aa r X i v : . [ m a t h . S G ] J u l Polytope Novikov Homology
Alessio Pellegrini
Department of Mathematics, ETH Z¨urich, Switzerland [email protected] 21, 2020
Abstract
Let M be a closed manifold and A ⊆ H ( M ) a polytope. For each a ∈ A we definea Novikov chain complex with a multiple finiteness condition encoded by the polytope A . The resulting polytope Novikov homology generalizes the ordinary Novikov homology.We prove that any two cohomology classes in a prescribed polytope give rise to chainhomotopy equivalent polytope Novikov complexes over a Novikov ring associated to saidpolytope. As applications we present a novel approach to the (twisted) Novikov MorseHomology Theorem and prove a new polytope Novikov Principle. The latter generalizesthe ordinary Novikov Principle and a recent result of Pajitnov in the abelian case. Given a closed manifold M and a cohomology class a ∈ H ( M ) , one can define theso called Novikov homology HN ● ( a ) , introduced by Novikov [12, 13]. Roughly speaking,HN ● ( a ) is defined by picking a Morse representative α ∈ a and a cover on which α pullsback to an exact form d ˜ f , and then mimicking the definition of Morse homology using ˜ f as the underlying Morse function. The groups HN ● ( a ) enjoy three distinctive features: • (Novikov-module) The Novikov homology HN ● ( a ) is a finitely-generated moduleover the so called Novikov ring
Nov ( a ) . • (Cohomology-invariance) The Novikov homology HN ● ( a ) does not depend onthe choice of Morse representative α of the prescribed cohomology class a . • (Ray-invariance) Morse forms on the same positive half-ray induce identical Novikovhomologies: HN ● ( r ⋅ a ) ≅ HN ● ( α ) for all r > ● ( a ) is isomorphic tothe twisted singular homology H ● ( M, Nov ( a )) . By using the Novikov Morse Homology See Corollary 3.4 for a precise statement. heorem one can thus investigate the relation between HN ● ( a ) and HN ● ( b ) , when a ≠ b ,by studying H ● ( M, Nov ( a )) and H ● ( M, Nov ( b )) and their respective twisted coefficientsystems Nov ( a ) and Nov ( b ) instead. The latter is a “purely” algebraic task.In this article we refine the construction of Novikov homology HN ● ( a ) and define whatwe call polytope Novikov homology HN ● ( a, A) by including multiple finiteness conditionsimposed by a polytope A = ⟨ a , . . . , a k ⟩ ⊆ H ( M ) containing a . These polytope Novikovhomology groups HN ● ( a, A) retain the three features of HN ( a ) mentioned above, moduloreplacing Nov ( a ) by a “smaller” Novikov ring Nov (A) . The Main Theorem in Section 2gives a dynamical relation between HN ● ( a, A) and HN ● ( b, A) , i.e by staying in the realmof Novikov homology and not resorting to the algebraic counterpart of twisted singularhomology. Theorem (Main Theorem ) . For every subpolytope B ⊆ A and two cohomology classes a, b ∈ A there exists a commutative diagram HN ● ( a, A) HN ● ( b, A) HN ● ( a, A∣ B ) HN ● ( b, A∣ B ) . ≅≅ induced by continuation on the chain level. The statement of the Main Theorem might be known to some experts in the field, butlacks a proof in the literature. Similar variants of the Main Theorem have been proved indifferent settings, most noteworthy are [14, 8, 6, 26]. For example, in [14] Ono considersFloer-Novikov homology on a closed symplectic manifold and proves: Theorem (Ono [14]) . If two symplectic isoptopies have fluxes that are close to each other,then their respective Novikov-Floer homologies are isomorphic.
The Novikov-Floer homologies mentioned in Ono’s Theorem are defined over a com-mon Novikov ring that takes into account several finiteness conditions simultaneously –this modification is analogous to our implementation of polytopes. Within this analogy,the upper isomorphism in the Main Theorem corresponds to the isomorphism in Ono’sTheorem, but with less assumptions: the nearby assumption of the fluxes in Ono’s resultwould translate to a smallness assumption on A , which is not needed. Let us mentionthat the formulation and setup of the Main Theorem comes closest to a recent result dueto Groman and Merry [6, Theorem 5.1].At the end of the paper we present two applications of the Main Theorem. In thefirst application we recover the aforementioned Novikov Morse Homology Theorem: Theproof, modulo details, goes as follows: taking A = ⟨ , a ⟩ , setting B = ⟨ a ⟩ , invoking thelower isomorphism in the Main Theorem, and unwinding the definitions revealsHN ● ( a ) ≅ HM ● ( f, Nov ( a )) , See Theorem 2.24 for the precise statement. Also note that the actual theorem contains a stronger chainlevel statement. For the sake of simplicity we omit the precise conditions. This is not a circular argument, since the Novikov Morse Homology Theorem is not used in Section 2. here the right hand side is Morse homology with local coefficients Nov ( a ) . The latteris known to be isomorphic to singular homology with twisted coefficients, for a quickproof see [2, Theorem 4.1], and thus we recover the Novikov Morse Homology Theorem.This line of reasoning is analogous to the proof of [6, Theorem 5.3] and seems to bea novel approach to the Novikov Morse Homology Theorem: the proof draws a directconnection between Novikov and twisted Morse homology instead of using the NovikovPrinciple and/or equivariant Morse homology, see [9, 20, 5] for proofs of the NovikovMorse Homology Theorem using the latter.The second application is concerned with a general polytope Novikov Principle : Theorem (Polytope Novikov Principle) . Let B ⊆ A be a subpolytope. Then for every a ∈ A there exists a Morse representative α ∈ a such that CN ● ( α, A∣ B ) ≃ C ● ( ̃ M A ) ⊗ Z [ Γ A ] Nov (A∣ B ) , as Novikov-modules. The proof idea is similar to the sketch above – one relates the polytope Novikovcomplex to a a twisted Morse complex by including the 0-vertex in the polytope A andusing the Main Theorem. We call this the (cf. Lemma 3.1). To get from thetwisted Morse complex to the equivariant singular chain complex we use a Morse-Eilenbergtype result (cf. Lemma 3.2) and a chain homotopy equivalence CM ( ˜ h ) ≃ C ● ( ̃ M A ) overthe group ring of deck transformations Z [ Γ A ] .Immediate consequences of the polytope Novikov Principle include the ordinary NovikovPrinciple (cf. Corollary 3.8) and a recent “conical” Novikov Principle [18, Theorem 5.1] in the abelian case (cf. Corollary 3.10). Remark.
Symplectic homology is a version of Floer homology well-suited to certain non-compact symplectic manifolds. In [19] we combine ideas of Ono’s Theorem, the magneticcase [6], and of the present paper to construct a polytope Novikov symplectic homology,which is related to Ritter’s twisted symplectic homology [21]. The analogue of the MainTheorem remains true. Applications include Novikov number-type bounds on the numberof fixed points of symplectomorphisms with prescribed flux on the boundary, and the studyof symplectic isotopies of such maps.
Acknowledgements.
I would like to thank Will Merry for all the helpful discussions andfor encouraging me to flesh out the polytope picture in the case of Novikov homology. Thiswork has been supported by the Swiss National Science Foundation (grant This is a slightly imprecise formulation, see Theorem 3.6 for the precise statement See Theorem 3.9 for a statement below. Novikov Homology and Polytopes
In this subsection we quickly recall the (ordinary) definition of the Novikov chaincomplex and its homology, together with some well known properties. The main purposeis to fix the notation for the remainder of the section. For a thorough treatment of Novikovhomology, we recommend [23, 20, 5] and the recently published [1]. For more details onthe construction of the Novikov ring see for instance [7, Chapter 4].Fix once and for all a closed smooth oriented and connected finite-dimensional manifold M . For any Morse-Smale pair ( α, g ) one can define the Novikov chain complex ( CN ● ( α, g ) , ∂ ● ) , whose homology is called Novikov homology of ( α, g ) HN i ( α, g ) ∶ = ker ∂ i / im ∂ i + , i ∈ N . It is a standard fact that two Morse-Smale pairs with cohomologous Morse forms induceisomorphic Novikov homologies, thus we shall write HN ● ( a ) with a = [ α ] to denote theNovikov homology of pairs ( α, g ) . Notation.
