Reidemeister torsion

A polynomial invariant of a homology 3-sphere defined by Reidemeister torsion

Teruaki Kitano

Soka Univ.

26th Jan., 2017

Contents

Johnson theory

Reidemeister torsion Casson invariant

Polynomial defined by Reidemeister torsion Brieskorn homology sphere

1/n-surgery along the figure-eight knot SL(2;C)-Casson invariant

Witten’s topolgical field theory

Reidemeister torsion

1930’s Reidemeister, Franz, de Rham: Classification of lens space

1961-2 Milnor:

Hauptvermutung for polyhedrons

Alexander polynomial=Reidemeister torsion 1980’s asymptotic behavior of quantum invariants 1990’s twisted Alexander polynomial=Reidemeister torsion

1996 Meng-Taubes: SW=Milnor torsion

Casson invariant and Reidemeister torsion

1985 Casson, an integral lift of Rochlin invariant Late 1980’s D. Johnson, A geometric form of Casson’s invariant and its connection to Reidemeister torsion, unpublished lecture notes (hand written).

A generalization of the determinant

Roughly speaking,

Reidemeister torsion= ”det”(an acyclic based chain complex).

acyclic: a linear isomorphism

based: a matrix is given under a basis

determinant for a linear isomorphism

V: l-dimensional vector space over a field F(=C,R).

f :V →V: a linear isomorphism How to define det(f) ?

e1, . . . ,el: a basis of V

F ∈M(l;F): the matrix defined by f undere₁, . . . ,e_l Fore₁, . . . ,e_l, we take their image f(e₁), . . . ,f(e_l).

det(f,(e1, . . . ,el)) = det(F)

= det (f(e₁), . . . ,f(e_l)/e₁, . . . ,e_l).

Under fixing a basis ofV, e₁, . . . ,e_l, we have an identification

∧^lV ∼=F.

f(e₁)∧· · ·∧f(e_l) = det (f(e₁), . . . ,f(e_l)/e₁, . . . ,e_l)e₁∧· · ·∧e_l

determinant for a short exact sequence

0→V₁ →^f1 V₂ →^f2 V₃ →0: an exact sequence of finite dimensional vector spaces overF.

b_i ⊂V_i: a basis of V_i for i = 1,2,3 f₂⁻¹(b₃)⊂V₂: a lift of b₂

(f1(b1),f₂⁻¹(b3)): another basis of V2

τ = det(

f₁(b₁),f₂⁻¹(b₃)/b₂) Remark

τ does not depend on a choice of a lift f₂⁻¹(b₃).

Reidemeister torsion for a chain complex

F: a field

a chain complex of finite dimensional vector spaces overF C_∗:

0−→^∂m+1 C_m −→^∂m C_m−1 ^∂−→^m−1. . .−→^∂2 C₁ −→^∂1 C₀ −→^∂0 0 c_i: a basis of C_i

Definition

A chain complexC_∗ isacyclic if

Im∂_i+1 =Ker∂_i (Z_i(C_∗) =B_i(C_∗),H_i(C_∗) ={0})

Assumption: C_∗ is acyclic.

Fix bi for Bi(C_∗) =Zi(C_∗).

Exact sequence

0−→Z_i(C_∗)−→C_i −→^∂i B_i−1(C_∗)−→0 ebi−1: a lift of bi−1 in Ci

(b_i,eb_i−1): another basis ofC_i (

b_i,eb_i−1/c_i )

: the transformation matrix from c_i to (bi,ebi−1)

[b_i,eb_i−1/c_i] = det (

b_i,eb_i−1/c_i )

:

Torsion of a based chain complex

Definition

The torsionτ(C_∗) of a based chain complex C_∗ with c_∗ τ(C_∗) = Π_i:odd[b_i,eb_i−1/c_i]

Π_i:even[b_i,eb_i−1/c_i] ∈F\ {0}. Lemma

τ(C_∗) does not depend on choices of lifts {b_i}.

Lemma

τ(C_∗) does not depend on choices of {b_i}. Proof.

Assumeb^′_q is another basis ofB_q.

