On the leading coefficients of higher-order Alexander polynomials Takahiro KITAYAMA JSPS research fellow (DC) Graduate School of Mathematical Sciences, the University of Tokyo
0. Introduction M : compact orientable 3 -manifold w/ empty or troidal ∂ ψ : π 1 M ։ � t � ∆ M ,ψ ( t ) = ord H 1 ( M ; Z [ H 1 ( M ) / torsion ]) ∈ Z [ H 1 ( M ) / torsion ] / ± H 1 ( M ) / torsion . [Cochran ′ 04 , Harvey ′ 05 ] ∆ ( n ) M ,ψ ( t ) “ = ord H 1 ( M ; Z [ π 1 M / ( π 1 M ) ( n + 1) ]) ” : Higher-order Alexander polynomial of order n : big indeterminacy deg ∆ ( n ) M ,ψ ∈ Z : Cochran-Harvey invariant of order n Extract another kind of information on ∆ ( n ) M ,ψ .
0. Introduction M : compact orientable 3 -manifold w/ empty or troidal ∂ ψ : π 1 M ։ � t � ∆ M ,ψ ( t ) = ord H 1 ( M ; Z [ H 1 ( M ) / torsion ]) ∈ Z [ H 1 ( M ) / torsion ] / ± H 1 ( M ) / torsion . [Cochran ′ 04 , Harvey ′ 05 ] ∆ ( n ) M ,ψ ( t ) “ = ord H 1 ( M ; Z [ π 1 M / ( π 1 M ) ( n + 1) ]) ” : Higher-order Alexander polynomial of order n : big indeterminacy deg ∆ ( n ) M ,ψ ∈ Z : Cochran-Harvey invariant of order n Extract another kind of information on ∆ ( n ) M ,ψ .
Aim of the talk Today we consider non-commutative Reidemeister torsion τ ρ n ( M ) “associated to π 1 M ։ π 1 M / ( π 1 M ) ( n + 1) ” and introduce “the leading coefficient” c n ( ψ ) . [Friedl ′ 07 ] ∆ ( n ) M ,ψ ∼ τ ρ n ( M ) . τ ρ n ( M ) has smaller indeterminacy than ∆ ( n ) M ,ψ . Aim · Fiberedness obstruction on c n ( ψ ) · Computations for metabelian cases · Difference between c n ( ψ ) and c n + 1 ( ψ )
Aim of the talk Today we consider non-commutative Reidemeister torsion τ ρ n ( M ) “associated to π 1 M ։ π 1 M / ( π 1 M ) ( n + 1) ” and introduce “the leading coefficient” c n ( ψ ) . [Friedl ′ 07 ] ∆ ( n ) M ,ψ ∼ τ ρ n ( M ) . τ ρ n ( M ) has smaller indeterminacy than ∆ ( n ) M ,ψ . Aim · Fiberedness obstruction on c n ( ψ ) · Computations for metabelian cases · Difference between c n ( ψ ) and c n + 1 ( ψ )
Outline Non-commutative Reidemeister torsion 1 The leading coefficient and fiberedness 2 Metabelian examples 3 Monotonicity and realization 4
1. Non-commutative Reidemeister torsion F : skew field det: GL ( n , F ) → F × ab (: = F × / [ F × , F × ]) ρ : Z [ π 1 M ] → F : homomorphism s. t. ∗ ( M ; F ) (: = H ∗ ( C ∗ ( � M ) ⊗ Z [ π 1 M ] F )) = 0 H ρ � τ ρ ( M ) ∈ F × ab / ± ρ ( π 1 M ) : Reidemeister torsion Lemma. (Turaev) C i ( � M ) ⊗ Z [ π 1 M ] F = C ′ i ⊕ C ′′ i s. t. (i) C ′ i , C ′′ i are spanned by lifts of cells, and i − 1 ◦ ∂ i : C ′ (ii) pr C ′′ i → C ′′ i − 1 is an isomorphism. � i − 1 ◦ ∂ i ) ( − 1) i . ⇒ τ ρ ( X ) = (det pr C ′′ i