Sometimes we will also omit the g in the notation of the chain complex.Moreover, Latin lowercase letters, e.g. a, b , will typically denote cohomology classes, whilethe respective lowercase Greek letters are representatives in the corresponding cohomologyclasses, e.g. α ∈ a, β ∈ b .Let us quickly recall the relevant definitions. Each cohomology class a determines aperiod homomorphism Φ a ∶ π ( M ) → R defined by integrating any representative α ∈ a over loops γ in M . Denote by ker ( a ) the kernel of the period homomorphism Φ a and let π ∶ ̃ M a → M be the associated abelian cover, i.e. a regular covering with Γ a ∶ = Deck ( ̃ M a ) ≅ π ( M ) / ker ( a ) . Then α pulls back to an exact form on ̃ M a , i.e. π ∗ α = d ˜ f α for some˜ f α ∈ C ∞ ( ̃ M a ) . Define V i ( α ) ∶ = ⊕ ˜ x ∈ Crit i ( ˜ f α ) Z ⟨ ˜ x ⟩ , i ∈ N , where Crit i ( ˜ f α ) denotes the critical points of ˜ f α with Morse index i . The i -th Novikovchain group CN i ( α ) can then be defined as the downward completion of V i ( α ) with respectto ˜ f α , which shall be denoted by ̂ V i ( α ) ˜ f α or more concisely ̂ V i ( α ) α . (1)Explicitly, elements ξ ∈ CN i ( α ) are infinite sums with a finiteness condition determinedby ˜ f α : ξ = ∑ ˜ x ∈ Crit i ( ˜ f α ) ξ ˜ x ˜ x ∈ CN i ( α ) ⇐⇒ ∀ c ∈ R ∶ { ˜ x ∣ ξ ˜ x ≠ ∈ Z , ˜ f α ( ˜ x ) > c } is finite . Cohomologous one-forms induce the same period homomorphism by Stokes’ Theorem. he boundary operator is defined by counting Morse trajectories of ˜ f α : ∂ ∶ CN i ( α ) → CN i − ( α ) , ∂ξ ∶ = ∑ ˜ x, ˜ y ξ ˜ x ⋅ alg M ( ˜ x, ˜ y ; ˜ f α ) ˜ y, where M ( ˜ x, ˜ y ; ˜ f α ) = { ˜ γ ∈ C ∞ ( R , ̃ M ) ∣ ˙˜ γ + ∇ ˜ g ˜ f α ( ˜ γ ) = , ˜ γ ( −∞ ) = ˜ x, ˜ γ ( +∞ ) = ˜ y } is the usual moduli space and M ( ˜ x, ˜ y ; ˜ f α ) = M ( ˜ x, ˜ y ; ˜ f α ) / R . Similarly, we denote by M ( x, y ; α ) and M ( x, y ; α ) the moduli spaces downstairs. The alg indicates the alge-braic count, i.e. counting the Novikov trajectories with signs determined by a choice oforientation of the underlying unstable manifolds.The Novikov ring Λ α associated to α ∈ a is defined as the upward completion of thegroup ring Z [ Γ a ] with respect to the period homomorphism Φ a , therefore λ = ∑ A ∈ Γ a λ A A ∈ Λ α ⇐⇒ ∀ c ∈ R ∶ { A ∣ λ A ≠ ∈ Z , Φ a ( A ) < c } is finite . The Novikov ring Λ α does not depend on the choice of representative α ∈ a , thus we shallwrite Λ a . Moreover, Λ a acts on CN ● ( α ) in the obvious way. By fixing a preferred lift˜ x j in each fiber of the finitely many zeros x j ∈ Z ( α ) ∶ = { x ∈ M ∣ α ( x ) = } , one can viewCN ● ( α ) as a finitely-generated Λ a -module:CN i ( α ) ≅ ⊕ ˜ x j ∈ Crit i ( ˜ f α ) Λ a ⟨ ˜ x j ⟩ as Novikov ring modules . (2)Another standard fact asserts that the boundary operator ∂ is Λ a -linear and consequentlythe Novikov homology HN ● ( a ) carries a Λ a -module structure. The latter is implicitly usingthe fact that isomorphism of Novikov homologies for cohomologous Morse forms, whichis suppressed in the notation HN ● ( a ) , is also Λ a -linear. Remark 2.1. If M is not orientable one can still define a Novikov homology by replacing Z with Z in all the definitions above. We are now ready to refine the Novikov chain complex using polytopes – this notionwill be key for the proof all incoming theorems.
Definition 2.2.
Given a , . . . , a k ∈ H ( M ) , denote by A = ⟨ a , . . . , a k ⟩ ⊂ H ( M ) the polytope spanned by the vertices { a l } l = ,...,k , i.e. the set of all convex combinations a = k ∑ l = c l ⋅ a l with c l ∈ [ , ] and k ∑ l = c l = . o any polytope A we associate a regular cover π ∶ ̃ M A → M determined by Deck ( ̃ M A ) ≅ π ( M ) / k ⋂ l = ker ( a l ) , and we shall abbreviate Γ A ∶ = Deck ( ̃ M A ) . Example 2.3.
For the polytope A = ⟨ a ⟩ the covering ̃ M A agrees with the abelian cover ̃ M a associated to a ∈ H ( M ) . The same is true for any polytope A , whose other vertices a l satisfy ker ( a ) ⊆ ker ( a l ) .The defining condition of ̃ M A ensures that each vertex a l pulls back to the trivialcohomology class, and so does every a ∈ A , see Lemma 2.7. We write ˜ f α ∈ C ∞ ( ̃ M A ) todenote some primitive of π ∗ α for α a representative of a ∈ A . Now we fix a smooth section θ ∶ A Ð→ Ω ( M ) , a ↦ θ a of the projection of closed one-forms to their cohomology class. In other words, θ a is arepresentative of a . This enables us to talk about a “preferred” representative of eachcohomology class in the polytope.For every polytope a ∈ A we define V i ( θ a , A ) ∶ = ⊕ ˜ x ∈ Crit i ( ˜ f θa ) Z ⟨ ˜ x ⟩ . The subtle but crucial difference to V i ( θ a ) is that ̃ M A does not necessarily coincide withthe abelian cover ̃ M a . Definition 2.4.
Let A be a polytope with section θ ∶ A → Ω ( M ) . Then the (polytope)Novikov chain complex groups CN i ( θ a , A ) , i ∈ N , are defined as the intersections of the downward completions of V i ( θ a , A ) with respect toany ˜ f β ∶ ̃ M A → R for b ∈ A . In other words, with the notation of (1):CN i ( θ a , A ) ∶ = ⋂ b ∈ A ̂ V i ( θ a , A ) β . Remark 2.5.
Let β ∈ b be any representative. The choice of primitive ˜ f β of π ∗ β is uniqueup to adding constants and hence does not affect the finiteness condition. Additionally,two primitives ˜ f β and ˜ f β ′ induce the same finiteness condition for β, β ′ ∈ b . Indeed,˜ f β ′ − ˜ f β = h ○ π for some smooth h ∶ M → R with dh = β ′ − β . Since M is compact we get˜ f β ( ˜ x ) > c Ô⇒ ˜ f β ′ ( ˜ x ) > min z ∣ h ( z )∣ + c and ˜ f β ′ ( ˜ x ) > d Ô⇒ ˜ f β ( ˜ x ) > d − max z ∣ h ( z )∣ , hence the two finiteness conditions are equivalent. This justifies Definition 2.4. npacking Definition 2.4 we see ξ = ∑ ˜ x ∈ Crit i ( ˜ f θa ) ξ ˜ x ˜ x ∈ CN i ( θ a , A ) ⇐⇒ ∀ b ∈ A , ∀ c ∈ R ∶ { ˜ x ∣ ξ ˜ x ≠ , ˜ f β ( ˜ x ) > c } < +∞ , (3)where it does not matter which primitives ˜ f β we use, cf. Remark 2.5. The right hand sidedescribes a finiteness condition that has to hold for all b ∈ A , hence we will refer to it asthe multi finiteness condition . Notation.
In view of Remark 2.5 we shall writeCN i ( θ a , A ) = ⋂ b ∈ A ̂ V i ( θ a , A ) b from now on.In a similar fashion we can define yet another completion of V i ( θ a , A ) by taking thecompletion with respect to less one-forms. Definition 2.6.
Let B ⊆ A be a subpolytope. Then we defineCN i ( θ a , A ∣ B ) ∶ = ⋂ b ∈ B ̂ V i ( θ a , A ) b the restricted (polytope) Novikov chain complex groups of B ⊆ A .By definition we get the inclusionCN ● ( θ a , A ) ⊆ CN ● ( θ a , A ∣ B ) , for all subpolytopes B ⊆ A . The next lemma asserts that CN ● ( θ a , A ) is uniquely determined by the vertices of A . Inother words, one only needs to check the multi finiteness condition for the finitely manyvertices a l . This is a straightforward adaptation of [26, Lemma 7.3]. Lemma 2.7.
Let θ ∶ A → Ω ( M ) be as above. Then CN i ( θ a , A ) = ⋂ b ∈ A ̂ V i ( θ a , A ) b = k ⋂ l = ̂ V i ( θ a , A ) a l = k ⋂ l = CN i ( θ a , A ∣ a l ) , ∀ a ∈ A . More generally, for every subpolytope B ⊆ A spanned by b j = a l j : CN i ( θ a , A ∣ B ) = ⋂ j CN i ( θ a , A ∣ b j ) , ∀ a ∈ A . One can play a similar game with the Novikov rings:
Definition 2.8.
Define the (polytope) Novikov ring Λ A = ⋂ b ∈ A ̂ Z [ Γ A ] b , where ̂ Z [ Γ A ] b denotes the upward completion of the group ring Z [ Γ A ] with respect to theperiod homomorphism Φ b ∶ Γ A → R . Analogously, for every subpolytope B ⊆ A we definethe restricted polytope Novikov ringΛ A∣ B = ⋂ b ∈ B ̂ Z [ Γ A ] b . s before we get Λ A ⊆ Λ A∣ B , for all subpolytopes B ⊆ A . The obvious analogue to Lemma 2.7 holds for Novikov rings as well. These rings enableus to view the polytope Novikov chain complexes as finite Novikov-modules just as in theordinary setting (2).Next we try to equip the groups CN ● ( θ a , A ) with a boundary operators that turnsthem into a genuine chain complex. The obvious candidate would be ∂ θ a ∶ CN ● ( θ a , A ) → CN ●− ( θ a , A ) , ∂ θ a ξ ∶ = ∑ ˜ x, ˜ y ξ ˜ x ⋅ alg M ( ˜ x, ˜ y ; ˜ f θ a ) ˜ y. (4)Note that the moduli space above actually also depends on a choice of metric g , and sodoes the boundary operator ∂ θ a . When we want to keep track of the metric we will writeCN ● ( θ a , g, A ) . For restrictions A ∣ B we define the boundary operator analogously.Formally, the definition of ∂ θ a looks identical to the definition of ∂ on CN ● ( α ) , andmorally it is. However, there are two major differences. Firstly, the cover ̃ M A might differfrom the abelian cover ̃ M a of a . Secondly, it is not clear whether ∂ = ∂ θ a preserves themulti finiteness condition, i.e. whether ∂ξ lies in CN ● ( θ a , A ) . Luckily, we will achieve thisby replacing the original section θ with a perturbed section ϑ ∶ A → Ω ( M ) (cf. Theorem2.14). Whenever the chain complex is defined we make the following definition. Definition 2.9.
