Non-commutative Reidemeister torsion and Morse-Novikov theory
aa r X i v : . [ m a t h . G T ] J u l NON-COMMUTATIVE REIDEMEISTER TORSION AND MORSE-NOVIKOVTHEORY
TAKAHIRO KITAYAMAA bstract . Given a circle-valued Morse function of a closed oriented manifold, we prove thatReidemeister torsion over a non-commutative formal Laurent polynomial ring equals the prod-uct of a certain non-commutative Lefschetz-type zeta function and the algebraic torsion of theNovikov complex over the ring. This paper gives a generalization of the result of Hutchingsand Lee on abelian coe ffi cients to the case of skew fields. As a consequence we obtain a Morsetheoretical and dynamical description of the higher-order Reidemeister torsion.
1. I ntroduction
In this paper let X be a closed connected oriented Riemannian d -manifold with χ ( X ) = f : X → S a Morse function such that the stable and unstable manifolds of the critical points of f transversely intersect and the closed orbits of flows of ∇ f are all nondegenerate. (See Section2.2 and 3.1.)For a generic closed 1-form, for instance d f , we can define the Lefschetz-type zeta functionwhich counts closed orbits of flows induced by the 1-form. In [7], [8], [9] Hutchings and Leeshowed that the product of the zeta function and the algebraic torsion of the abelian Novikovcomplex associated to the 1-form is a topological invariant and is equal to the abelian Reide-meister torsion of X . In [19], [20] Pazhitnov also proved a similar theorem in terms of thetorsion of a canonical chain homotopy equivalence map between the abelian Novikov complexand the completed simplicial chain complex of the maximal abelian covering of X . In the casewhere X is a fiber bundle over a circle and f is the projection these results give Milnor’s theo-rem in [13], which claims that the Lefschetz zeta function of a self map is equal to the abelianReidemeister torsion of the mapping torus of the map.In fixed point theory there is a non-commutative substitute for the Lefschetz zeta functionwhich is called the total Lefschetz-Nielsen invariant, and in [3] Geoghegan and Nicas showedthat the invariant has similar properties to these of torsion and determines the Reidemeistertraces of iterates of a self map. In [16] Pajitnov considered the eta function associated to −∇ f which lies in a suitable quotient of the Novikov ring of π X and whose abelianization coincideswith the logarithm of the Lefschetz-type zeta function. He also proved a formula expressing theeta function in terms of the torsion of a chain homotopy equivalence map between the Novikovcomplex and the completed simplicial chain complex of the universal covering of X . Theseworks were generalized to the case of generic closed 1-forms by Sch¨utz in [21] and [22].Non-commutative Alexander polynomials which are called the higher-order Alexander poly-nomials were introduced, in particular for 3-manifolds, by Cochran in [1] and Harvey in [6],and are known by Friedl in [2] to be essentially equal to Reidemeister torsion over certain skew Mathematics Subject Classification.
Primary 57Q10, Secondary 57R70.
Key words and phrases.
Reidemeister torsion, Morse-Novikov complex, derived series. fields. We call it higher-order Reidemeister torsion. The aim of this paper is to give a gener-alization of Hutchings and Lee’s theorem to the case where the coe ffi cients are skew fields byusing Dieudonn´e determinant and to obtain a Morse theoretical and dynamical description ofhigher-order Reidemeister torsion. Note that it is known by Goda and Pajitnov in [4] that thetorsion of a chain homotopy equivalence between the twisted Novikov complex and the twistedsimplicial complex by a linear representation equals the twisted Lefschetz zeta function whichwas introduced by Jiang and Wang in [10]. This work is closely related to twisted Alexanderpolynomials which were introduced first by Lin in [11] and later generally by Wada in [26].Our objects and approach considered here are di ff erent from theirs.Let Λ f be the Novikov completion of Z [ π X ] associated to f ∗ : π X → π S . We first con-sider a certain quotient group ( Λ × f ) ab of the abelianization of the unit group Λ × f and introducenon-commutative Lefschetz-type zeta function ζ f ∈ ( Λ × f ) ab of f . Taking a poly-torsion-free-abelian group G and group homomorphisms ρ : π X → G , α : G → π S such that α ◦ ρ = f ∗ ,we construct a certain Novikov-type skew field K θ (( t l )). Similar to ( Λ × f ) ab we define a certainquotient group K θ (( t l )) × ab of the abelianization K θ (( t l )) × ab of K θ (( t l )) × . We can check that ρ nat-urally extends to a ring homomorphism Λ f → K θ (( t l )) and also denote it by ρ . There is anaturally induced homomorphism ρ ∗ : ( Λ × f ) ab → K θ (( t l )) × ab by ρ . If the twisted homology group H ρ ∗ ( X ; K θ (( t l ))) of X associated to ρ vanishes, then we can define the Reidemeister torsion τ ρ ( X )of X associated to ρ and the algebraic torsion τ Nov ρ ( f ) of the Novikov complex over K θ (( t l ))as elements in K θ (( t l )) × ab / ± ρ ( π X ). Here is the main theorem which can be applied for thehigher-order Reidemeister torsion. Theorem 1.1 (Theorem 3.7) . For a given pair ( ρ, α ) as above, if H ρ ∗ ( X ; K θ (( t l ))) = , then τ ρ ( X ) = ρ ∗ ( ζ f ) τ Nov ρ ( f ) ∈ K θ (( t l )) × ab / ± ρ ( π X ) . To prove the theorem we use a similar approach to that of Hutchings and Lee in [9], butwe need more subtle argument because of the non-commutative nature, especially in the secondhalf, which is the heart of the proof. We can check that the non-commutative zeta function ζ f canbe seen as a certain reduction of the eta function associated to −∇ f , and this theorem can alsobe deduced from the results of Pajitnov in [16] by a purely algebraic functoriality argument. In[5, Theorem 5.4] Goda and Sakasai showed another splitting formula for Reidemeister torsionover skew fields.This paper is organized as follows. In the next section we review some of the standard factsof Reidemeister torsion and the Novikov complex of f . In Section 3 we introduce the non-commutative Lefschetz-type zeta function ζ f and construct the skew field K θ (( t l )). There wealso set up notation for higher-order Reidemeister torsion. Section 4 is devoted to the proof ofthe main theorem. 2. P reliminaries Reidemeister torsion.