In the definition of τ(C_∗), the difference between b_q and b^′_q is related to the followings only two parts:

[b^′_q,b_q−1/c_q]

= [b_q,b_q−1/c_q][

b^′_q/b_q] [b_q+1,b^′_q/c_q+1] = [b_q+1,b_q/c_q+1][

b^′_q/b_q] Since[

b^′_q/b_q]

appears in the both of the denominator and the numerator of the definition, they are cancelled.

Reidemeister torsion for a CW-complexe

X:a finite(=compact) CW-complex X˜ →X:universal covering

π1(X) acts on ˜X (from the right), cellularly.

C_i( ˜X;Z):spanned by lifts of i-cells of X overZ.

C_∗( ˜X;Z): a chain complex of right Z[π₁(X)]-modules．

local coefficients

ρ:π1(X)→GL(V): a finite dimensional linear representation over F

V:l-dimensional vector space over F (=C²,sl(2;C),su(2))

C_∗(X;V_ρ) =C_∗( ˜X;Z)⊗π1(X)V_ρ Assume: C_∗(X;V_φ) is acyclic.

Definition

τ_ρ(X) = τ(C_∗(X;V_ρ))∈F\ {0}.

well-definedness

Choices of lifts of cells Orders of cells． Choice of a basis for V Proposition

τρ(X)∈F/±det(ρ(π1(X))) does not depend on choices of lifts and orders of cells.

It depends on a choice of basis e₁, . . . ,e_l. Then we fix e₁ ∧ · · · ∧e_l ∈∧l

V．

topological invariant

Under same assumption, we have the following.

Theorem (Kirby-Siebenmann, Chapman)

If X is a closed manifold, thenτ_ρ(M)∈F/±det(ρ(π₁(X))) is a topological invariant of M．

Remark

For any acyclic even-dimensional unimodular representation ρ:π₁(X)→SL(2l;F)，τ_ρ(X)∈F\ {0} is well defined．

Remark

In the case of V =C², we take e1 =

(1 0

) ,e2 =

(0 1

) .

In the case of V =su(2), under a fixed Killing form, we take an orthonormal basis.

If ρ is not acyclic, then we defineτ_ρ(X) = 0.

Definition

We callρ an acyclic representation ifC_∗(X;V_φ) is acyclic．

Condition for acyclicity

If C_∗(X;V_φ) is acyclic, then Euler number of H_∗(X;V_φ) is 0 =∑

i

(−1)ⁱdimH_i(X;V_ρ)

=∑

i

(−1)ⁱdimCi(X;Vρ)

=∑

i

(−1)ⁱdimC_i(X;Z)·dimV

=lχ(X).

Hereχ(X) is Euler characteristic of X. Hence we have χ(X) = 0.

There does not exist an acyclic representation for X with χ(X)̸= 0.

Johnson Theory

M : homology 3-sphere λ(M)∈Z : Casson invariant

R(X) ={ρ:π₁X →SU(2);irreducible representation} Rˆ(X) =R(X)/SU(2)

M =A∪Σg B : Heegard decomposition Rˆ(M) = ˆR(A)∩Rˆ(B)⊂Rˆ(Σ_g)

Assume:

Rˆ(M) is a finite set.

Any point of ˆR(M) isnon degenerate.

For anyρ∈ R(M), compose it with

Adj :SU(2)→Aut(su(2)), we have

Adj ◦ρ:π₁(M)→Aut(su(2)).

dim_RH¹(Σ_g;su(2)_Adj◦ρ) = 6g −6 dim_RH¹(A;su(2)_Adj◦ρ) = 3g −3 dim_RH¹(B;su(2)_Adj◦ρ) = 3g−3 H¹(M;su(2)Adj◦ρ) = {0}

volume forms

Johnson constructed volume forms vol_Σg onH¹(Σ_g;su(2)_Adj◦ρ), vol_A on H¹(A;su(2)_Adj◦ρ), vol_B on H¹(B;su(2)_Adj◦ρ).

By the assumption, he definedt_[ρ]∈R\ {0} by

volΣg =t_[ρ](volA ∧volB)∈

6g∧−6

H¹(Σg;su(2)Adj◦ρ).