Let ϑ ∶ A → Ω ( M ) be a section such that ( CN ● ( ϑ a , g ϑ a , A ) , ∂ ) definesa chain complex. Then we call the induced homology (polytope) Novikov homology and denote it by HN ● ( ϑ a , g ϑ a , A ) or more abusively HN ● ( ϑ a , A ) . Analogously, we define HN ● ( ϑ a , g ϑ a , A ∣ B ) = HN ● ( ϑ a , A ∣ B ) . Remark 2.10.
Analogously to ordinary Novikov homology, one can show that the Novikovhomologies HN ● ( ϑ a , A ) and HN ● ( ϑ a , A ∣ B ) are both finitely-generated modules over theNovikov rings Λ A and Λ A∣ B , respectively, thus generalizing the Novikov-module property.This follows from the fact that the boundary operator (4) is Λ A -linear (and similarly forthe restricted case). In this subsection we state and prove all the technical auxiliary results needed forthe proof of Theorem 2.14, which roughly speaking asserts the well-definedness of thepolytope chain complexes and their respective homologies after modifying the section θ ∶ A → Ω ( M ) to a new section ϑ ∶ A → Ω ( M ) . Notation.
For any (closed) one-form ρ we will denote by ∇ g ρ the dual vector field to ρ with respect to the metric g . Note that with this notation we have ∇ g H = ∇ g dH for anysmooth function H ∶ M → R . roposition 2.11. Let ( ρ, g ) be a Morse-Smale pair. Then for every δ > there exists aconstant C ρ = C ρ ( δ, g ) > such that ∥ ∇ g ρ ( z )∥ < C ρ Ô⇒ ∃ x ∈ Z ( ρ ) with d ( x, z ) < δ, where both ∥ ⋅ ∥ and d ( ⋅ , ⋅ ) are induced by g .Proof. Suppose the assertion does not hold. Then there exists a δ >
0, a positive sequence C k → ( z k ) ⊂ M such that ∥ ∇ g ρ ( z k )∥ < C k and z k ∈ M ∖ ⋃ x ∈ Z ( ρ ) B δ ( x ) . By compactness of M we can pass to a subsequence ( z k ) converging to some z ∈ M . Theabove however implies ∥ ∇ g ρ ( z )∥ =
0, which is equivalent to z ∈ Z ( ρ ) . At the same time z lies in M ∖ ⋃ x ∈ Z ( ρ ) B δ ( x ) , which is a contradiction. This concludes the proof. Notation.
Such a constant C ρ > Palais-Smale constant (short:
PS-constant ). The main case of interest is the exact one, i.e. ρ = dH , for which we willabbreviate C dH = C H . Sometimes we will also abbreviate C ρ = C .The next Lemma builds the main technical tool of Subsection 2.4. The idea is toperturb one-forms α close to a given reference Morse-Smale pair ( ρ, g ) so that the per-tubations, say α ′ , maintain their cohomology classes of α , become Morse, have the samezeros as ρ , and are still relatively close to ρ . This is reminiscent of Zhang’s arguments[26, Section 3]. Lemma 2.12.
Let ( ρ, g ) be a Morse-Smale pair, δ > so small that the balls B δ ( x ) ,with x ∈ Z ( ρ ) , are geodisically convex and lie in pairwise disjoint charts of M , and C = C ρ ( δ, g ) > as in Proposition 2.11.Let α ∈ Ω ( M ) with ∥ α − ρ ∥ < C and a = [ α ] , where ∥ ⋅ ∥ is the norm induced by g . Then there exists a Morse-Smale pair ( α ′ , g ′ ) , with α ′ ∈ a , satisfying • ∥ α ′ − ρ ∥ ≤ ⋅ ∥ α − ρ ∥ , • α ′ ∣ B δ ( x ) = ρ ∣ B δ ( x ) for all x ∈ Z ( ρ ) and • Z ( α ′ ) = Z ( ρ ) .Moreover ∥ ∇ g ′ α ′ ( z )∥ ′ < C implies z ∈ B δ ( x ) for some zero x ∈ Z ( α ′ ) , where ∥ ⋅ ∥ ′ is thenorm induced by g ′ . This is a well known result in Riemannian geometry, see [25] This is still the ball of radius δ with respect to the distance metric induced by g . roof. Since ρ is a Morse form, there are only finitely many zeros x ∈ Z ( ρ ) . Around eachsuch x we will perturb α without changing its cohomology class: Enumerate the finitelymany zeros of ρ by { x i } i = ,...,k and pick a bump functions h i ∶ M → R with ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ h i ≡ , on M ∖ B δ ( x i ) h i ≡ , on B δ ( x i )∥ ∇ g h i ∥ ≤ δ . Since every B δ ( x i ) is simply connected, there exist unique smooth functions f i ∶ B δ ( x i ) → R satisfying f i ( x i ) = df i = ( α − ρ )∣ B δ ( x i ) . (5)We set α ′ = α − k ∑ i = d ( h i ⋅ f i ) . (6)By construction we have α ′ ∈ a , α ′ = ρ on B δ ( x ) for x ∈ Z ( ρ ) , and that α ′ agrees with ρ outisde of ⋃ ki = B δ ( x i ) . Consequently α ′ − ρ and α − ρ agree outside of ⋃ ki = B δ ( x i ) . Thismeans that for the inequality in the first bullet point it suffices to argue why the boundholds inside each ball B δ ( x i ) . Inserting the definitions grants ∥ α ′ − ρ ∥ B δ ( x i ) = ∥ α − ρ − h i ⋅ df i − f i ⋅ dh i ∥ B δ ( x i ) ≤ ( − h i )∥ α − ρ ∥ B δ ( x i ) + ∥ f i ∥ B δ ( x i ) ⋅ ∥ ∇ g h i ∥ B δ ( x i ) ≤ ∥ α − ρ ∥ B δ ( x i ) + δ ⋅ ∥ f i ∥ B δ ( x i ) Recall that f i was chosen such that f i ( x i ) =
0. Due to the geodesic convexity of the balls B δ ( x i ) we can apply the mean value inequality ∣ f i ( y )∣ = ∣ f i ( x i ) − f i ( y )∣ ≤ ∥ ∇ g f i ∥ B δ ( x i ) ⋅ d ( x, y ) ≤ ∥ α − ρ ∥ B δ ( x i ) ⋅ δ, ∀ y ∈ B δ ( x i ) . All in all this implies ∥ α ′ − ρ ∥ B δ ( x i ) ≤ ∥ α − ρ ∥ B δ ( x i ) + δδ ∥ α − ρ ∥ B δ ( x i ) = ⋅ ∥ α − ρ ∥ B δ ( x i ) . This proves the first inequality in the first bullet point. From this we will deduce that Z ( ρ ) = Z ( α ′ ) : the inclusion Z ( ρ ) ⊆ Z ( α ′ ) is clear as ρ agrees with α ′ around Z ( ρ ) . Thereverse inclusion is obtained by observing that for y ∈ Z ( α ′ ) we have ∥ ∇ g ρ ( y )∥ = ∥ ∇ g ρ ( y ) − ∇ g α ′ ( y )´¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¶ = ∥ ≤ ∥ ρ − α ′ ∥ ≤ ⋅ ∥ α − ρ ∥ < C, by assumption on ρ and the inequality above. Proposition 2.11 then implies that z hasto be a zero of ρ as well. This proves Z ( ρ ) = Z ( α ′ ) , in particular that α ′ is a Morse form.To get a Riemannian metric g ′ that turns ( α ′ , g ′ ) into a Morse-Smale pair it sufficesto perturb g on an open set that intersects all the Novikov trajectories of ( α ′ , g ) , see [20, age 38-40] for more details. Since Z ( ρ ) = Z ( α ′ ) , we can take a perturbation g ′ thatagrees with g on M ∖ ⋃ ki = B δ ( x i ) and is close to g in the C ∞ -topology.The last assertion of the statement follows from the observation that, for α ′ fixed, themap g ′ ↦ ∥ ∇ g ′ α ′ ∥ ′ is continuous, thus for g ′ close to g we get ∥ ∇ g α ′ ( z )∥ ≤ ∣∥ ∇ g α ′ ( z )∥ − ∥ ∇ g ′ α ′ ( z )∥ ′ ∣´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ < ε + ∥ ∇ g ′ α ′ ( z )∥ ′ . Assuming ∥ ∇ g ′ α ′ ( z )∥ ′ < C we thus end up with ∥ ∇ g ρ ( z )∥ ≤ ∥ ∇ g ρ ( z ) − ∇ g α ′ ( z )∥ + ∥ ∇ g α ′ ( z )∥ ≤ ∥ ρ − α ′ ∥ + ε + C , ≤ ⋅ ∥ ρ − α ∥ + ε + C , < ⋅ C + ε + C ε ≤ C and invoking Proposition 2.11 then concludes the proof.Lemma 2.12 can be applied to a whole section θ ∶ A → Ω ( M ) nearby a reference Morse-Smale pair ( ρ, g ) and give rise to a perturbed section ϑ ∶ A → Ω ( M ) that is still relativelyclose to ρ , so that each ϑ a agrees with ρ near the zeros x ∈ Z ( ρ ) . Proposition 2.13.