1. Non-commutative Reidemeister torsion F : skew field det: GL ( n , F ) → F × ab (: = F × / [ F × , F × ]) ρ : Z [ π 1 M ] → F : homomorphism s. t. ∗ ( M ; F ) (: = H ∗ ( C ∗ ( � M ) ⊗ Z [ π 1 M ] F )) = 0 H ρ � τ ρ ( M ) ∈ F × ab / ± ρ ( π 1 M ) : Reidemeister torsion Lemma. (Turaev) C i ( � M ) ⊗ Z [ π 1 M ] F = C ′ i ⊕ C ′′ i s. t. (i) C ′ i , C ′′ i are spanned by lifts of cells, and i − 1 ◦ ∂ i : C ′ (ii) pr C ′′ i → C ′′ i − 1 is an isomorphism. � i − 1 ◦ ∂ i ) ( − 1) i . ⇒ τ ρ ( X ) = (det pr C ′′ i
Rational derived series π : group � π (0) : = π, r π ( n + 1) : = { γ ∈ π ( n ) ; ∃ k ∈ Z \ 0 s. t. γ k ∈ [ π ( n ) r , π ( n ) r ] } . r r : rational derived series π ( n ) r /π ( n + 1) : torsion free abelian r � π/π ( n + 1) : poly-torsion-free-abelian r 1 ⊳ π ( n ) r /π ( n + 1) ⊳ · · · ⊳ π (1) r /π ( n + 1) ⊳ π/π ( n + 1) . r r r Theorem. (Passman) Γ : poly-torsion-free-abelian ⇒ the quotient skew field Q ( Γ ) : = Z [ Γ ]( Z [ Γ ] \ 0) − 1 is defined. So we have ρ n : Z [ π 1 M ] → Q ( π 1 M / ( π 1 M ) ( n + 1) ) . r
Rational derived series π : group � π (0) : = π, r π ( n + 1) : = { γ ∈ π ( n ) ; ∃ k ∈ Z \ 0 s. t. γ k ∈ [ π ( n ) r , π ( n ) r ] } . r r : rational derived series π ( n ) r /π ( n + 1) : torsion free abelian r � π/π ( n + 1) : poly-torsion-free-abelian r 1 ⊳ π ( n ) r /π ( n + 1) ⊳ · · · ⊳ π (1) r /π ( n + 1) ⊳ π/π ( n + 1) . r r r Theorem. (Passman) Γ : poly-torsion-free-abelian ⇒ the quotient skew field Q ( Γ ) : = Z [ Γ ]( Z [ Γ ] \ 0) − 1 is defined. So we have ρ n : Z [ π 1 M ] → Q ( π 1 M / ( π 1 M ) ( n + 1) ) . r
The degree ψ : π 1 M ։ � t � Γ n : = π 1 M / ( π 1 M ) ( n + 1) r n : = Ker ψ/ ( π 1 M ) ( n + 1) Γ ′ r θ ∈ Aut( Γ ′ n ) n ⋊ θ � t � , Γ n = Γ ′ � Q ( Γ n ) = Q ( Γ ′ n )( t ) ( t · x = θ ( x ) · t ) ∗ ( M ; Q ( Γ ′ n )( t )) = 0 , then τ ρ n ( M ) ∈ Q ( Γ ′ n )( t ) × If H ρ n n · � t � and the ab / ± Γ ′ degree are defined. δ n ( ψ ) : = deg τ ρ n ( M ) ∈ Z ∼ deg ∆ ( n ) M ,ψ
The degree ψ : π 1 M ։ � t � Γ n : = π 1 M / ( π 1 M ) ( n + 1) r n : = Ker ψ/ ( π 1 M ) ( n + 1) Γ ′ r θ ∈ Aut( Γ ′ n ) n ⋊ θ � t � , Γ n = Γ ′ � Q ( Γ n ) = Q ( Γ ′ n )( t ) ( t · x = θ ( x ) · t ) ∗ ( M ; Q ( Γ ′ n )( t )) = 0 , then τ ρ n ( M ) ∈ Q ( Γ ′ n )( t ) × If H ρ n n · � t � and the ab / ± Γ ′ degree are defined. δ n ( ψ ) : = deg τ ρ n ( M ) ∈ Z ∼ deg ∆ ( n ) M ,ψ
2. The leading coefficient and fiberedness For a fibered knot K , ∆ K is a monic polynomial with degree 2 g ( K ) . Theorem. (Cochran, Friedl, Harvey) If M is fibered, M � S 1 × S 2 , S 1 × D 2 , and ψ : π 1 M ։ � t � ( ∈ H 1 ( M ; Z )) is induced by the fibration, then for all n , δ n ( ψ ) = || ψ || T . How about a generalization for monicness? What is the leading coefficient of τ ρ n ( M ) ∈ Q ( Γ ′ n )( t ) × n · � t � ? ab / ± Γ ′