We begin with the definition of Reidemeister torsion over a skewfield K . See [12] and [24] for more details.For a matrix over K , we mean by an elementary row operation the addition of a left multipleof one row to another row. After elementary row operations we can turn any matrix A ∈ GL k ( K ) ORSION AND MORSE-NOVIKOV THEORY 3 into a diagonal matrix ( d i , j ). Then the Dieudonn´e determinant det A is defined to be [ Q ki = d i , i ] ∈ K × ab : = K × / [ K × , K × ].Let C ∗ = ( C n ∂ n −→ C n − → · · · → C ) be a chain complex of finite dimensional right K -vectorspaces. If we have bases b i of Im ∂ i + and h i of H i ( C ∗ ) for i = , , . . . n , we can take a basis b i h i b i − of C i as follows. Picking a lift of h i in Ker ∂ i and combining it with b i , we first obtaina basis b i h i of C i . Then picking a lift of b i − in C i and combining it with b i h i , we can obtain abasis b i h i b i − of C i . Definition 2.1.
For given bases c = { c i } of C ∗ and h = { h i } of H ∗ ( C ∗ ), we choose a basis { b i } ofIm ∂ ∗ and define τ ( C ∗ , c , h ) : = n Y i = [ b i h i b i − / c i ] ( − i + ∈ K × ab , where [ b i h i b i − / c i ] is the Dieudonn´e determinant of the base change matrix from c i to b i h i b i − .If C ∗ is acyclic, then we write τ ( C ∗ , c ).It can be easily checked that τ ( C ∗ , c , h ) does not depend on the choices of b i and b i h i b i − .Torsion has the following multiplicative property. Let0 → C ′∗ → C ∗ → C ′′∗ → K -vector spacesand let c = { c i } , c ′ = { c ′ i } , c ′′ = { c ′′ i } and h = { h i } , h ′ = { h ′ i } , h ′′ = { h ′′ i } be bases of C ∗ , C ′∗ , C ′′∗ and H ∗ ( C ∗ ) , H ∗ ( C ′∗ ) , H ∗ ( C ′′∗ ). Picking a lift of c ′′ i in C i and combining it with the image of c ′ i in C i , we obtain a basis c ′ i c ′′ i of C i . We denote by H ∗ the corresponding long exact sequence inhomology and by d the basis of H ∗ obtained by combining h , h ′ , h ′′ . Lemma 2.2. ( [12, Theorem 3. 1] ) If [ c ′ i c ′′ i / c i ] = for all i, then τ ( C ∗ , c , h ) = τ ( C ′∗ , c ′ , h ′ ) τ ( C ′′∗ , c ′′ , h ′′ ) τ ( H ∗ , d ) . The following lemma is a certain non-commutative version of [24, Theorem 2.2]. Turaev’sproof can be easily applied to this setting.
Lemma 2.3.
If C ∗ is acyclic and we find a decomposition C ∗ = C ′∗ ⊕ C ′′∗ such that C ′ i and C ′′ i are spanned by subbases of c i and the induced map pr C ′′ i − ◦ ∂ i | C ′ i : C ′ i → C ′′ i − is an isomorphismfor each i, then τ ( C ∗ , c ) = ± n Y i = (det pr C ′′ i − ◦ ∂ | C ′ i ) ( − i . Let ϕ : Z [ π X ] → K be a ring homomorphism. We take a cell decomposition of X and definethe twisted homology group associated to ϕ as follows: H ϕ i ( X ; K ) : = H i ( C ∗ ( e X ) ⊗ Z [ π X ] K ) , where e X is the universal covering of X . Definition 2.4. If H ϕ ∗ ( X ; K ) =
0, then we define the
Reidemeister torsion τ ϕ ( X ) associated to ϕ as follows. We choose a lift ˜ e in e X for each cell e of X . Then τ ϕ ( X ) : = [ τ ( C ∗ ( e X ) ⊗ Z [ π X ] K , h ˜ e ⊗ i e )] ∈ K × ab / ± ϕ ( π X ) . We can check that τ ϕ ( X ) does not depend on the choice of ˜ e . It is known that Reidemeistertorsion is a simple homotopy invariant of a finite CW-complex. T. KITAYAMA
The Novikov complex.
Next we review the Novikov complex of f , which is the simplestversion of Novikov’s construction for closed 1-forms in [14]. See also [17] and [18].We can lift f to a function ˜ f : e X → R . If p is a critical point of f or ˜ f , the unstable manifold D ( p ) is the set of all points x such that the upward gradient flow starting at x converges to p .Similarly, the stable manifold A ( p ) is the set of all points x such that the downward gradientflow starting at x converges to p . Recall that we chose a Riemann metric such that D ( p ) ⋔ A ( p )for any critical points p , q of f .We take the “downward” generator t of π S . Definition 2.5.
We define the
Novikov completion Λ f of Z [ π X ] associated to f ∗ : π X → h t i tobe the set of a formal sum P γ ∈ π X a γ γ such that a γ ∈ Z and for any k ∈ Z , the number of γ suchthat a γ , f ∗ ( γ ) ≤ k is finite. Definition 2.6.
