The torsion of a based chain complex over a field #
Let C be a ℤ-graded chain complex of vector spaces over a field F, with an ordered basis
c i : Basis (Fin (n i)) F (C.X i) in each degree. Write Bⱼ ⊆ Cⱼ for the space of boundaries,
the image of d : Cⱼ₊₁ → Cⱼ. When C is acyclic, every sequence
0 → Bⱼ₊₁ → Cⱼ₊₁ → Bⱼ → 0
is short exact, so a basis bⱼ₊₁ of Bⱼ₊₁, a basis bⱼ of Bⱼ and lifts sⱼ of bⱼ through
d give an ordered basis (bⱼ₊₁, sⱼ) of Cⱼ₊₁. The bracket [bⱼ₊₁ sⱼ / cⱼ₊₁] is its
determinant with respect to cⱼ₊₁ (HomologicalComplex.torsionBracket), and the torsion of C is
the alternating product
τ(C) = ∏ⱼ [bⱼ₊₁ sⱼ / cⱼ₊₁] ^ ((-1) ^ j),
so that the bracket in degree i carries the exponent (-1) ^ (i + 1). With this convention a
complex concentrated in degrees 1 and 0 has torsion the inverse of the determinant of its
differential (HomologicalComplex.reidemeisterTorsion_eq_det_zpow); for instance multiplication
by a ≠ 0 on F, from degree 1 to degree 0, has torsion a⁻¹. The torsion of a complex
that is not acyclic is defined to be 0 (HomologicalComplex.reidemeisterTorsion), and so is
that of an unbounded complex, for which the alternating product is not defined.
For a bounded complex the alternating product does not depend on the boundary bases nor on the
lifts (HomologicalComplex.reidemeisterTorsion_eq_finprod): changing the lifts does not change
any bracket, and changing bⱼ multiplies the brackets in degrees j + 1 and j by the same
determinant, which enters with opposite exponents. The torsion of a bounded complex vanishes
exactly when the complex is not acyclic (HomologicalComplex.reidemeisterTorsion_eq_zero_iff).
Replacing the chain bases c i by c' i multiplies the torsion by ∏ᵢ [c'ᵢ / cᵢ] ^ ((-1) ^ i),
where [c'ᵢ / cᵢ] is the determinant of c' i with respect to c i
(HomologicalComplex.reidemeisterTorsion_eq_mul_finprod_det). In particular reordering the chain
bases multiplies the torsion by the signs of the permutations, and rescaling the basis vectors by
units multiplies it by the alternating product of their products.
The torsion is invariant under isomorphisms of based complexes: transporting the bases of C
along an isomorphism e : C ≅ D gives D the same torsion
(HomologicalComplex.reidemeisterTorsion_map).
Combined with the basis-change formula, if Aᵢ is the matrix of eᵢ in given bases of Cᵢ and
Dᵢ, then τ(C) = τ(D) · ∏ᵢ det (Aᵢ) ^ ((-1) ^ i)
(HomologicalComplex.reidemeisterTorsion_eq_mul_finprod_det_toMatrix). This is the formula
τ(A⁻¹ d A) = τ(d) · ∏ᵢ det (Aᵢ) ^ ((-1) ^ i) for a differential d conjugated by degreewise
automorphisms Aᵢ.
The torsion is multiplicative in short exact sequences
(HomologicalComplex.reidemeisterTorsion_eq_mul_of_shortExact): if 0 → C₁ → C₂ → C₃ → 0 is a
short exact sequence of chain complexes with C₁ and C₃ acyclic, and the bases of C₂ are the
images of the bases of C₁ followed by lifts of the bases of C₃, then
τ(C₂) = ε · τ(C₁) · τ(C₃), where ε = ∏ⱼ (-1) ^ (dim B₃,ⱼ₊₁ · dim B₁,ⱼ) is the sign of the
exchange of the boundaries of C₃ in degree j + 1 with the lifts of the boundaries of C₁ in
degree j. To prove it, the boundaries of C₂ in each degree are given the basis made of the
boundaries of C₁ followed by lifts of the boundaries of C₃, using that the boundaries form a
short exact sequence 0 → B₁,ⱼ → B₂,ⱼ → B₃,ⱼ → 0 when C₃ is acyclic.