Geometric form of Casson invariant

Theorem (Johnson)

λ(M) = ∑

[ρ]∈Rˆ(M)

sgn(t_[ρ]) t_[ρ] =τ_Adj◦ρ(M)

Remark

Adj ◦ρ :π₁M →SU(2) →Aut(su(2)) τAdj◦ρ(M) =τ(C_∗(M;su(2)Adj◦ρ)) Ageometric form of Casson invariant:

∑

[ρ]∈Rˆ(M)

t_[ρ]

A polynomial invariant defined by Reidemeister torsion

Johnson proposed to study

a polynomial whose zeros are {t_[ρ]}:

∏

[ρ]

(t −t_[ρ]),

a polynomial for Reidemeister torsion τ_ρ(M) for an SL(2;C)-representation.

In particular he studied ∏

[ρ]

(t −τρ) for 1/n-surgeried manifold along the trefoil knot.

A polynomial defined by Reidemeister torsion

M : homology 3-sphere

τρ(M)∈C : Reidemeister torsion forM with an irreducibleSL(2;C)-representation ρ

τ_ρ^′(M) = 1/τ_ρ(M)

Assume : ˆRSL(2;C)(M) is afinite set.

By assumption, {τ_ρ(M) | τ_ρ(M)̸= 0} is a finite set.

Definition

σ_M(t)∈C[t] is a polynomial whose roots are given by the set {

τ_ρ^′(M) = _τ ¹

ρ(M) | τ_ρ(M)̸= 0

} .

Assumption : ˆRSL(2;C)(M) is an algebraic variety of dimension zero.

By using resultants, it is given as a set of algebraic numbers.

Any representation ρ in ˆRSL(2;C)(M) can be realized over an algebraic extension fieldF over Q.

τ_ρ(M) and τ_ρ^′(M) are also algebraic numbers inF. Any Galois conjugate ofρ (under the action of Gal(F/Q)) is a representation.

If the action of Gal(F/Q) gives all Galois conjugate of τ_ρ^′ by Gal(Q({τ_ρ^′})/Q), then

σ_M(t)∈Q[t].

It is the minimal polynomial ofτ_ρ^′ ∈Q(τ_ρ^′).

minimal polynomial

F ⊃Q is an extension ofQ. θ∈F

Definition

The minimal polynomial ofθ is the monic polynomial of least degree among all polynomials inQ[x] having θ as a root.

Remark

The minimal polynomial ofθ exists when θ is algebraic over Q, that is, f(θ) = 0 for some non-zero polynomial f(x)∈Q[x].

Brieskorn homology 3-sphere

Σ(p,q,r) : Brieskorn homology 3-sphere {z₁^p+z₂^q+z₃^r = 0} ∩S⁵ ⊂C³ =R⁶ p,q,r ∈Z : pairwise coprime positive integers We may assumeq and r are odd numbers.

Σ(p,q,r) can be given by Dehn surgery along (p,q)-torus knot

Σ(p, q, | pqn + 1 | )

Now consider the case of r =|pqn+ 1|. Then

Σ(p,q,|pqn+ 1|) can be obtained by 1/n-surgery along T(p,q).

In this case any conjugacy class of irreducible representations can be represented by

ρ_(a,b,k) :π₁(Σ(p,q,|pqn+ 1|))→SL(2;C) with

1 0<a <p,0<b <q,a ≡b mod 2,

2 0<k <|pqn+ 1|,k ≡na mod 2,

3 tr(ρ_(a,b,k)(x)) = 2 cos^aπ_p ,

4 tr(ρ_(a,b,k)(y)) = 2 cos^bπ_q ,

5 tr(ρ_(a,b,k)(µ)) = 2 cos_|pqn+1^kπ _|.

τ

(Σ(p , q , | pqn + 1 | ))

Johnson computed Reidemeister torsion τ_ρ(a,b,k)(Σ(p,q,|pqn+ 1|)) as follows.

Theorem (Johnson)

1 A representation ρ_(a,b,k) is acyclic if and only if a≡b ≡1,k ≡n mod 2.

2 For an acyclic representation ρ_(a,b,k), τ_ρ(a,b,k)(Σ(p,q,|pqn+ 1|))

= 1

2 (

1−cos ^aπ_p ) (

1−cos^bπ_q ) (

1 + cos_|pqn+1^pqkπ_| ).

τ

(Σ(p , q , r ))

In general case, we have the following.