The
Novikov complex ( C Nov ∗ ( f ) , ∂ f ∗ ) of f is defined as follows. For each criticalpoint p of f , we choose a lift ˜ p ∈ e X . Then we define C Novi ( f ) to be the free right Λ f -modulegenerated by the lifts ˜ p of index i . If p is a critical point of index i , then ∂ fi ( ˜ p · γ ) : = X q of index i − ,γ ′ ∈ π X n ( ˜ p · γ, ˜ q · γ ′ ) ˜ q · γ ′ , where n ( ˜ p · γ, ˜ q · γ ′ ) is the algebraic intersection number of D ( ˜ p · γ ), A ( ˜ q · γ ′ ) and an appropriatelevel set, which can be seen as the signed number of negative gradient flow lines from ˜ p · γ to˜ q · γ ′ . By the linear extension we obtain the di ff erential ∂ fi : C Novi ( f ) → C Novi − ( f ).Obviously, the definition dose not depend on the choices of ˜ f and { ˜ p } . It is known thatappropriate orientations of stable and unstable manifolds ensure that ∂ fi − ◦ ∂ fi = Theorem 2.7 ([18]) . The Novikov complex C
Nov ∗ ( f ) is chain homotopic to C ∗ ( e X ) ⊗ Z [ π X ] Λ f .
3. T he main theorem
Non-commutative zeta functions.
First we introduce a non-commutative zeta function ζ f associated to f , which is closely related to the total Lefschetz-Nielsen invariant of a self mapin [3].Let Λ × f be the group of units of Λ f . Namely Λ × f consists of elements of Λ f having left andright inverses. For x , y ∈ ( Λ × f ) ab : = Λ × f / [ Λ × f , Λ × f ], we write x ∼ y if for any k ∈ Z , there exist representatives P γ ∈ π X a γ γ, P γ ∈ π X b γ γ ∈ Λ × f of x , y respectively suchthat for any γ ∈ π X with deg f ∗ ( γ ) ≤ k , a γ = b γ . Lemma 3.1.
The relation ∼ is an equivalence relation in ( Λ × f ) ab .Proof. We only need to show the transitivity. We assume that x ∼ y and y ∼ z for x , y , z ∈ ( Λ × f ) ab and for any k ∈ Z take representatives P γ a γ γ, P γ b γ γ and P γ b ′ γ γ, P γ c ′ γ γ of x , y and y , z respectively such that for any γ with deg f ∗ ( γ ) ≤ k , a γ = b γ and b ′ γ = c ′ γ . There exists λ = P γ d γ γ ∈ [ Λ × f , Λ × f ] such that P γ b γ γ = ( P γ b ′ γ γ ) λ . Note that for any γ with deg f ∗ ( γ ) < d γ =
0. We set c γ so that P γ c γ γ = ( P γ c ′ γ γ ) λ . Then P γ c γ γ is also a representative of z , and forany γ with deg f ∗ ( γ ) ≤ k , a γ = c γ . We thus get x ∼ z . (cid:3) ORSION AND MORSE-NOVIKOV THEORY 5
We define ( Λ × f ) ab to be the quotient set by the equivalence relation. The abelian group struc-ture of ( Λ × f ) ab naturally induces that of ( Λ × f ) ab .A closed orbit is a non-constant map o : S → X with dods = −∇ f . Two closed orbits arecalled equivalent if they di ff er by linear parameterization. We denote by O the set of the equiv-alence classes of closed orbits. The period p ( o ) is the largest integer p such that o factorsthrough a p -fold covering S → S . We assume that all the closed orbits are nondegener-ate , namely the determinant of id − d φ : T x X / T x o ( S ) → T x X / T x o ( S ) does not vanish for any[ o ] ∈ O , where φ is a p ( o )th return map around a point x ∈ o ( S ). The Lefschetz sign ǫ ( o ) isthe sign of the determinant. We denote by i + ( o ) and i − ( o ) the numbers of real eigenvalues of d φ : T x X / T x o ( S ) → T x X / T x o ( S ) for a return map φ which are > < − Definition 3.2.
We number [ o ] ∈ O with p ( o ) = { [ o i ] } ∞ i = and choose a path σ o i from thebase point of X to a point of o i ( S ) for each [ o i ]. Then we have [ σ o i o i ¯ σ o i ] ∈ π X , where ¯ σ o i isthe inverse path of σ o i . We define ζ f : = ∞ Y i = (1 − ( − i − ( o i ) [ σ o i o i ¯ σ o i ]) ( − i + ( oi ) + i − ( oi ) + ∈ ( Λ × f ) ab . By the completeness of Λ f we can easily check that the infinite product Q ∞ i = (1 − ( − i − ( o i ) [ σ o i o i ¯ σ o i ]) ( − i + ( oi ) + i − ( oi ) + ∈ Λ × f makes sense. Lemma 3.3.