2. The leading coefficient and fiberedness For a fibered knot K , ∆ K is a monic polynomial with degree 2 g ( K ) . Theorem. (Cochran, Friedl, Harvey) If M is fibered, M � S 1 × S 2 , S 1 × D 2 , and ψ : π 1 M ։ � t � ( ∈ H 1 ( M ; Z )) is induced by the fibration, then for all n , δ n ( ψ ) = || ψ || T . How about a generalization for monicness? What is the leading coefficient of τ ρ n ( M ) ∈ Q ( Γ ′ n )( t ) × n · � t � ? ab / ± Γ ′
Key homomorphism n · � p − 1 θ ( p ) � p ∈ Z [ Γ ′ c : Q ( Γ ′ n )( t ) × n · � t � → Q ( Γ ′ n ) × ab / ± Γ ′ ab / ± Γ ′ n ] \ 0 : ( a l t l + a l − 1 t l − 1 + . . . )( b m t m + b m − 1 t m − 1 + . . . ) − 1 �→ a l b − 1 m Lemma. The map c is a well-defined homomorphism. Definition. ∗ ( M ; Q ( Γ ′ n )( t )) = 0 , then we set If H ρ n n · � p − 1 θ ( p ) � p ∈ Z [ Γ ′ c n ( ψ ) : = c ( τ ρ n ( M )) ∈ Q ( Γ ′ n ) × ab / ± Γ ′ n ] \ 0 .
Key homomorphism n · � p − 1 θ ( p ) � p ∈ Z [ Γ ′ c : Q ( Γ ′ n )( t ) × n · � t � → Q ( Γ ′ n ) × ab / ± Γ ′ ab / ± Γ ′ n ] \ 0 : ( a l t l + a l − 1 t l − 1 + . . . )( b m t m + b m − 1 t m − 1 + . . . ) − 1 �→ a l b − 1 m Lemma. The map c is a well-defined homomorphism. Definition. ∗ ( M ; Q ( Γ ′ n )( t )) = 0 , then we set If H ρ n n · � p − 1 θ ( p ) � p ∈ Z [ Γ ′ c n ( ψ ) : = c ( τ ρ n ( M )) ∈ Q ( Γ ′ n ) × ab / ± Γ ′ n ] \ 0 .
Fiberedness obstruction Theorem. If M is fibered and ψ : π 1 M ։ � t � ( ∈ H 1 ( M ; Z )) is induced by the fibration, then for all n , c n ( ψ ) = 1 . Problem. If c n ( ψ ) = 1 for all n and δ 0 ( ψ ) = || ψ || T , then is M fibered? For what class of 3 -manifolds is this true? cf. [Friedl-Vidussi ′ 08 ] Twisted Alexander polynomials associated to representations onto finite groups detect fiberedness of M .
Fiberedness obstruction Theorem. If M is fibered and ψ : π 1 M ։ � t � ( ∈ H 1 ( M ; Z )) is induced by the fibration, then for all n , c n ( ψ ) = 1 . Problem. If c n ( ψ ) = 1 for all n and δ 0 ( ψ ) = || ψ || T , then is M fibered? For what class of 3 -manifolds is this true? cf. [Friedl-Vidussi ′ 08 ] Twisted Alexander polynomials associated to representations onto finite groups detect fiberedness of M .
3. Metabelian examples K ⊂ S 3 : tame knot with monic ∆ K M = E : exterior ψ : π 1 E ։ � t � : abelianization Γ 1 = π 1 E / ( π 1 E ) (2) 1 = ( π 1 E ) (1) / ( π 1 E ) (2) � Z d , d : = deg ∆ K Γ ′ Z [ Γ ′ 1 ] = Z [ s ± 1 1 , . . . , s ± 1 d ] : UFD d ] : prime / ±� s 1 , . . . s d �·� θ ( p ) Q ( Γ ′ 1 ) × / ± Γ ′ 1 ·� p − 1 θ ( p ) � p ∈ Z [ Γ ′ 1 ] \ 0 = � p � p ∈ Z [ s ± 1 p � 1 ,..., s ± 1 � Z ∞ (if d > 0 ) Using prime decomposition, we can determine an element in Q ( Γ ′ 1 ) × / ± Γ ′ 1 · � p − 1 θ ( p ) � p ∈ Z [ Γ ′ 1 ] \ 0 is 1 or not.
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