Proposition (Johnosn, Kitano-Tran)

τ_ρ(a,b,c)(Σ(p,q,r)) = 1

2 (

1−cos^aπ_p ) (

1−cos^bπ_q ) (

1−cos^cπ_r )

= 1

16 sin² (πa

2p

) sin²

(πb 2q

)

sin²(_πc

2r

)

where(0,0,0)<(a,b,c)<(p,q,r), a≡b≡c ≡1mod 2.

σ

(t )

From hereσ_Σ(p,q,r)(t) is simply written as σ_(p,q,r)(t).

σ_(p,q,r)(t) = C ∏

(a,b,c)

(

t −16 sin² (πa

2p )

sin² (πb

2q )

sin² (πc

2r ))

where the above product can be taken over the same condition (0,0,0)<(a,b,c)<(p,q,r), a ≡b ≡c ≡1 mod 2.

Now put

C_(p,q,a,b)= 2 sin (πa

2p )

sin (πb

2q )

.

Normalized Chebyshev polynomial of first kind

Normalized Chebyshev polynomial of the first kind:

T_n(2cosθ) =2cosnθ.

Recursive definition:

T0(x) = 2, T1(x) =x.

T_n+1(x) =xT_n(x)−T_n−1(x).

T0(x) = 2,T1(x) =x,T2(x) = x²−2,T3(x) = x³−3x.

Normalized Chebyshev polynomial of second kind

Normalized Chebyshev polynomial of the second kind:

S_n(2cosθ) = sin((n+ 1)θ) sinθ . Recursive definition:

S0(x) = 1, S1(x) = x.

S_n+1(x) =xS_n(x)−S_n−1(x).

S₀(x) = 1,S₁(x) =x,S₂(x) =x²−1,S₃(x) =x³−2x. A Relation:

−

Polynomial of Σ(p , q , r )

Theorem (Kitano-Tran)

σ_(p,q,r)(t) = ∏

(a,c)

S_q−1

( √ t C_(p,r,a,c)

)

= ∏

(a,b)

S_r−1

( √ t C_(p,q,a,b)

)

=













∏

(b,c)

Sp−1

( √ t C_(q,r,b,c)

)

(p :odd)

∏

(b,c)

S¯_2p−1

( √ t C_(q,r,b,c)

)

(p :even) .

HereS¯_2p−1(x) =S_2p−1(x)/x .

Examples

σ_(2,3,5)(t) =4t² −6t + 1

σ_(2,3,7)(t) =8t³ −20t²+ 12t −1

σ_(2,3,11)(t) =32t⁵−144t⁴+ 224t³−140t²+ 30t −1

σ_(2,3,13)(t) =64t⁶−352t⁵+ 720t⁴−672t³+ 280t²−42t + 1 σ(2,3,17)(t) =256t⁸−1920t⁷+ 5824t⁶−9152t⁵+ 7920t⁴

−3693t³+ 840t²−72t + 1

σ_(2,3,19)(t) =512t⁹−4352t⁸+ 15360t⁷−29120t⁶+ 32032t⁵

−20592t⁴ + 7392t³−1320t²+ 90t −1 They are irreducible over Q.

Remark

Example (reducible example)

σ_(2,5,9)(T) =256t⁸ −2688t⁷+ 9856t⁶−15840t⁵ +12192t⁴−4608t³+ 820t²−60t + 1 This is not irreducible as

σ_(2,5,9)(T)

=(4t²−6t+ 1)

×(64t⁶−576t⁵+ 1584t⁴−1440t³+ 492t²−54t + 1)

=σ_(2,3,5)(T)

×(64t⁶−576t⁵+ 1584t⁴−1440t³+ 492t²−54t + 1)

SL(2; C )-Casson invariant

ForM = Σ(2,3,6n+ 1)(n>0),

♯RˆSL(2;C)(Σ(2,3,6n+ 1)) =λ_SL(2;C)(Σ(2,3,6n+ 1))

=3n (Boden−Curtis) On the other hand,

λ_SU(2)(Σ(2,3,6n+ 1)) =n.

λ_SL(2;C)(Σ(2,3,6n+ 1))−2λ_SU(2)(Σ(2,3,6n+ 1)) =n What is the difference ?