The zeta function ζ f does not depend on the choices of { [ o i ] } ∞ i = and σ o i .Proof. We take another sequence { [ o ′ i ] } ∞ i = and another path σ ′ o i for each [ o i ]. For any k ∈ Z ,since { [ o i ] ∈ O ; deg f ∗ ([ o i ]) ≤ k } , which equals { [ o ′ i ] ∈ O ; deg f ∗ ([ o ′ i ]) ≤ k } , is a finite set and[1 ± [ σ o i o i ¯ σ o i ]] = [1 ± [ σ ′ o i o i ¯ σ ′ o i ]]in ( Λ × f ) ab , ∞ Y i = (1 ± [ σ o i o i ¯ σ o i ]) ± = Y ≤ i ≤∞ , deg f ∗ ([ o i ]) ≤ k (1 ± [ σ o i o i ¯ σ o i ]) ± Y ≤ i ≤∞ , deg f ∗ ([ o i ]) > k (1 ± [ σ o i o i ¯ σ o i ]) ± = Y ≤ i ≤∞ , deg f ∗ ([ o ′ i ]) ≤ k (1 ± [ σ o ′ i o ′ i ¯ σ o ′ i ]) ± Y ≤ i ≤∞ , deg f ∗ ([ o i ]) > k (1 ± [ σ o i o i ¯ σ o i ]) ± = Y ≤ i ≤∞ , deg f ∗ ([ o ′ i ]) ≤ k (1 ± [ σ ′ o ′ i o ′ i ¯ σ ′ o ′ i ]) ± Y ≤ i ≤∞ , deg f ∗ ([ o i ]) > k (1 ± [ σ o i o i ¯ σ o i ]) ± . in ( Λ × f ) ab . Therefore for any k ∈ Z , the products hQ ∞ i = (1 ± [ σ o i o i ¯ σ o i ]) ± i and (cid:20)Q ∞ i = (1 ± [ σ ′ o ′ i o ′ i ¯ σ ′ o ′ i ]) ± (cid:21) have representatives in Λ × f such that for any γ ∈ π X with deg f ∗ ( γ ) ≤ k ,the coe ffi cients of γ are same, and the lemma follows. (cid:3) T. KITAYAMA
By the above lemma we can write ζ f = Y [ o ] ∈O , p ( o ) = [1 − ( − i − ( o ) [ σ o o ¯ σ o ]] ( − i + ( o ) + i − ( o ) + . Let Λ + f be the subring of Λ f whose element P γ ∈ π X a γ γ ∈ Λ f satisfies the fact that a γ = f ∗ ( γ ) ≤
0. There is a formal exponential exp : Λ + f → Λ f given by exp( λ ) : = P ∞ n = λ n n ! . Since ǫ ( o j ) = ( − i + ( o ) + ( j + i − ( o ) and exp ∞ X j = ( ± γ ) j j = (1 ∓ γ ) − for γ ∈ π X with deg f ∗ ( γ ) > ζ f = Y [ o ] ∈O , p ( o ) = exp ∞ X j = ǫ ( o j ) j [ σ o o j ¯ σ o ] , where o j is the composition of a j -fold covering S → S and o .3.2. Novikov-type skew fields.
Here we proceed to construct Novikov-type non-commutativecoe ffi cients for torsion and formulate the main theorem.A group G is called poly-torsion-free-abelian (PTFA) if there exists a filtration1 = G ⊳ G ⊳ · · · ⊳ G n − ⊳ G n = G such that G i / G i − is torsion free abelian. Proposition 3.4 ([15]) . If G is a PTFA group, then Q [ G ] is a right (and left) Ore domain;namely Q [ G ] embeds in its classical right ring of quotients Q [ G ]( Q [ G ] \ − . Let G be a PTFA group and ρ : π X → G , α : G → h t i be group homomorphisms such that α ◦ ρ = f ∗ . Then Ker α is also PTFA, and so we have the classical ring of quotients K of Q [Ker α ]. We denote by l the nonnegative integer such that t l generates Im α . We pick µ ∈ G such that α ( µ ) = t l and let θ : K → K be the automorphism given by θ ( k ) = µ k µ − for k ∈ K .Now we have a Novikov type skew field K θ (( t l )). More precisely, the elements of K θ (( t l )) areformal sums P ∞ i = n a i t li with n ∈ Z and a i ∈ K , and the multiplication is defined by using the rule t l k = θ ( k ) t l . Note that the isomorphism type of the ring K θ (( t l )) dose not depend on the choiceof µ , and we can regard Z [ G ] as a subring of K θ (( t l )).For x , y ∈ K θ (( t )) × ab , we write x ∼ y if for any k ∈ Z , there exist representatives P i ∈ Z a i t li , P i ∈ Z b i t li ∈ K θ (( t l )) × of x , y respectivelysuch that for any i ≤ k , a i = b i .The following lemma can be similarly proved as Lemma 3.1 and so we omit the proof. Lemma 3.5.
The relation ∼ is an equivalence relation in K θ (( t l )) × ab . We define K θ (( t l )) × ab to be the quotient set by the equivalence relation, which is also an abeliangroup. Note that if K θ (( t l )) is commutative, then K θ (( t l )) × ab = K θ (( t l )) × ab . Moreover We canshow that the natural map K × ab → K θ (( t l )) × ab is injective as follows. Let a ∈ K × and assume ORSION AND MORSE-NOVIKOV THEORY 7 [ a ] = ∈ K θ (( t l )) × ab . Then there exists λ ∈ [ K θ (( t l )) × , K θ (( t l )) × ] such that a is equal to the degree0 part of λ , which is in [ K × , K × ]. Therefore [ a ] = ∈ K × ab .The group homomorphism ρ naturally extends to a ring homomorphism Λ f → K θ (( t l )). Byabuse of notation, we also denote it by ρ . By virtue of Theorem 2.7 H ρ ∗ ( X ; K θ (( t l ))) is isomorphicto H ∗ ( C Nov ∗ ( f ) ⊗ Λ K θ (( t l ))). Definition 3.6. If H ρ ∗ ( X ; K θ (( t l ))) =
0, then we define the
Novikov torsion τ ϕ ( X ) associated to ρ as τ Nov ρ ( f ) : = [ τ ( C Nov ∗ ( f ) ⊗ Λ f K θ (( t l )) , h ˜ p ⊗ i p )] ∈ K θ (( t l )) × ab / ± ρ ( π X ) . The ring homomorphism ρ : Λ f → K θ (( t l )) naturally induces a group homomorphism ρ ∗ : ( Λ × f ) ab → K θ (( t l )) × ab and K θ (( t l )) × ab / ± ρ ( π X ) is a quotient group of K θ (( t l )) × ab / ± ρ ( π X ). Theorem 3.7 (Main theorem) . For a given pair ( ρ, α ) as above, if H ρ ∗ ( X ; K θ (( t l ))) = , then τ ρ ( X ) = ρ ∗ ( ζ f ) τ Nov ρ ( f ) ∈ K θ (( t l )) × ab / ± ρ ( π X ) . Remark . More generally, the similar construction makes sense and the theorem also holdsunder the assumption that Q [Ker α ] is a right Ore domain instead of that G is PTFA. Moreoverit is expected that we can eliminate the ambiguity of multiplication by an element of ρ ( π X ),carefully considering Euler structures by Turaev [24], [25] as in [9].An important example of a pair ( ρ, α ) is provided by Harvey’s rational derived series in [6].