When the short exact sequence splits, no acyclicity is needed
(HomologicalComplex.reidemeisterTorsion_eq_mul_of_splitting): the homology of C₁ and C₃ is a
retract of that of C₂ (CategoryTheory.ShortComplex.Splitting.acyclic_X₁,
CategoryTheory.ShortComplex.Splitting.acyclic_X₃), so C₂ is acyclic exactly when C₁ and C₃
are, and otherwise both sides are 0. In particular the torsion of a direct sum C₁ ⊞ C₃, based
in each degree by the basis of C₁ followed by the basis of C₃
(HomologicalComplex.biprodXBasis), is ε · τ(C₁) · τ(C₃) with the same sign
(HomologicalComplex.reidemeisterTorsion_biprod), since the biproduct is the middle term of the
split short exact sequence 0 → C₁ → C₁ ⊞ C₃ → C₃ → 0.
The torsion is compatible with scalar extension along a homomorphism of fields φ : F →+* K
(HomologicalComplex.map_reidemeisterTorsion): if D is a complex over K with bases e i in
which the matrices of the differentials are the images under φ of those of C in the bases
c i, then τ(D) = φ(τ(C)). The ranks of the differentials do not change under φ, so D is
acyclic exactly when C is (HomologicalComplex.acyclic_iff_of_toMatrix_eq), and the brackets
of D are computed from the images of the boundary bases and lifts of C under the semilinear map
Module.Basis.semilinearMap φ (c i) (e i), whose determinants are the images under φ of those
in C. Thus the torsion of a based complex over a ring with several residue or fraction fields can
be compared componentwise along the field homomorphisms between them.
Main declarations #
HomologicalComplex.torsionBracket: the bracket[bⱼ₊₁ sⱼ / cⱼ₊₁]of a degree.HomologicalComplex.torsionBracket_eq_det_append: the bracket is the determinant of the boundary basis followed by the lifts.HomologicalComplex.reidemeisterTorsion: the torsion of a based chain complex.HomologicalComplex.reidemeisterTorsion_eq_finprod: for a bounded acyclic complex, the torsion is the alternating product of the brackets for any boundary bases and lifts.HomologicalComplex.reidemeisterTorsion_eq_zero_iff: the torsion of a bounded complex vanishes if and only if the complex is not acyclic.HomologicalComplex.reidemeisterTorsion_eq_det_zpow: the torsion of an acyclic complex concentrated in two consecutive degrees.HomologicalComplex.reidemeisterTorsion_eq_mul_finprod_det: the change of the torsion under a change of the chain bases.HomologicalComplex.reidemeisterTorsion_reindex: the change of the torsion under a reordering of the chain bases.HomologicalComplex.reidemeisterTorsion_unitsSMul: the change of the torsion under a rescaling of the chain basis vectors by units.HomologicalComplex.reidemeisterTorsion_map: the torsion is invariant under isomorphisms of based complexes.HomologicalComplex.reidemeisterTorsion_eq_mul_finprod_det_toMatrix: the change of the torsion under an isomorphism of complexes, in terms of the determinants of its matrices.HomologicalComplex.reidemeisterTorsion_eq_mul_of_shortExact: the multiplicativity of the torsion in a based short exact sequence of acyclic complexes.HomologicalComplex.reidemeisterTorsion_eq_mul_of_splitting: the multiplicativity of the torsion in a based split short exact sequence, with no acyclicity hypothesis.HomologicalComplex.reidemeisterTorsion_biprod: the torsion of a direct sum of based complexes.HomologicalComplex.map_reidemeisterTorsion: the torsion of the scalar extension of a based complex along a homomorphism of fields is the image of its torsion.
References #
- J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), §3.
- V. Turaev, Introduction to combinatorial torsions, Birkhäuser 2001, §1.