Let ( ρ, g ) and C = C ρ > as in Lemma 2.12, denote N = ⋃ i B δ ( x i ) with x i ∈ Z ( ρ ) , and let θ ∶ A → Ω ( M ) be a section satisfying ∥ θ a − ρ ∥ < C . (7) Then there exists a section ϑ = ϑ ( θ, ρ, g ) ∶ A Ð→ Ω ( M ) and a positive constant D = D ( N, g ) > with the following significance: • Z ( ϑ a ) = Z ( ρ ) for all a ∈ A , • ϑ a ∣ B δ ( x i ) = ρ ∣ B δ ( x i ) for all x i ∈ Z ( ρ ) , a ∈ A .Moreover, for every ϑ a there exists a Riemannian metric g ϑ a close to g with g ϑ a ∣ M ∖ N = g ∣ M ∖ N such that • ( ϑ a , g ϑ a ) is Morse-Smale, • ∥ ϑ b − ρ ∥ ϑ a ≤ D ⋅ ∥ ϑ b − ρ ∥ ≤ ⋅ D ⋅ ∥ θ b − ρ ∥ for all a, b ∈ A , where ∥ ⋅ ∥ ϑ a is the operator norm induced by g ϑ a . The choice of D > roof. Since the whole section θ ∶ A → Ω ( M ) is C -close to ( ρ, g ) , we can take ( ρ, g ) asa reference pair and apply Lemma 2.12 to every θ a and denote ϑ a the correspondingperturbation. Recall from the proof of Lemma 2.12 that ϑ a is obtained by an exactperturbation of θ a around the zeros of ρ – a closer inspection reveals that this exactperturbation varies smoothly along θ a , in particular that ϑ defines a smooth section. Thefirst two bullet points follow immediately from Lemma 2.12.We choose g ϑ a = g ( θ a ) ′ just as g ′ in Lemma 2.12, i.e. by means of a small perturbationof g inside N . The argument in [20] shows that sufficiently small perturbations give riseto Riemannian metrics that are uniformly equivalent to the original g , in other words wemay choose g ϑ a such that ( ϑ a , g ϑ a ) is Morse-Smale and1 D g ϑ a ( v, v ) ≤ g ( v, v ) ≤ D g ϑ a ( v, v ) , ∀ v ∈ T M, ∀ a ∈ A , with D > N and g . Using this inequality and invokingthe first bullet point of Lemma 2.12 concludes the proof. We can finally state and prove Theorem 2.14 by applying the previous results in thespecial case of exact reference pairs:
Theorem 2.14.
Let θ ∶ A → Ω ( M ) be a section and a reference Morse-Smale pair ( H, g ) on M . Then there exists a perturbed section ϑ = ϑ ( θ, H, g ) ∶ A Ð→ Ω ( M ) and a choice of Riemannian metrics g ϑ a with the following significance: • (Morse-Smale property) Each pair ( ϑ a , g ϑ a ) is Morse-Smale, for all a ∈ A , • (Chain-complex) The chain complex ( CN ● ( ϑ a , g ϑ a , A ) , ∂ ϑ a ) is well defined forevery pair ( ϑ a , g ϑ a ) as above, • (Ray-invariance) The chain complexes are equal upon scaling, i.e. CN ● ( ϑ a , g ϑ a , A ) = CN ● ( r ⋅ ϑ a , g ϑ a , r ⋅ A ) for all r > , a ∈ A . The rough idea is to “shift-and-scale” : we shift and scale the polytope A so that itis sufficiently close to a given exact one-form dH in the operator norm ∥ ⋅ ∥ coming from g . Then one can perturb the scaled section by means of Proposition 2.13 and scale back.This will be the desired section ϑ on A . By construction we will then see that the threebullet points are satisfied. The choices involved (i.e. choice of section θ , reference pair ( H, g ) and perturbation coming from Theorem 2.14) will prove harmless – they result inchain homotopy equivalent complexes. This is proven in the next subsection (cf. Theorem2.19). Note that here the metric is not scaled. t the cost of imposing a smallness condition on the underlying section, we get thesame results for perturbations associated to non-exact reference pairs (cf. Corollary 2.17)and the same independence of auxiliary data holds (cf. Theorem 2.22). Proof of Theorem 2.14.
As a first candidate for ϑ , we pick ϑ ∶ A Ð→ Ω ( M ) , ϑ a ∶ = θ a + dH. This is still a section, but does not satisfy the bullet points above. Since θ is smooth,there exists ε = ε ( θ, H, g, δ ) > , such that ε ⋅ θ a + dH is C H D ⋅ dH, with respect to ∥ ⋅ ∥ induced by g , C H = C H ( δ, g ) > D = D ( N, g ) > ε ⋅ a ↦ ε ⋅ θ a + dH andobtain a new section ϑ ε ∶ ε ⋅ A Ð→ Ω ( M ) . . Finally we scale back and redefine ϑ = ϑ ( θ, H, g, ε ) ∶ A Ð→ Ω ( M ) , a ↦ ε ⋅ ϑ ε ( ε ⋅ a ) . Thus we have ε ⋅ ϑ a = ϑ ε ( ε ⋅ a ) , ∀ a ∈ A . For each ϑ ε ( ε ⋅ a ) we choose a Riemannian metric denoted by g a as in Proposition 2.13.Thus ( ϑ ε ( ε ⋅ a ) , g a ) is Morse-Smale, and so is ( ϑ a , g a ) since scaling does not affect theMorse-Smale property. This proves the first bullet point. Claim. CN ● ( ϑ ε ( ε ⋅ a ) , ε ⋅ A ) = CN ● ( ε ⋅ ϑ a , ε ⋅ A ) is a well-defined chain complex for any a ∈ A .Indeed, assume for contradiction that there exists a Novikov-chain ξ = ∑ ˜ x ξ ˜ x ˜ x ∈ CN ● ( ϑ ε ( ε ⋅ a ) , ε ⋅ A ) such that ∂ξ ∉ CN ● ( ϑ ε ( ε ⋅ a ) , ε ⋅ A ) . This means that there is some ε ⋅ b ∈ ε ⋅ A , c ∈ R and sequences ˜ x n with ξ ˜ x n ≠
0, ˜ y n pairwisedistinct, ˜ γ n ∈ M ( ˜ x n , ˜ y n ; ˜ f ε ⋅ ϑ a ) , and˜ f ϑ ε ( ε ⋅ b ) ( ˜ y n ) = ˜ f ε ⋅ ϑ b ( ˜ y n ) ≥ c, Explicitly, this section is of the form ε ⋅ a ↦ ε ⋅ θ a + dH + ∑ i d ( f i ⋅ h i ) , where f i depends smoothly on θ a and satisfies df i = ε ⋅ θ a , f i ( x i ) = x i of H , see Lemma2.12, (5) and (6) applied to ε ⋅ θ a + dH and ρ = dH . ee Remark 2.5. Denote by γ n = π ○ ˜ γ n the Novikov trajectories downstairs. The energyexpression can then be massaged as follows:0 ≤ E ( ˜ γ n ) = E ( γ n ) = − ∫ γ n ϑ ε ( ε ⋅ a ) = − ∫ γ n ϑ ε ( ε ⋅ b ) + ∫ γ n ϑ ε ( ε ⋅ b ) − ϑ ε ( ε ⋅ a ) = ˜ f ϑ ε ( ε ⋅ b ) ( ˜ x n ) − ˜ f ϑ ε ( ε ⋅ b ) ( ˜ y n ) + ∫ γ n ϑ ε ( ε ⋅ b ) − ϑ ε ( ε ⋅ a ) ≤ ˜ f ϑ ε ( ε ⋅ b ) ( ˜ x n ) − c + ∫ γ n ϑ ε ( ε ⋅ b ) − ϑ ε ( ε ⋅ a ) . Showing that the rightmost term is bounded by m ⋅ E ( γ n ) , m ∈ ( , ) suffices to obtain acontradiction: admitting such a bound leads to0 ≤ E ( γ n ) ≤ ( − m ) − ⋅ ( ˜ f ϑ ε ( ε ⋅ b ) ( ˜ x n ) − c ) . In particular, c ≤ ˜ f ϑ ε ( ε ⋅ b ) ( ˜ x n ) for all n . But ξ belongs to CN ● ( ε ⋅ ϑ a , ε ⋅ A ) and ξ ˜ x n ≠ x n .Up to passing to a subsequence