Definition 3.9.
For a group Π , let Π (0) r = Π and we inductively define Π ( i ) r = { γ ∈ Π ( i − r ; γ k ∈ [ Π ( i − r , Π ( i − r ] for some k ∈ Z \ } . Lemma 3.10 ([6]) . For any group Π and any n, Π ( n − r / Π ( n ) r = ( Π ( n − r / [ Π ( n − r , Π ( n − r ]) / torsionand Π / Π ( n ) r is a PTFA group. For any n , we have the natural surjection ρ ( n ) : π X → π X / ( π X ) ( n + r and the induced ho-momorphism α ( n ) : π X / ( π X ) ( n + r → h t i by f ∗ . By the above construction we obtain the grouphomomorphism ρ ( n ) : Z [ π X ] → K ( n ) θ ( t l ) and the extended one ˜ ρ ( n ) : Λ f → K ( n ) θ (( t l )), where K ( n ) θ ( t l ) is the subfield of K ( n ) θ (( t l )) consisting of rational elements. Definition 3.11. If H ρ ( n ) ∗ ( X ; K ( n ) θ ( t l )) =
0, then τ ρ ( n ) ( X ) ∈ K ( n ) θ ( t l ) × ab / ± ρ ( n ) ( π X ) is defined. Wecall it the higher-order Reidemeister torsion of order n . Remark . It is known by Friedl that τ ρ ( n ) ( X ) equals an alternating product of non-commutative Alexander polynomials. See [2] for the details.If H ρ ( n ) ∗ ( X ; K ( n ) θ ( t l )) =
0, then we have H ˜ ρ ( n ) ∗ ( X ; K ( n ) θ (( t l ))) =
0, and we can also define τ ˜ ρ ( n ) ( X ) ∈K ( n ) θ (( t l )) × ab / ± ρ ( n ) ( π X ). By the functoriality of Reidemeister torsion the image of τ ρ ( n ) ( X ) by thenatural map K ( n ) θ ( t l ) × ab / ± ρ ( n ) ( π X ) → K ( n ) θ (( t l )) × ab / ± ρ ( n ) ( π X ) equals τ ˜ ρ ( n ) ( X ). Thus for the pair( ρ ( n ) , α ( n ) ), the main theorem gives a Morse theoretical and dynamical presentation of τ ρ ( n ) ( X ) in K ( n ) θ (( t l )) × ab / ± ρ ( n ) ( π X ) as a corollary. T. KITAYAMA
4. P roof
The proof of the main theorem is divided into two parts. In the first part we construct an“approximate” CW complex X ′ which is adapted to ∇ f , and we show that the Reidemeistertorsion of X ′ equals that of X . The second part is devoted to computation of the torsion of X ′ ,and we see that it has the desired form.4.1. An approximate CW-complex.
Let Σ be a level set of a regular value of f and let Y bethe compact Riemannian manifold obtained by cutting X along Σ . We can pick a Morse function f : Y → R induced by f . We write ∂ Y = Σ ⊔ Σ , where Σ , Σ are the cutting hypersurfaces and −∇ f points outward along Σ . We denote by A ( p ) , D ( p ) the stable and unstable manifoldsof a critical point p of f .We take a smooth triangulation T of Σ such that each simplex is transverse to A ( p ) for eachcritical point p of f . For σ ∈ T , let us denote by F ( σ ) the set of all y ∈ Y such that the flowof ∇ f starting at y hits σ . It is well-known that the submanifolds D ( p ) and F ( σ ) have naturalcompactifications D ( p ) and F ( σ ) respectively by adding broken flow lines of −∇ f . (See forinstance [8].) We choose a cell decomposition T of Σ such that D ( p ) ∩ Σ and F ( σ ) ∩ Σ aresubcomplexes for each critical point p and each simplex σ . Then we can check that the cells in T , T , D ( p ) and F ( σ ) give a cell decomposition T Y of Y .Let h : ( Σ , T ) → ( Σ , T ) be a cellular approximation to the canonical identification Σ → Σ . We consider the mapping cylinder M h of h : M h : = (( Σ × [0 , ⊔ Σ ) / ( x , ∼ h ( x ) . It has a natural cell decomposition induced by T and T . Definition 4.1.
Let X ′ be the space obtained by gluing Y and M h along Σ ⊔ Σ .For a cell ∆ in T Y of the form D ( p ) and F ( σ ), we define b ∆ to be the set obtained by gluing ∆ and ((( ∆ ∩ Σ ) × [0 , ⊔ h ( ∆ ∩ Σ )) / ( x , ∼ h ( x )along ∆ ∩ Σ . Cells of the form [ D ( p ), σ and [ F ( σ ) for a critical point p of f and σ ∈ T givea cell decomposition of X ′ .We pick a homotopy equivalent map X ′ → X and identify π X ′ with π X . Lemma 4.2.
Under the assumptions of Theorem 3.7 we have τ ρ ( X ) = τ ρ ( X ′ ) . Proof.