In a degree j + 1 where C is exact, the dimension n (j + 1) of Cⱼ₊₁ is the sum of the
dimensions p and q of the boundaries Bⱼ₊₁ and Bⱼ.
The bracket [b₁ s / cⱼ₊₁]: the determinant, with respect to the basis c (j + 1) of Cⱼ₊₁,
of the family made of a basis b₁ of the boundaries Bⱼ₊₁ ⊆ Cⱼ₊₁ followed by lifts s of a
basis b₀ of the boundaries Bⱼ ⊆ Cⱼ.
Instances For
The bracket [b₁ s / cⱼ₊₁] is the determinant, with respect to the basis c (j + 1) of
Cⱼ₊₁, of the family (b₁, s) obtained by appending the lifts s to the basis b₁ of the
boundaries Bⱼ₊₁.
Replacing the boundary bases and the lifts in a bracket multiplies it by the determinants of the new boundary bases with respect to the old ones; in particular the bracket does not depend on the lifts.
Replacing the basis c (j + 1) of Cⱼ₊₁ by c' (j + 1) multiplies a bracket by the
determinant of c (j + 1) with respect to c' (j + 1).
A bracket is nonzero.
The bracket in a degree where the complex vanishes is 1.
In a degree j + 1 where the boundaries Bⱼ₊₁ fill Cⱼ₊₁ (that is, Bⱼ = 0, as the
dimension n (j + 1) is p), the bracket is the determinant of the basis b₁ of Bⱼ₊₁.
In a degree j + 1 where Cⱼ₊₁ has no nonzero boundaries (that is, Bⱼ₊₁ = 0, as the
dimension n (j + 1) is q), the bracket is the determinant of the lifts s of the basis b₀
of Bⱼ.
The basis Module.finBasis of the boundaries Bⱼ, used to define the torsions of a based
complex.
Instances For
Lifts of chosenBoundaryBasis C c j through the differential, chosen by Function.surjInv.
Instances For
The lifts chosenBoundaryLift C c j are mapped by the differential to the basis
chosenBoundaryBasis C c j.
The torsion of a chain complex of vector spaces with ordered bases c i of each C.X i: the
alternating product ∏ⱼ [bⱼ₊₁ sⱼ / cⱼ₊₁] ^ ((-1) ^ j) of the brackets when C is acyclic and
bounded (only finitely many n i are nonzero), for a choice of bases bⱼ of the boundaries and
lifts sⱼ of bⱼ, and 0 otherwise. The product does not depend on these choices
(reidemeisterTorsion_eq_finprod).
Instances For
The torsion of a complex that is not acyclic is 0.
The torsion of an unbounded complex, one with infinitely many nonzero n i, is 0.
The torsion of a bounded based complex vanishes if and only if the complex is not acyclic.
For a bounded complex, the determinants of one family of bases of the boundaries with respect
to another have finite multiplicative support: they are 1 where the complex vanishes.
For a bounded complex, the alternating brackets have finite multiplicative support.
For a bounded acyclic complex, the torsion is the alternating product
∏ⱼ [bⱼ₊₁ sⱼ / cⱼ₊₁] ^ ((-1) ^ j) of the brackets, for any bases bⱼ of the boundaries and any
lifts sⱼ of bⱼ; the bases bⱼ may be indexed by any Fin (r j).
The basis-change formula for the torsion: replacing the bases c i of C by c' i
multiplies the torsion by ∏ᵢ [c'ᵢ / cᵢ] ^ ((-1) ^ i), where [c'ᵢ / cᵢ] is the determinant of
c' i with respect to c i. This holds for every complex, both sides being 0 when C is
not acyclic or is unbounded.
Reordering the bases c i of C by permutations σ i multiplies the torsion by the product
of the signs of the σ i.
Rescaling the vectors of the bases c i of C by units w i k multiplies the torsion by
∏ᵢ (∏ₖ w i k) ^ ((-1) ^ i).
An acyclic complex concentrated in degrees j + 1 and j + 2 has bases of the same size in
both degrees, as its differential Cⱼ₊₂ → Cⱼ₊₁ is then an isomorphism.