we can therefore assume ˜ x n = ˜ x and also ˜ y n ∈ π − ( y ) . The corresponding Novikov trajectories γ n ∈ M ( x, y ; ϑ ε ( ε ⋅ a )) have uniformly bounded energy E ( γ n ) ≤ ( − m ) − ⋅ ( ˜ f ϑ ε ( ε ⋅ b ) ( ˜ x ) − c ) , therefore γ n has a C ∞ loc -convergent subsequence. At the same time M ( x, y ; ϑ ε ( ε ⋅ a )) isa 0-dimensional manifold, which means that the convergent subsequence γ n eventuallydoes not depend on n . This contradicts our assumption that the endpoints ˜ y n upstairsare pairwise disjoint.Therefore we are only left to show the bound A ( γ n ) ∶ = ∫ γ n ϑ ε ( ε ⋅ b ) − ϑ ε ( ε ⋅ a ) ≤ E ( γ n ) to conclude the Claim. For this purpose we define S n ∶ = { s ∈ R ∣ ∥ ∇ g a ( ϑ ε ( ε ⋅ a )) ( γ n ( s ))∥ g a ≥ C H } . The crucial observation is that both ϑ ε ( ε ⋅ b ) and ϑ ε ( ε ⋅ a ) agree with dH around Crit ( H ) ,by choice of ϑ ε via Proposition 2.13. In particular ϑ ε ( ε ⋅ b ) − ϑ ε ( ε ⋅ a )∣ B δ ( z ) = , ∀ z ∈ Crit ( H ) = Z ( ϑ ε ( ε ⋅ b )) = Z ( ϑ ε ( ε ⋅ a )) . The latter is possible since Z ( ε ⋅ ϑ a ) = Z ( ε ⋅ dH ) = Crit ( H ) is finite. emma 2.12 says that for s ∈ R ∖ S n we get γ n ( s ) ∈ ⋃ z ∈ Z ( ϑ ε ( ε ⋅ a )) B δ ( z ) , consequently ∫ R ∖S n ( ϑ ε ( ε ⋅ b ) − ϑ ε ( ε ⋅ a )) ˙ γ n ( s ) ds = . (8)The Lebesgue measure µ ( S n ) can be bounded using the energy: E ( γ n ) = − ∫ γ n ϑ ε ( ε ⋅ a ) = ∫ R ∥ ∇ g a ( ϑ ε ( ε ⋅ a )) ( γ n ( s ))∥ g a ds ≥ µ ( S n ) ⋅ ( C H ) , thus µ ( S n ) ≤ ( C H ) ⋅ E ( γ n ) . (9)And finally ∣ A ( γ n )∣ ≤ ∥ ϑ ε ( ε ⋅ b ) − ϑ ε ( ε ⋅ a )∥ g a ⋅ ∫ S n ∥ ˙ γ n ( s )∥ g a ds by (8) , ≤ (∥ ϑ ε ( ε ⋅ b ) − dH ∥ g a ++ ∥ dH − ϑ ε ( ε ⋅ a )∥ g a ) ⋅ µ ( S n ) ⋅ E ( γ n ) ≤ D ⋅ (∥ ε ⋅ ϑ b − dH ∥ ++ ∥ dH − ε ⋅ ϑ a ∥) ⋅ C H ⋅ E ( γ n ) Proposition 2 . , (9) , ≤ ⋅ D ⋅ C H D ⋅ ⋅ C H ⋅ E ( γ n ) by choice of scaling ε > , < E ( γ n ) . This proves the Claim.Now we observe that scaling ϑ a by r > ( ϑ a , g a ) are in one to one correspondence with those of ( r ⋅ ϑ a , g a ) .It is also clear that the multi finiteness condition imposed by A is equivalent to that of r ⋅ A . All in all this means that for any r > ( r ⋅ ϑ a , g a ) agree with each other. This proves the ray-invariance. Setting r = ε and usingthe Claim proves the remaining first bullet point. Remark 2.15.
Instead of running the argument for the sections ϑ ε ∶ ε ⋅ A → Ω ( M ) wecould also work with ϑ = ε ⋅ ϑ ε ∶ A → Ω ( M ) by directly by applying Proposition 2.13 tothe section a ↦ θ a + d ( ε − H ) and ( ε − H, g ) . These two approaches are equivalent, the only difference is psychological:we find it more natural to visualize the shrinking of the polytope opposed to the scaling f Morse functions. Note that the analogous bound at the end of the proof of Theorem2.14 holds upon replacing H by ε − H . This follows from the nice scaling behaviour of thePS-constants: C ε − H ( δ, g ) = ε − C H ( δ, g ) , therefore “ ε ⋅ θ a + dH is C H D ⋅ -close to dH ” if and only if “ θ a + d ( ε − H ) is C ε − H D ⋅ = C H ε ⋅ D ⋅ -close to d ( ε − H ) ”. Remark 2.16.
The skeptical reader might wonder whether ∂ ϑ a = ● ( ϑ a , A) as a certain twisted chain group allows for a quick and simpleproof – see Remark 2.35.The key in the proof of Theorem 2.14 was to obtain control over the energy by per-turbing the section θ ∶ A → Ω ( M ) via Proposition 2.13. The perturbation was chosen sothat there would be no contribution to the energy near the zeros. Similar ideas to controlthe energy can be found in [3, Subsection 3.6.2], [26].The question remains why we used an (exact) reference pair ( H, g ) instead of a moregeneral Morse-Smale pair ( ρ, g ) in Theorem 2.14. The answer is simple: the given ar-gument already breaks down in the very first line – the corresponding ϑ is not a sectionanymore, since the ρ -shift changes the cohomology class. However, whenever the sec-tion θ ∶ A → Ω ( M ) is already sufficiently close to ( ρ, g ) in terms of the correspondingPS-constant C ρ >
0, we do not need to shift and scale θ , and can perturb θ directly: Corollary 2.17.
Let ( ρ, g ) be a Morse-Smale pair, C = C ρ > , N = ⋃ i B δ ( x i ) with x i ∈ Z ( ρ ) , and D = d ( N, δ ) > as in Proposition 2.13. Let θ ∶ A → Ω ( M ) be a smoothsection such that ∥ θ a − ρ ∥ < C ρ D ⋅ , with ∥ ⋅ ∥ the operator norm induced by g . Then there exists a perturbed section ϑ = ϑ ( θ, ρ, g ) ∶ A Ð→ Ω ( M ) , (10) and g ϑ a such that the same conclusions as in Theorem 2.14 hold.Proof. Upon replacing ε ⋅ θ a + dH and dH with θ a and ρ , the proof is word for word thesame as the one of Theorem 2.14. The section ϑ = ϑ ( θ, H, g ) constructed in Theorem 2.14 does not only depend on ( θ, H, g ) , but also comes with a choice of scaling ε ( θ, H, g, δ ) >
0. We shall prove that anyvalid perturbation ϑ i = ϑ i ( θ i , H i , g i , ε i ) in the sense of Theorem 2.14 gives rise to chainhomotopy equivalent chain complexes. The same is true for perturbations coming fromCorollary 2.17 and at the end of the subsection we will show that both perturbations leadto chain homotopy equivalent Novikov complexes. emark 2.18. All the chain maps and chain homotopy equivalences constructed fromhere on are Novikov-module morphisms, i.e. linear over the Novikov ring. We will notexplicitly state this every time for better readability.
Theorem 2.19.
For i = , , let θ i ∶ A → Ω ( M ) be sections, ( H i , g i ) Morse-Smale pairsand δ i > , ε i ( θ i , H i , g i , δ i ) > as in proof of Theorem 2.14.Then any two perturbed sections ϑ i = ϑ i ( θ i , H i , g i , ε i ) ∶ A → Ω ( M ) , (11) in the sense of Theorem 2.14, induce chain homotopy equivalent polytope complexes: CN ● ( ϑ a , A) ≃ CN ● ( ϑ a , A) , ∀ a ∈ A . (12) Proof of Theorem 2.19.