In all of the calculations below, we implicitly tensor the chain complexes with the skewfield K θ (( t l )), and brackets mean that they are in K θ (( t l )) × ab / ± ρ ( π X ).We regard X as the union of Y and Σ × [0 ,
1] along Σ ⊔ Σ , then we have short exact sequences0 → C ∗ ( e Σ ) ⊕ C ∗ ( e Σ ) → C ∗ ( e Σ × [0 , ⊕ C ∗ ( e Y ) → C ∗ ( e X ) → , → C ∗ ( e Σ ) ⊕ C ∗ ( e Σ ) → C ∗ ( f M h ) ⊕ C ∗ ( e Y ) → C ∗ ( e X ) → . The natural surjection Σ × [0 , → M h induces an isomorphism between H ρ ∗ ( Σ × [0 , K θ (( t l ))) and H ρ ∗ ( M h ; K θ (( t l ))), and there is an isomorphism between the long exact se-quences in homology for the above sequences. Let c and c ′ be the bases of C ∗ ( e Σ × [0 , C ∗ ( f M h ) which are obtained by T and the product cell structure from T . We pick bases h and ORSION AND MORSE-NOVIKOV THEORY 9 h ′ of H ρ ∗ ( Σ × [0 , K θ (( t l ))) and H ρ ∗ ( M h ; K θ (( t l ))) such that the isomorphism maps h to h ′ . Thenfrom Lemma 2.2 we obtain(4.1) τ ρ ( X ) τ ρ ( X ′ ) = [ τ ( C ∗ ( e Σ × [0 , , c , h )][ τ ( C ∗ ( e M h ) , c ′ , h ′ )] . We have short exact sequences0 → C ∗ ( e Σ × → C ∗ ( e Σ × [0 , → C ∗ ( e Σ × [0 , , e Σ × → , → C ∗ ( e Σ ) → C ∗ ( e M h ) → C ∗ ( e M h , e Σ ) → . The surjection Σ × [0 , → M h also induces an isomorphism between the long exact sequencesin homology for the above sequences. Let d and d ′ be bases of C ∗ ( e Σ × [0 , , e Σ ×
1) and C ∗ ( e M h , e Σ )induced by c and c ′ . Then again from Lemma 2.2 we obtain(4.2) [ τ ( C ∗ ( e Σ × [0 , , c , h )][ τ ( C ∗ ( e M h ) , c ′ , h ′ )] = [ τ ( C ∗ ( e Σ × [0 , , e Σ × , d )][ τ ( C ∗ ( e M h , e Σ ) , d ′ )] . By direct computations we have[ τ ( C ∗ ( e Σ × [0 , , e Σ × , d )] = [ τ ( C ∗ ( e M h , e Σ ) , d ′ )] = [1] . Now the lemma follows from (4.1), (4.2) and these equalities. (cid:3)
Computation of the torsion.
We decompose C i ( e X ′ ) ⊗ Z [ π X ′ ] K θ (( t l )) = D i ⊕ E i ⊕ F i , where D i , E i and F i are generated by elements of the form [ D ( p ), σ and [ F ( σ ) for a criticalpoint p of f and σ ∈ T respectively. There are natural identifications D i (cid:27) C Novi ( f ) ⊗ Λ f K θ (( t l ))and F i (cid:27) E i − . Then the matrix for the di ff erential ∂ i can be written as D i E i F i ∂ i = D i − E i − F i − N i W i − M i ∂ Σ i I − φ i − − ∂ Σ i , where ∂ Σ i is the di ff erential on C ∗ ( e Σ ) ⊗ Z [ π Σ ] K θ (( t l )) and φ i − can be interpreted as the return mapof the gradient flow in e X after perturbation by h . We set K i : = N i + W i ( I − φ i − ) − M i : D i → D i − . Since C ∗ ( e X ) ⊗ Z [ π X ] K θ (( t l )) is acyclic, the Novikov complex ( D ∗ , ∂ f ∗ ) is also acyclic by The-orem 2.7, and we can choose a decomposition D i = D ′ i ⊕ D ′′ i such that D ′ i and D ′′ i are spannedby lifts of the critical points of f and ∂ f induces an isomorphism D ′ i → D ′′ i − . We denote by K i : D ′ i → D ′′ i − the induced map by K i . Lemma 4.3.
Under the assumptions of Theorem 3.7, if K i is non-singular for each i, then τ ρ ( X ′ ) = d Y i = [det( I − φ i − ) det K i ] ( − i . Proof.
We consider the matrix D ′ i F i Ω i : = D ′′ i − E i − N i W i − M i I − φ i − ! , where M i : D ′ i → E i − , N i : D ′ i → D ′′ i − and W i : F i → D ′′ i − be the induced maps by M i , N i and W i respectively. After elementary row operations we can turn Ω i into the matrix K i − M i I − φ i − ! . Since K i is nonsingular, Ω i is also nonsingular anddet Ω i = det( I − φ i − ) det K i . By Lemma 2.3 we have τ ρ ( X ′ ) = d Y i = [det Ω i ] ( − i , which proves the lemma. (cid:3) For a positive integer k and x , y ∈ K θ (( t l )) × ab / ± ρ ( π X ), we write x ∼ k y if there exist representatives P ∞ i = a i t li , P ∞ i = b i t li ∈ K θ (( t )) × of x , y respectively such that a b , a i = b i for i = , , . . . , k . Note that x = y if and only if for any positive integer k , x ∼ k y . Lemma 4.4.