♯RˆSL(2;R)(Σ(2,3,6n+ 1)) =n

Proposition (K.-Yamaguchi)

λ_SL(2;C)(Σ(p,q,r)) = 2λ_SU(2)(Σ(p,q,r))+♯RˆSL(2;R)(Σ(p,q,r)) ρ:π₁(Σ(p,q,r))→SL(2;C) induces

¯

ρ:π₁^orbS²(p,q,r)→PSL(2;C).

π^orb₁ S²(p,q,r) =⟨ x,y,z | x^p =y^q =z^r =xyz = 1 ⟩ Jenkins-Neumann:

Classification of PSL(2;R)-representations of triangle groups

AnySU(2)-representation that is conjugate to SL(2;R)-representation is an abelian representation.

”SL(2; R )-Casson invariant”

Problem

Can we define an SL(2;R)-Casson invariant ?

Dehn surgery along the figure-eight knot

K = 4₁

π₁(S³\K) =⟨a,b|w⁻¹a =bw⁻¹⟩wherew =ba⁻¹b⁻¹a.

M_n(K): 1/n-Dehn surgery along K．

ρ:π₁(M_n(K))→SL(2,C): irreducible representation.

µ=a∈π₁(M_n(K)): meridian of K x =tr(ρ(µ))

Reidemeister torsion for M

(K )

Reidemeister torsion ofM_n(K) for ρ is given as follows．

Theorem (K.)

τ_ρ(M_n(K)) =− 2(x−1) x⁴−9x²+ 4 Here x =tr(ρ(µ)).

Remark

The explicit value x is determined by the surgery condition． Remark that any complex value x can not be realized.

Letρ:π₁(S³\K)→SL(2;C) be an irreducible representaion.

By taking a conjugation, we may assume ρ(a) =

[ s 1 0 s⁻¹

]

, ρ(b) =

[ s 0

−u s⁻¹ ]

.

Here (s,u)∈(C^∗)² is a solution of

ϕ_K(s,u) =u²−(u+ 1)(s²+s⁻²−3).

Compute the image of longitude λ= (ab⁻¹a⁻¹b)⁻¹)(ba⁻¹b⁻¹a⁻¹)⁻¹,

ρ(λ) =

[ λ₁₁ λ₁₂ λ₂₁ λ₂₂

]

where

λ11= (−s⁻²−1)u³+ (s²−s⁻²)u+ (s⁻⁴+s²−s⁻²−1)u²+ 1, λ₁₂=u(s +s⁻¹)(s²+s⁻²−1−u),

λ₂₁=−u²(s+s⁻¹)ϕ_K(s,u),

λ₂₂= (−s²−1)u³+ (s⁴−s²+s⁻²−1)u²+ (s⁻²−s²)u+ 1.

The conditionϕ_K(s,u) = 0 implies

trρ(λ) =s⁴+s⁻⁴−s²−s⁻²−2

=x⁴−5x²+ 2 wherex =s+s⁻¹.

Here this representation π₁(S³\K)→SL(2;C) can be extend as ρ:π₁(M_n(K))→SL(2;C) if and only if

ρ(λⁿ) = ρ(µ⁻¹).

By using L=trρ(λ) =x⁴−5x²+ 2, ρ(λⁿ) =

[ λⁿ₁₁ λ₁₂S_n−1(L) 0 λⁿ₂₂

] .

Further the condition ρ(λⁿ) = ρ(µ⁻¹) is equivalent to the following three conditions of

1 λⁿ₁₁=s⁻¹,

2 λⁿ₂₂=s,

3 λ₁₂S_n−1(L) = −1.

By direct computation, we have

1 u² −(u+ 1)(s²+s⁻²−3) = 0,

2 λⁿ₁₁=s⁻¹,

3 L̸=−2 (i.e. x² ̸= 4).

Lemma

The set of s satisfying the following

1 u² −(u+ 1)(s²+s⁻²−3) = 0,

2 λⁿ₁₁=s⁻¹

is coincided wth the set of solutions for

s +s⁻¹ =T_n(s⁴+s⁻⁴−s²−s⁻²−2).

By the formula of Reideimeister torsion τ_ρ = 2x −2

x²(x²−5), we consider the resultant

Pn(t) = Resx

(Tn(x⁴−5x²+ 2)−x,tx²(x²−5)−(2x −2)) .