The torsion of an acyclic complex concentrated in two consecutive degrees j + 1 and j + 2
is det (d) ^ ((-1) ^ j), where det (d) is the determinant of the differential
Cⱼ₊₂ → Cⱼ₊₁ in the bases c (j + 2) and c (j + 1). For j = -1 this is the normalization
of the torsion: a complex concentrated in degrees 1 and 0 has torsion det (d)⁻¹, so that
multiplication by a ≠ 0 on F, from degree 1 to degree 0, has torsion a⁻¹.
Multiplicativity in short exact sequences #
Multiplicativity of the torsion. Let 0 → C₁ → C₂ → C₃ → 0 be a short exact sequence of
chain complexes of vector spaces, with C₁ and C₃ acyclic, and with compatible bases: the basis
c₂ i of C₂ᵢ is the image of the basis c₁ i of C₁ᵢ followed by lifts of the basis c₃ i of
C₃ᵢ. Then τ(C₂) = ε · τ(C₁) · τ(C₃), with the sign ε = ∏ⱼ (-1) ^ (dim B₃,ⱼ₊₁ · dim B₁,ⱼ)
given by the dimensions of the boundaries Bₖ,ⱼ ⊆ Cₖ,ⱼ. This also holds when the complexes are
unbounded, both sides being 0.
Split short exact sequences and direct sums #
Torsion of a split short exact sequence. Let 0 → C₁ → C₂ → C₃ → 0 be a split short exact
sequence of chain complexes of vector spaces, with compatible bases: the basis c₂ i of C₂ᵢ is
the image of the basis c₁ i of C₁ᵢ followed by lifts of the basis c₃ i of C₃ᵢ. Then
τ(C₂) = ε · τ(C₁) · τ(C₃), with the sign ε = ∏ⱼ (-1) ^ (dim B₃,ⱼ₊₁ · dim B₁,ⱼ) given by the
dimensions of the boundaries Bₖ,ⱼ ⊆ Cₖ,ⱼ. This holds for all complexes: when C₁ or C₃ is
not acyclic, neither is C₂, and both sides are 0.
Torsion of a direct sum. The torsion of the biproduct C₁ ⊞ C₃ of two based complexes,
based in each degree by the basis of C₁ followed by the basis of C₃, is
τ(C₁ ⊞ C₃) = ε · τ(C₁) · τ(C₃), with the sign ε = ∏ⱼ (-1) ^ (dim B₃,ⱼ₊₁ · dim B₁,ⱼ) given by
the dimensions of the boundaries Bₖ,ⱼ ⊆ Cₖ,ⱼ. This holds for all complexes: C₁ ⊞ C₃ is acyclic
exactly when C₁ and C₃ both are, and both sides are 0 otherwise.
Scalar extension along a homomorphism of fields #
Scalar extension of the torsion. Let φ : F →+* K be a homomorphism of fields, and let
D be a complex of vector spaces over K with bases e i in which the matrices of the
differentials are the images under φ of the matrices of the differentials of C in the bases
c i; that is, (D, e) is the scalar extension of (C, c) along φ. Then the torsion of D is
the image under φ of the torsion of C. This holds for every complex: D is acyclic exactly when
C is, and both torsions are 0 otherwise.
The torsion is invariant under isomorphisms of based complexes: transporting the bases c i
of C along an isomorphism e : C ≅ D gives bases of D for which D has the same torsion as
C.
The torsion under an isomorphism e : C ≅ D of complexes with bases c i and c' i: if
Aᵢ is the matrix of eᵢ : Cᵢ → Dᵢ in the bases c i and c' i, then
τ(C, c) = τ(D, c') · ∏ᵢ det (Aᵢ) ^ ((-1) ^ i). When C and D have the same chain spaces and
bases, the differential of C is the conjugate A⁻¹ d A of the differential d of D by the
degreewise automorphisms Aᵢ, and this is the formula
τ(A⁻¹ d A) = τ(d) · ∏ᵢ det (Aᵢ) ^ ((-1) ^ i).