For any positive integer k, if we choose T su ffi ciently fine and h su ffi ciently closeto the identity, then d Y i = [det( I − φ i − )] ( − i ∼ k [ ρ ∗ ( ζ f )] . We prepare some notation and a lemma for the proof.Let ϕ : Σ \ ⊔ p A ( p ) → Σ \ ⊔ p D ( p ) be the di ff eomorphism defined by the downward gradientflows and let H : Σ × [0 , → Σ be the homotopy from id to h . We can consider the i times iteratemaps ϕ i and ( H ( · , t ) ◦ ϕ ) i for t ∈ [0 ,
1] which are partially defined. A natural compactification Γ it of the graph Γ it ∈ Σ × Σ of ( H ( · , t ) ◦ ϕ ) i is defined by attaching pairs ( x , H ( y , t )), where x isthe starting point and y is the end point of a broken flow line of −∇ f . (See [8], [9] for moredetails.)It is known that there exists a positive integer N such that if the simplexes in T are allcontained in balls of radius ǫ , then H can be chosen so that the distance between x and H ( x , t ) is < N ǫ for all x ∈ Σ and t ∈ [0 , ϕ i liesin the interior of Γ i under the diagonal map Σ → Σ × Σ and is compact from the nondegenerateassumption, it follows for any positive integer k that we can choose ǫ so that Γ it does not crossthe diagonal in Σ × Σ for all i ≤ k . ORSION AND MORSE-NOVIKOV THEORY 11
Lemma 4.5.
Let k be a positive integer and suppose that l = . Let ( a i , j ) be the n-dimensionalmatrix over K θ (( t )) such that a i , j = c i , j t, where c i , j ∈ K . If a , i a i , i . . . a i j − , = for anysequence i , i , . . . , i j − with j ≤ k, then [det( δ i , j − a i , j ) ≤ i , j ≤ n ] ∼ k [det( δ i , j − a i , j ) ≤ i , j ≤ n ] , where δ i , j is Kronecker’s delta.Proof. We set b (0) i , j = δ i , j − a i , j and inductively define b ( m ) i , j for m = , . . . , n as follows: b ( m ) i , j = b ( m − i , j − b ( m − i , m ( b ( m − m , m ) − b ( m − m , j . This is an elementary row operation with respect to the i th row and b ( m ) i , j = i , j ≤ m . Wehave [det( b (0) i , j ) ≤ i , j ≤ n ] = [det( b ( n ) i , j ) ≤ i , j ≤ n ] = n Y i = b ( n ) i , i . By induction on m we first show the following observation concerning any nonzero term in b ( m ) i , j − δ i , j :(i) The term has positive degree.(ii) The term has elements a i , i , a i , i , . . . , a i j ′− , j as factors for a sequence i , i , . . . , i j ′ − .(iii) If the term has a i ′ , as a factor for some i ′ , then then the degree of the term is > k or we canmake such a sequence in (ii) contains 1.It is easy to check them for m =
0. We assume them for m = m ′ −
1. Since( b ( m ′ − m ′ , m ′ ) − = + ∞ X i = ( − i ( b ( m ′ − m ′ , m ′ − i , (i) for m = m ′ follows from (i) for m = m ′ −
1. By (ii) for m = m ′ − m = m ′ . We take any nonzero term c in ( b ( m ′ − m ′ , m ′ ) − which has a i ′ , as a factor. Then thereis a nonzero term c ′ in b ( m ′ − m ′ , m ′ which has a i ′ , as a factor such that c has c ′ as a factor. If c ′ haselements a m ′ , i , a i , i , . . . , a i j ′− , m ′ as factors for a sequence i , i , . . . , i j ′ − containing 1, then j ′ ≥ k by the assumption, and deg c ′ > k . Hence by (ii) and (iii) for m = m ′ −
1, deg c ≥ deg c > k .Now we can immediately check (iii) for m = m ′ .As a consequence of the above argument the degree of any nonzero term in b ( n ) i , i which has a i ′ , as a factor is > k , and hQ ni = b ( n ) i , i i is invariant even if we erase such terms. Therefore inconsidering the equivalence class we can regard a i , as 0 for all i , which deduce the lemma. (cid:3) Proof of Lemma 4.4.
We only consider the case where l =
1. If l =
0, then φ i = i andthere is no closed orbit, and so there is nothing to prove. If l >
1, then we can prove it by asimilar argument.We set I k : = { [ o ] ∈ O ; p ( o ) = , deg f ∗ ([ o ]) ≤ k } . Then we see that(4.3) [ ρ ∗ ( ζ f )] ∼ k Y [ o ] ∈I k [1 − ( − i − ( o ) ρ ([ σ o o ¯ σ o ])] ( − i + ( o ) + i − ( o ) + . For [ o ] ∈ I k , there is a sequence x , x , . . . , x deg f ([ o ]) − of fixed points of ϕ deg f ∗ ([ o ]) such that ϕ ( x j − ) = x j for j = , , . . . , deg f ([ o ]) −
1. Choosing T su ffi ciently fine and H as above, we canpick mutually disjoint contractible subcomplexes N x j of T satisfying the following conditions:(i) Each N x j contains all the fixed points of ( H ( · , t ) ◦ ϕ ) deg f ∗ ([ o ]) which are close to x j for all t ∈ [0 , N x j − and ones of N x j by h ◦ ϕ and h ◦ ϕ ( N deg f ∗ ([ o ]) − ) ∩ N j = ∅ for j = , , . . . , deg f ∗ ([ o ]) − σ = σ , σ , . . . , σ i = σ ∈ T so that σ j ⊂ h ◦ ϕ ( σ j − ) for a simplex σ not contained in any N x j , then i > k .We denote by N [ o ] and N ′ [ o ] for [ o ] ∈ I k the union of all N x j for a fixed point x j ∈ o ( S ) andthat of all N x for such a sequence of [ o ]. Then we define φ [ o ] , i : C i ( e N [ o ] ) ⊗ K θ (( t )) → C i ( e N [ o ] ) ⊗ K θ (( t )) φ ′ [ o ] , i : C i ( e N ′ [ o ] ) ⊗ K θ (( t )) → C i ( e N ′ [ o ] ) ⊗ K θ (( t ))to be the maps induced by h ◦ ϕ and ( h ◦ ϕ ) deg f ∗ ([ o ]) respectively.Note that all the entries of φ i are monomials. By condition (iii) we can apply Lemma 4.5repeatedly and obtain(4.4) [det( I − φ i )] ∼ k Y [ o ] ∈I k [det( I − φ [ o ] , i )] . By condition (ii) we can take simplexes σ j ⊂ N x j such that h ◦ ϕ ( σ j − ) = σ j for j = , , . . . , deg f ([ o ]) −