We may assume ε ≥ ε . Denote by ϑ i ( σ i , H i , g i , ε i ) , i = , , the respective sections on A as in the first part of the proof of Theorem 2.14. Let h ∶ [ , ] → R (13)be a smooth function with h ≡ ( −∞ , e ) and h ≡ ( − e, +∞ ) , for some small e > ϑ s = ( − h ( s )) ⋅ ϑ + h ( s ) ⋅ ϑ . Fix a ∈ A and pick g i ∶ = g ϑ ia , i = , g s = g s ( a ) be ahomotopy of Riemannian metrics connecting g to g and assume that ( ϑ sa , g s ) is regular– this is rectified by Remark 2.20 below. Note that here g s actually depends on a .To this regular homotopy we can now associate a chain continuationΨ ∶ CN ● ( ϑ a , A) Ð→ CN ● ( ϑ a , A) , ξ = ∑ ˜ x ξ ˜ x ˜ x ↦ ∑ ˜ x, ˜ y ξ ˜ x ⋅ alg M ( ˜ x, ˜ y ; ˜ f ϑ sa ) ˜ y. (14)Analogously to the case of the boundary operator in Theorem 2.14, proving that Ψ defines a well defined Novikov chain map essentially boils down to proving that it respectsthe multi finiteness condition – the rest follows by standard Novikov-Morse techniques.Thus, proceeding as in Theorem 2.14 reveals that it suffices to bound ∫ γ n ϑ sb − ϑ sa (15) This is also implicitly using that ˜ f ϑ sb = ˜ f ϑ b + h s ○ π for a smooth family h s ∈ C ∞ ( M ) since ϑ sb are cohomologousfor all s . Hence − ∫ γ n ϑ sb = − ˜ f ϑ b ( ˜ y n ) + ˜ f ϑ b ( ˜ x n ) + ∫ [ , ] ∂h s ∂s ( γ n ( s )) ds ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ≤ C , for some uniform constant C . y either a multiple m ∈ ( , ) of the energy E ( γ n ) , where b is some cohomology class in A , or a uniform bound altogether. Set S n ∶ = { s ∈ ( −∞ , ) ∣ ∥ ∇ g ϑ a ( γ n ( s ))∥ g ≥ C H ε ⋅ } , S n ∶ = { s ∈ ( , +∞ ) ∣ ∥ ∇ g ϑ a ( γ n ( s ))∥ g ≥ C H ε ⋅ } . This time around we need to divide by ε i as we are running the continuation directlyon the original polytope A instead of the scaled polytope (see Remark 2.15). As in theprevious proof of Theorem 2.14, the s ∈ R ≤ ∖ S n and s ∈ R ≥ ∖ S n do not contribute to (15)as γ n ( s ) will be near the zeros of H i , where ϑ ib = ϑ ia . On the other hand, using similararguments we obtain ∣ ∫ S n ∪S n ( ϑ sb − ϑ sa ) ˙ γ n ( s ) ds ∣ ≤ ⋅ max i = , ∥ ϑ ib − ϑ ia ∥ i ⋅ µ ( S in ) ⋅ E ( γ n ) ≤ ⋅ max i = , (∥ ϑ ib − dH i ∥ i + ∥ dH i − ϑ ia ∥ i ) ⋅ ε i ⋅ C H ⋅ E ( γ n ) ≤ max i = , ⋅ ⋅ D ⋅ ε i ⋅ ⋅ C H D ⋅ ε i ⋅ ⋅ C H ⋅ E ( γ n ) ≤ ⋅ E ( γ n ) , where we have used Proposition 2.13 as in Theorem 2.14. We are left to bound (15) for s ∈ [ , ] . For this we compute via Cauchy-Schwarz: ∣ ∫ [ , ] ( ϑ sb − ϑ sa ) ˙ γ n ( s ) ds ∣ ≤ max s ∈ [ , ] ∥ ϑ sb − ϑ sa ∥ s ⋅ E ( γ n ) . By compactness of [ , ] , A and continuity of ϑ ∶ [ , ] × A → Ω ( M ) , we may boundmax s ∈ [ , ] ∥ ϑ sb − ϑ sa ∥ s ≤ F, where F > s ∈ [ , ] and b ∈ A – recall that g s depends on a ,but that does not matter. In particular, this proves ∣ ∫ γ n ϑ sb − ϑ sa ∣ ≤ F ⋅ E ( γ n ) + ⋅ E ( γ n ) , ∀ n ∈ N , b ∈ A . A case distinction now does the job: for any n ∈ N we either have E ( γ n ) ≥ F ⋅ E ( γ n ) or E ( γ n ) < F ⋅ E ( γ n ) . In the first case we can bound the norm of (15) by ⋅ E ( γ n ) ,whereas in the second case we get ⋅ E ( γ n ) < F and thus we may bound the norm of(15) by 10 F . This proves ∣ ∫ γ n ϑ sb − ϑ sa ∣ ≤ max { F , E ( γ n )} . Uniform in b and n ∈ N , that is. s explained before, this suffices to conclude that Ψ defines a well defined Novikov chainmap, which defines the desired chain homotopy equivalence (see proof of Theorem 2.14for more details). This concludes the proof. Remark 2.20.
The (linear) homotopy ( ϑ sa , g s ) chosen in the proof of Theorem 2.19might be non-regular. One can replace ( ϑ sa , g s ) with an arbitrarily close regular homo-topy (( ϑ sa ) ′ , g ′ s ) connecting the same data. The only bit where this affects the previousargument in Theorem 2.19 is when trying to bound max s ∈ [ , ] ∥ ϑ sb − ( ϑ sa ) ′ ∥ ′ s . By using that ϑ sa is smooth in s and close to ( ϑ sa ) ′ , we still get the desired uniform bound b .As a consequence of Theorem 2.14 and Theorem 2.19 we obtain the analogue resultsfor restrictions to subpolytopes B ⊆ A : Corollary 2.21.
Let θ ∶ A → Ω ( M ) be a section. Then for every perturbed section ϑ ∶ A → Ω ( M ) coming from Theorem 2.14 and subpolytope B ⊆ A we obtain a well defined polytopechain complex ( CN ● ( ϑ a , A∣ B ) , ∂ ϑ a ) , ∀ a ∈ A , (16) satisfying all the bullet points of Theorem 2.14. Any other choice ϑ ′ = ϑ ′ ( θ ′ , H ′ , g ′ ) doesnot affect the chain complexes up to chain homotopy equivalence.Moreover, the inclusion ι B ∶ CN ● ( ϑ a , A) Ð→ CN ● ( ϑ a , A∣ B ) (17) defines a Novikov-linear chain map for all a ∈ A .Proof. The proof of the first part is literally the same as in Theorem 2.14 and Theorem2.19. To see that the inclusion defines a chain map it suffices to observe that both bound-ary operators in (17) are identical upon restricting to the smaller complex CN ● ( ϑ a , A) .In the preceding subsection we also defined a polytope chain complex variant usingperturbed sections with respect to non-exact reference pairs (cf. Corollary 2.17). Whilethis variant requires the underlying section to satisfy some a priori smallness conditions,it does agree with the polytope chain complex variant of Theorem 2.14. Theorem 2.22.
Let ( ρ, g ) and θ ∶ A → Ω ( M ) be as in Corollary 2.17, and denote by ϑ ρ = ϑ ( θ, ρ, g ) the corresponding perturbed section. Let ϑ H = ϑ ( θ, H, g H , ε ) be any perturbedsection as in Theorem 2.14. Then CN ● ( ϑ ρa , A) ≃ CN ● ( ϑ Ha , A) , ∀ a ∈ A . (18) Let θ i ∶ A → Ω ( M ) be any other two sections i = , with reference pairs ( ρ i , g i ) satisfyingthe conditions of Corollary 2.17. Then for any two choices ϑ ρ i = ϑ ( θ i , ρ i , g i ) one has CN ● ( ϑ ρ a , A) ≃ CN ● ( ϑ ρ a , A) , ∀ a ∈ A . (19) Moreover, both (18) and (19) continue to hold in the restricted case B ⊆ A . roof. The proof idea is again arguing via continuations as in Theorem 2.19 above – wewill use the latter as carbon copy and adapt the same notation. Define ϑ s = ( − h ( s )) ⋅ ϑ ρ + h ( s ) ⋅ ϑ H . By the same logic as in Theorem 2.19, it suffices to control the expression ∫ γ ϑ sb − ϑ sa , (20)for all b ∈ A , in order to get the desired continuation chain map to conclude (18). For thispurpose we define S ρ ∶ = { s ∈ ( −∞ , ) ∣ ∥ ∇ g ϑ ρa ( γ ( s ))∥ g ≥ C ρ } , S H ∶ = { s ∈ ( , +∞ ) ∣ ∥ ∇ g ϑ H a ( γ ( s ))∥ g ≥ C H ε ⋅ } . Here g and g (abusively) denote Riemannian metrics g ϑ ρa and g ϑ Ha coming from Propo-sition 2.13.Observe that by assumption and choice of ( ϑ ρa , g ) we have ∥ ϑ ρb − ρ ∥ g ≤ ⋅ D ⋅ ∥ θ b − ρ ∥ g ≤ ⋅ D ⋅ C ρ D ⋅ , see proof of Corollary 2.17 and Proposition 2.13.Just as in the proof of Theorem 2.19 there is no contribution to (20) for s in thecomplement of [ , ] ∪ S ρ ∪ S H . At the same time, we can again bound ∣ ∫ S α ( ϑ sb − ϑ sa ) ˙ γ ( s ) ds ∣ < ⋅ E ( γ ) , ∣ ∫ S α ( ϑ sb − ϑ sa ) ˙ γ ( s ) ds ∣ < ⋅ E ( γ ) , and ∣ ∫ [ , ] ( ϑ sb − ϑ sa ) ˙ γ ( s ) ds ∣ ≤ F ⋅ E ( γ ) . This suffices to obtain the desired control over (20) and proves (18), see proof of Theorem2.19 for more details. Last but not least, (19) follows by applying (18) twice:CN ● ( ϑ ρ a , A) ≃ CN ● ( ϑ Ha , A) ≃ CN ● ( ϑ ρ a , A) , ∀ a ∈ A . Theorem 2.19 and Theorem 2.22 readily imply:
Corollary 2.23.
Let θ i ∶ A → Ω ( M ) with i = , be two sections and ϑ i associatedperturbations as in Theorem 2.14 (or Corollary 2.17). Then the resulting polytope Novikovhomologies are isomorphic: HN ● ( ϑ a , A) ≅ HN ● ( ϑ a , A) , ∀ a ∈ A . Corollary 2.23 is the analogue to the independence of Morse-Smale pairs ( α, g ) in thecase of ordinary Novikov homology. The latter can also be recovered from the formerby taking the trivial polytope A = ⟨ a ⟩ . Nevertheless, keeping track of the section θ , orrather its perturbations, will prove useful, especially when establishing the commutativediagram in the Main Theorem 2.24. .6 Non-exact deformations and proof of the Main Theo-rem The power of the polytope machinery will become evident in this subsection – roughlyspeaking, the notion of polytopes allows us to compare the Novikov homologies comingfrom two different cohomology classes, see Main Theorem 2.24. In Section 3 we presentsome applications of the Main Theorem 2.24.
