H_q(Y; k) ⟶ Hₘ₊₁(X × Y; k) ⟶ ⨁_{p + q = m} Tor₁(Hₚ(X; k), H_q(Y; k)),
formed by the Künneth map and TopCat.singularHomologyToTor through the unitor k ⊗ k ≅ k.
Instances For
The Künneth short exact sequence over a hereditary commutative ring. For a commutative
ring k whose modules have projective dimension at most one, for instance ℤ,
0 ⟶ ⨁_{p + q = m + 1} Hₚ(X; k) ⊗ H_q(Y; k) ⟶ Hₘ₊₁(X × Y; k)
⟶ ⨁_{p + q = m} Tor₁(Hₚ(X; k), H_q(Y; k)) ⟶ 0
is exact, the first map being the Künneth map followed by the map induced by the unitor
k ⊗ k ≅ k.
The Künneth short exact sequence over a hereditary commutative ring splits, so that
Hₘ₊₁(X × Y; k) is the direct sum of ⨁_{p + q = m + 1} Hₚ(X; k) ⊗ H_q(Y; k) and
⨁_{p + q = m} Tor₁(Hₚ(X; k), H_q(Y; k))
(CategoryTheory.ShortComplex.Splitting.isoBinaryBiproduct).
The splitting depends on chosen splittings of the cycles of the singular chain complexes, and need
not be natural in X and Y.
Instances For
The retraction of the Künneth splitting with coefficients in k is the chosen retraction of
the Künneth map, preceded by the map induced by the inverse unitor.
A number killing the homology of two spaces kills that of their product. Over a
hereditary commutative ring k, if a natural number e kills the positive-degree homology of
spaces X and Y, then it kills the positive-degree homology of X × Y.