1. The matrix of the restriction of I − φ [ o ] , i on these simplexes has theform . . . m ρ ( γ deg f ∗ ([ o ]) ) ± ρ ( γ ) 1 . . . ± ρ ( γ ) . . . ... ...... ... . . . . . . ± ρ ( γ deg f ∗ ([ o ]) − ) 1 , where m ∈ Z and γ j ∈ π X . Since Q deg f ∗ ([ o ]) − j = γ deg f ∗ ([ o ]) − j = [ σ o o ¯ σ o ] for a path σ o from the basepoint of X to x , the determinant of the matrix is (1 − ( ± m ρ ([ σ o o ¯ σ o ])) and the coe ffi cient of σ of φ [ o ] , i ( σ ⊗
1) is ± m ρ ([ σ o o ¯ σ o ]). According to the above argument, we have(4.5) [det( I − φ [ o ] , i )] = [det( I − φ ′ [ o ] , i )] . Since the entries of the matrix φ ′ [ o ] , i are all in an abelian ring ρ ( Z [[ σ o o ¯ σ o ]]), d − Y i = [det( I − φ ′ [ o ] , i )] = exp ∞ X j = d − X i = ( − i j tr( φ ′ [ o ] , i ) j ∼ k exp ∞ X j = Fix(( ϕ | N ′ [ o ] ) j deg f ∗ ([ o ]) ) j ρ ([ σ o o ¯ σ o ]) j = exp ∞ X j = ǫ ( o j ) j ρ ([ σ o o ¯ σ o ]) j = [1 − ( − i − ( o ) ρ ([ σ o o ¯ σ o ])] ( − i + ( o ) + i − ( o ) + , ORSION AND MORSE-NOVIKOV THEORY 13 where Fix(( ϕ | N ′ [ o ] ) j deg f ∗ ([ o ]) ) counts fixed points of ( ϕ | N ′ [ o ] ) j deg f ∗ ([ o ]) with sign. The second equiva-lence follows from the machinery used to prove the Lefschetz fixed point theorem. From (4.3),(4.4), (4.5) and this the lemma is proved. (cid:3) Lemma 4.6.
For any positive integer k, if we choose T su ffi ciently fine and h su ffi ciently closeto the identity, then K i is non-singular and d Y i = [det K i ] ( − i ∼ k [ τ Nov ρ ( f )] . Proof.
Suppose to begin that h = id . The unstable manifold D ( p ) of a critical point p of f hasa natural compactification D ( p ) such as D ( p ). The compactification D ( p ) can be representedas D ( p ) + ∞ X j = F ( φ ji − M i ( D ( p )) , where by abuse of notation we also denote by F the linear extension of F . So if we identify D i with C Novi ( f ) ⊗ Λ f K θ (( t l )), then ∂ fi ( D ( p )) = pr D i − ◦ ∂ i D ( p ) + ∞ X j = F ( φ ji − M i ( D ( p ))) = K i ( D ( p ))for a critical point p with index i . Hence ∂ fi induces K i : D ′ i → D ′′ i − , and K i is nonsingular.From Lemma 2.3 we have d Y i = [det K i ] ( − i = d Y i = [det pr D ′′ i ◦ ∂ fi | D ′ i ] ( − i = [ τ Nov ρ ( f )] . Next we consider the case where h , id .Let pr j , pr j : Γ jt → Σ be the restriction of the first and second projections of Σ × Σ . We define B jt ( p ) : = pr ( pr − ( H ( · , t )( D ( p ) ∩ Σ )))for j = , , . . . , k − t ∈ [0 ,
1] and a critical point p of f . Since the set of the intersectionpoints of B j ( p ) and A ( q ) ∩ Σ for any critical point q lies the interior of B j ( p ) and is compactfrom the transverse condition, it follows that we can choose T su ffi ciently fine and H as aboveso that B jt ( p ) does not cross A ( q ) ∩ Σ for all j < k , where we naturally identify Σ with Σ .By a similar argument to that of Lemma 4.4 we can check that the image of the hat of F ( φ ji − M i ( [ D ( p ))) by pr D i − ◦ ∂ i can be computed from the local intersection numbers of B jt ( p )and A ( q ) ∩ Σ and the elements of π X determined by the perturbed flows by h from p to q ,which are invariant on t for j < k . Hence from the computation of the case where h = id , weobtain ∂ fi ( [ D ( p )) ∼ k K i ( [ D ( p ))for all p with index i , and K i is non-singular. Again from Lemma 2.3 we analogously see thedesired relation. (cid:3) From the proofs of Lemma 4.4 and 4.6, if we choose appropriate T and H , then the conclu-sions of the lemmas simultaneously hold for any positive integer k , and so d Y i = [det( I − φ i − ) det K i ] ( − i ∼ k [ ρ ∗ ( ζ f )][ τ Nov ρ ( f )] . Now we can establish Theorem 3.7 at once from Lemma 4.2 and 4.3.
Acknowledgement.
The author wishes to express his gratitude to Toshitake Kohno for his en-couragement and helpful suggestions. He is greatly indebted to Andrei Pajitnov for a thoroughexplanation of the deduction of the theorem from his results and for many stimulating conver-sations. He would also like to thank Hiroshi Goda, Takayuki Morifuji, Takuya Sakasai andYoshikazu Yamaguchi for fruitful discussions and advice. This research was supported by JSPSResearch Fellowships for Young Scientists.R eferences [1] T. Cochran,
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