Theorem 2.24 (Main Theorem) . Let A ⊂ H ( M ) be a polytope and θ ∶ A → Ω ( M ) asection. Then there exists a perturbation ϑ ∶ A → Ω ( M ) of θ such that for every subpoly-tope B ⊆ A and any two cohomology classes a, b ∈ A there exists a commutative diagram CN ● ( ϑ a , A) CN ● ( ϑ b , A) CN ● ( ϑ a , A∣ B ) CN ● ( ϑ b , A∣ B ) . ι B ≃ ι B ≃ where the horizontal maps are Novikov-linear chain homotopy equivalences. In particular HN ● ( ϑ a , A) HN ● ( ϑ b , A) HN ● ( ϑ a , A∣ B ) HN ● ( ϑ b , A∣ B ) . ≅≅ with all the maps being Novikov-linear.Proof. Most of the ideas have already been established in the previous subsection, espe-cially in the proof of Theorem 2.14 and Theorem 2.19. Fix a reference Morse-Smale pair ( H, g ) and pick a perturbation ϑ = ϑ ( θ, H, g, ε ) as in Theorem 2.14. Let h ∶ [ , ] → R bea smooth function as in (13) and define a homotopy ϑ sab ∶ = ( − h ( s )) ⋅ ϑ a + h ( s ) ⋅ ϑ b , ∀ s ∈ R . Pick g s a smooth homotopy connecting the two metrics g ϑ a and g ϑ b and assume that ( ϑ sab , g s ) is regular (see Remark 2.20). The idea now is to show that the chain continuationmap Ψ ba ∶ CN ● ( ϑ a , g ϑ a , A) Ð→ CN ● ( ϑ b , g ϑ b , A) associated to the regular homotopy ( ϑ sab , g s ) is well defined and makes the desired diagramcommute. The argument that Ψ ba is a well defined Novikov chain map is the same as inTheorem 2.19 and follows by controlling terms of the form ∫ γ ϑ c − ϑ sab , with c ∈ A . Note that this time around we do not need to put an s -dependence on ϑ c (this correspondsto ϑ b in the proof of Theorem 2.19), since the endpoints of ϑ sab have the same zeros as ϑ c , namely Z ( ϑ c ) = Crit ( H ) . The s -dependence on ϑ c is the only bit that used the ohomologous assumption in Theorem 2.19, and indeed, the remaining part of the proofis verbatim the same and is thus omitted.We use the very same homotopy to define a chain continuationΨ ba ∣ B ∶ CN ● ( ϑ a , A∣ B ) Ð→ CN ● ( ϑ b , A∣ B ) . From this we obtain the following commutative diagram on the chain level:CN ● ( ϑ a , A) CN ● ( ϑ b , A) CN ● ( ϑ a , A∣ B ) CN ● ( ϑ b , A∣ B ) . ι B Ψ ba ι B Ψ ba ∣ B Here the ι B denote the inclusions (17), which are chain maps (cf. Corollary 2.21). Bysymmetry and the standard argument, we get continuations Ψ ab and Ψ ab ∣ B in the oppo-site direction by reversing the underlying regular homotopy. It is also easy to see thatcontinuation maps are linear over the underlying Novikov ring. This proves that the twohorizontal chain maps above define the desired chain homotopy equivalences. In particu-lar, the chain diagram above induces the desired diagram in homology and thus concludesthe proof. Remark 2.25.
In light of Theorem 2.19, Theorem 2.22 and Corollary 2.23 one canupgrade Theorem 2.24 and use different sections on the left and on the right of the thediagram, i.e. CN ● ( ϑ a , A) CN ● ( ϑ b , A) CN ● ( ϑ a , A∣ B ) CN ● ( ϑ b , A∣ B ) . ι B ≃ ι B ≃ The upper and lower chain homotopy equivalences however, do come from compositionsof chain continuations rather than genuine chain continuations.
Throughout this subsection we shall assume that θ ∶ A → Ω ( M ) has already beenperturbed as in Theorem 2.14. We present an alternative description of CN ● ( θ a , A) bymeans of local coefficients. For an extensive treatment of local coefficients we recommend[24] and [2, Chapter 2] in the case of Morse homology. Strictly speaking, we could also use perturbations coming from Corollary 2.17 at the expanse of workingwith small sections. For the sake of exposition we refrain from stating this explicitily. efinition 2.26. Let G ⊆ R be an additive subgroup. Then defineNov ( G ; Z ) ∶ = Nov ( G ) as the ring consisting of formal sums ∑ g ∈ G n g t g with n g ∈ Z , satisfying the finitenesscondition ∀ c ∈ R ∶ { g ∣ n g ≠ , g < c } is finite.Whenever G is the image of a period homomorphism Φ a ∶ π ( M ) → R we writeNov ( a ) ∶ = Nov ( G ) , G = im ( Φ a ) . It turns out that Nov ( a ) is isomorphic to Λ a , where the isomorphism is given by sendinga deck transformation A ∈ Γ a to t Φ a ( A ) – both finiteness conditions match and we obtain: Proposition 2.27.
For any cohomology class a ∈ H ( M ) we have Λ a ≅ Nov ( a ) , as rings. In view of Proposition 2.27 we will also refer to Nov ( a ) as Novikov ring of a . Inspiredby the definition of Nov ( a ) we will now define yet another ring Nov (A) , which will beisomorphic to Λ A almost by definition. Definition 2.28.
Let A ∶ = ⟨ a , . . . , a k ⟩ be a polytope. DefineNov (A ; Z ) ∶ = Nov (A) , as the ring consisting of elements ∑ A ∈ Γ A n A t Φ a ( A ) ⋯ t Φ ak ( A ) k , n A ∈ Z , (21)with a multi finiteness condition ∀ l = , . . . , k, ∀ c ∈ R ∶ { A ∣ n A ≠ , Φ a l ( A ) < c } is finite.Similarly, for any subpolytope B ⊆ A we define a (potentially) larger groupNov (A∣ B ) ⊇ Nov (A) consisting of the same formal sums (21), but with a (potentially) less restrictive multifiniteness condition:For all l such that a l ∈ B and c ∈ R ∶ { A ∣ n A ≠ , Φ a l ( A ) < c } is finite.Analogously to Proposition 2.27 we have: The addition and multiplication of Nov ( G ) are the obvious ones: n g t g + m g t g = ( n g + m g ) t g and ( n g t g ) ⋅ ( n h t h ) ∶ = n g ⋅ n h t g + h . Note that A = B in Γ A if and only if Φ a l ( A ) = Φ a l ( B ) for all l = , . . . , k . roposition 2.29. For any polytope A we have Λ A ≅ Nov (A) as rings. Similarly we obtain Λ A ∣ B ≅ Nov (A∣ B ) as well. Let us mention that these Novikov rings are commutative rings since Γ A is abelian.Indeed, the commutator subgroup of π ( M ) is contained in every kernel ker ( a l ) , in par-ticular ABA − B − ∈ k ⋂ l = ker ( a l ) , thus ABA − B − = ∈ Γ A . Each polytope A comes with a representation ρ A ∶ π ( M, x ) × Nov (A) → Nov (A) , ρ A ( η, λ ) = t − Φ a ( η ) ⋯ t − Φ ak ( η ) k ⋅ λ. Sometimes we will write out Φ a l ( η ) = ∫ η a l . The importance of the minus sign will becomeclear in Definition 2.31. To any such representation one can associate a local coefficientsystem Nov (A) ∶ Π ( M ) → mod Nov ( A ) , which is unique up to isomorphisms of local coefficients. We briefly recall the construc-tion: fix a basepoint x ∈ M and pick for every x ∈ M a homotopy class of paths { η x } relative to the endpoints x and x . SetNov (A)( x ) ∶ = Nov (A) , ∀ x ∈ M. For every homotopy class { γ } relative to the endpoints x and y , we first define a loop γ ∶ = η x ∗ γ ∗ η − y ∶ S → M, based at x , and then defineNov (A)( γ ) ∶ Nov (A)( y ) Ð→ Nov (A)( x ) , Nov (A)( γ )( ⋅ ) ∶ = ρ A ( γ, ⋅ ) . (22)Taking a closer look at (22) reveals that the Novikov ring isomomorphism Nov (A)( γ ) isgiven by multiplication with t − ∫ γ a ⋯ t − ∫ γ a k k ∈ Z [ Γ A ] ⊆ Nov (A) , hence we may also view it as a Z [ Γ A ] -module isomorphism. This stems from the construction of a category equivalence between the fundamental groupoid and thefundamental group of a sufficiently nice topological space. This procedure allows to switch back and forthbetween local coefficients and representations, see [11, Page 17], [2, Chapter 2] and [4] for more details. Note that Γ ( γ ∗ η ) = Γ ( γ ) ○ Γ ( η ) . emark 2.30. Usually local coefficients are considered to take values in the category ofabelian groups and are often called “bundle of abelian groups”. Mapping into mod
Nov ( A ) will allow us to obtain actual Novikov-module isomorphisms at times where working withbundle of abelian groups would merely grant group isomorphisms.With this we define the anticipated twisted Novikov complexes. Definition 2.31.