A resultant

f(x) =anxⁿ+an−1xⁿ⁻¹ +· · ·+a1x +a0

g(x) =b_mx^m+b_m−1x^m−1+· · ·+b₁x +b₀ Definition (the resultant off(x),g(x))

Res(f(x),g(x)) =

an an−1 · · · · · · a0

a_n a_n−1 · · · a₁ a₀

a_n · · ·

...

a_n · · · a₁ a₀

b_m b_m−1 · · · · · · b₀

bm bm−1 · · · b1 b0

b_m · · ·

...

bm · · · b1 b0

f(x,y),g(x,y):polynomials of 2-variable x,y By considering x is a variable andy is a constant, we can define the resultant Res_x(f,g).

Proposition

There exist polynomials F(x,y),G(x,y)such that f(x,y)G(x,y) +g(x,y)F(x,y) = Res_x(f,g).

Therefore a solution (x,y) off(x,y) =g(x,y) = 0 satisfies Res₍f,g) = 0.

Polynomial for surgeried manifold along 4

M_n(K) is a homology 3-sphere obtained by 1/n-surgery along K = 4₁.

Theorem (Kitano-Tran) σ_Mn(K)(t) =

{

P_n(t)/(2t−3) n is odd, P_n(t)/(2t+ 1) n is even.

Here

P_n(t) =Res_x(

T_n(x⁴−5x²+ 2)−x,tx²(x²−5)−(2x −2))

Examples

σ₁(t) =t³−12t²+ 20t −8

σ₂(t) =t⁷−56t⁶+ 660t⁵−3384t⁴+ 8720t³−11008t² + 5376t −128

σ₃(t) =t¹¹−124t¹⁰+ 3036t⁹−31696t⁸+ 161024t⁷

−364128t⁶ + 152640t⁵+ 426752t⁴−262144t³

−142336t² + 55296t −2048

σ₄(t) =t¹⁵−224t¹⁴+ 10320t¹³−211776t¹²+ 2296400t¹¹

−13570900t¹⁰+ 41172200t⁹−49672100t⁸−35529500t⁷ +156351000t⁶−113653000t⁵−58957800t⁴+ 115933000t³

−50004000t² + 5898240t−32768

Open Problems

Problem

How strong is this polynomial σ_M(t) ?

How about for Brieskorn homology sphere ?

σ

(t ) is a minimal polynomial?

Problem

How is it related with minimal poly ofτ_ρ^′ overQ?

Problem

If there exists an epimorphismφ:π₁(M)→π₁(M^′), then σ_M(t)can be divided by σ_M′(t)?

Problem

Does σ_M(t) know♯RˆSL(2;C)(M) and SL(2;C)-Casson invariant?

Does τ_ρ^′ with the action of a Galois group know

♯RˆSL(2;C)(M) and SL(2;C)-Casson invariant?

General case

Problem

How can we treat it in the case of ♯Rˆ(M) = ∞? Only consider the componets of dimension 0? How to take a perturbation ?

How to treat splicing manifolds ?

In the original definition of the Casson invariant, we need to take a perturbation to get a transverse intersection. The problem is that a representation after a perturbation does not give a representation of π₁(M).

Problem

Can we construct λ_SL(2;R)(M)for (some class of) homology 3-spheres satisfying

λ_SL(2;C)(Σ(p,q,r)) = 2λ(Σ(p,q,r)) +λ_SL(2;R)(Σ(p,q,r))?

Proposition

There exists a hyperbolic homology 3-sphere without an irreducible SL(2;R)-representation.

SL(2; R )-representations

M_n(K): 1/n-surgeried along the figure-eight knot Proposition

λ_SL(2;C)(M_n(K)) = 4n−1, λ(M_n(K)) = −n

λSL(2;C)(Mn(K))−2|λ(Mn(K))|= 2n−1 Checked by Mathematica:

Example (n=2,. . . ,6,8)

λ_SL(2;C)(Mn(K))−2|λ(Mn(K))|> ♯RˆSL(2;R)(Mn(K))>0 Example (n= 7,9)

TQFT

Problem

How can we construct TQFT for Johnson theory?

Witten’s theory

Description from the view point of topological quantum field theory: E. Witten, Topology-changing amplitudes in

(2 + 1)-dimensional gravity, Nuclear Phys. B 323(1989).