Let θ ∶ A → Ω ( M ) a section as above. We define the twisted Novikovchain complex groups byCN ● ( θ a , Nov (A)) ∶ = ⊕ x ∈ Z ( θ a ) Nov (A) ⟨ x ⟩ , ∀ a ∈ A . The twisted boundary oparator ∂ = ∂ θ a is defined by ∂ ( λ x ) ∶ = ∑ y, γ ∈ M ( x,y ; θ a ) Nov (A)( γ − )( λ ) y = ∑ y, γ ∈ M ( x,y ; θ a ) t ∫ γ a ⋯ t ∫ γ a k k ⋅ λ y, ∀ λ ∈ Nov (A) . The twisted chain complexes ( CN ● ( θ a , Nov (A∣ B )) , ∂ θ a ) , ∀ a ∈ A are defined analogously. The corresponding twisted Novikov homologies are denotedby HN ● ( θ a , Nov (A)) and HN ● ( θ a , Nov (A∣ B )) , ∀ a ∈ A , B ⊆ A . For every flow line γ ∈ M( x, y ; θ a ) we callNov (A)( γ − ) = t ∫ γ a ⋯ t ∫ γ a k k the Novikov twist of γ . Remark 2.32.
A priori it is not clear that ∂ maps into the prescribed chain complex.We will prove this in the next subsection, see Proposition 2.34.The Novikov twist of γ determines the lifting behaviour of γ . Indeed, if γ , γ are twopaths from x to y with unique lifts ˜ γ ( ) = ˜ γ ( ) = ˜ x , then:˜ γ ( ) = ˜ γ ( ) ⇐⇒ ∫ γ ∗ γ − a l = , ∀ l = , . . . , k ⇐⇒ ∫ γ a l = ∫ γ a l , ∀ l = , . . . , k ⇐⇒ ∫ γ a l = ∫ γ a l , ∀ l = , . . . , k ⇐⇒ t ∫ γ a ⋯ t ∫ γ a k k = t ∫ γ a ⋯ t ∫ γ a k k . This will be key in the next subsection.As the next examples shows, we recover the twisted Morse chain complex and itshomology as a special case. xample 2.33. Pick A = ⟨ , a ⟩ and a section θ ∶ A → Ω ( M ) . ThenNov (A∣ a ) ≅ Nov ( a ) ≅ Λ a , thanks to Proposition 2.27 and Example 2.3. Furthermore, the twisted chain complexCN ● ( θ , Nov (A∣ a )) = CN ● ( θ , Nov ( a )) agrees with the twisted Morse complexCM ● ( h, Nov ( a )) , where θ = dh. In general, the same issues as in subsection 2.2 arises when trying to prove thatthe twisted Novikov complexes are well defined: it is not clear whether ∂ maps into thedesired chain complex. This is a non-issue in the special case of θ = dh , i.e. twisted Morsehomology – the reason is that the 0-dimensional moduli spaces M( x, y ; h ) are compact,hence finite. Compare this to [21]. In the following however, we will see that the twistedchain groups can always be identified with CN ● ( θ a , A) so that ∂ and ∂ agree, which thenresolves the well definedness issue by Theorem 2.14. In other words, the twisted chaincomplex is an equivalent description of the polytope chain complex. Proposition 2.34.
Let θ ∶ A → Ω ( M ) be a section as above. Then the twisted andpolytope Novikov chain groups are isomorphic CN ● ( θ a , Nov (A)) ←→ CN ● ( θ a , A) , (23) as Novikov-modules. The isomorphism is preserved upon restrictions A∣ B , and ∂ = ∂ upto the identification (23) .In particular, the twisted Novikov chain complexes are well defined with HN ● ( θ a , Nov (A)) = HN ● ( θ a , A) and HN ● ( θ a , Nov (A∣ B )) = HN ● ( θ a , A∣ B ) , for all a ∈ A and subpolytopes B ⊆ A .Proof. First of all recall that we can view the i -th polytope Novikov chain groups asfinitely-generated Novikov-modules by fixing a finite set of preferred lifts ˜ x m ∈ π − ( x m ) ,for each zero x m of θ a of index i :CN i ( θ a , A) ≅ ⊕ m Λ A ⟨ ˜ x m ⟩ , as Novikov ring modules . Since Λ A ≅ Nov (A) (cf. Proposition 2.29), we end up withCN i ( θ a , A) ≅ ⊕ m Λ A ⟨ ˜ x m ⟩ ≅ ⊕ m Nov (A)⟨ x m ⟩ = CN i ( θ a , Nov (A)) . oth boundary operators ∂ and ∂ are Λ A - and Nov (A) -linear, thus it suffices to compare ∂ ˜ x m and ∂x m . On one hand we have ∂ ˜ x m = ∑ n λ m,n ˜ y n , with λ m,n = ∑ A ∈ Γ A alg M ( ˜ x m , A ˜ y n ; ˜ f θ a ) A ∈ Λ A , (24)and the other hand ∂x m = ∑ n ⎛⎝ ∑ γ ∈ M ( x m ,y n ,θ a ) t ∫ γ a ⋯ t ∫ γ a k k ⎞⎠ y n . (25)In the previous subsection we have seen that the Novikov twist t ∫ γ a ⋯ t ∫ γ a k k of γ uniquelydetermines the lifting behaviour of γ . Thus if ˜ γ denotes the unique lift which starts at ˜ x n and ends at some ˜ y , we get ˜ y = A ˜ y n , with A ∈ Γ A ⊂ Λ A corresponding to t ∫ γ a ⋯ t ∫ γ a k k ∈ Nov (A) . This proves that (24) and(25) agree up to identifying the respective isomorphic Novikov rings. The same proof alsoshows that the restricted complexes associated to A∣ B agree.With the identification of twisted and polytope complexes at hand, we can invokeTheorem 2.14 (recall that we already assumed that θ is perturbed accordingly) and deducethat the twisted chain complex is well defined. By the first part it follows that thecorresponding homologies agree. This finishes the proof. Remark 2.35.
The twisted complex can be used to deduce properties of the polytopecomplex and vice versa. For instance, trying to prove that ∂ = ∂ =
0, which has a far more pleasant proof – the reason is that the Novikov twists arenicely behaved with respect to the compactification of the moduli spaces, see for instance[21, Proposition 1].
All the applications we are about to present boil down to what we call the . The idea is to relate the polytope chain groups to twisted Morse chain groups byextending the underlying polytope A with 0 ∈ H ( M ) as an additional vertex, and thenusing the Main Theorem 2.24. This trick suffices to prove the (twisted) Novikov MorseHomology Theorem (cf. Theorem 3.3 and Corollary 3.4).For the polytope Novikov Principle (cf. Theorem 3.6) we shall need a Morse variantof the Eilenberg Theorem [4, Theorem 24.1], which has been proven in [2, Theorem 2.21].We will state and prove a slightly stronger version in the context of Novikov theory downbelow, see Lemma 3.2. emma 3.1 (0-vertex trick) . Let θ ∶ A → Ω ( M ) be a section, B ⊆ A a subpolytope, and A the polytope spanned by the vertices of A and . Let θ ∶ A → Ω ( M ) be a sectionextending θ . Then Z [ Γ A ] = Z [ Γ A ] and Nov (A∣ B ) = Nov (A ∣ B ) . In particular, there exists a perturbed section ϑ ∶ A → Ω ( M ) such that CN ● ( ϑ a , A∣ B ) = CN ● ( ϑ a , A ∣ B ) ≃ CN ● ( ϑ , A ∣ B ) = CM ● ( h, Nov (A∣ B )) , ∀ a ∈ A , as chain complexes, where dh = ϑ .Proof. Since ker ( ) = π ( M ) , adding 0 as vertex does not affect the underlying abeliancover, i.e. ̃ M A = ̃ M A , also see Example 2.3. Thus the deck transformation groups Γ A and Γ A are equal and so are the respective group rings. The finiteness conditions forboth Nov (A∣ B ) and Nov (A ∣ B ) is determined by the subpolytope B ⊆ A , and thus by thegroup ring equality we also deduceNov (A∣ B ) = Nov (A ∣ B ) . The equality as local coefficient systems then also follows by observing that the periodhomomorphism Φ is identically zero, hence t Φ ( A ) = A ∈ Γ A . Pick ϑ as inTheorem 2.14 (or Corollary 2.17). The first chain polytope equality follows from theNovikov rings being equal and the chain homotopy equivalence stems from the MainTheorem 2.24. The last equality follows from Example 2.33 and the equality of localcoefficient systems above.We conclude the subsection by stating and proving a chain-level Morse variant of theEilenberg Theorem [2, Theorem 2.21] Lemma 3.2 (Morse-Eilenberg Theorem) . Let h ∶ M → R be Morse function, A a polytopeand B ⊆ A a subpolytope. Then CM ● ( ˜ h ) ⊗ Z [ Γ A ] Nov (A∣ B ) ≅ CM ● ( h, Nov (A∣ B )) as chain complexes over Nov (A∣ B ) , where ˜ h = h ○ π . The proof of [2, Theorem 2.21] constructs a group chain isomorphism Ψ and a carefulinspection reveals that Ψ defines a Novikov-module chain isomorphism when working withthe according local coefficient system Nov (A∣ B ) . Nevertheless, we decided to give a fullproof of Lemma 3.2 using the tools developed in the previous sections. Proof of Lemma 3.2.
First of all observe that CM ● ( ˜ h ) is a finite Z [ Γ A ] -moduleCM ● ( ˜ h ) = ⊕ m Z [ Γ A ] ⟨ ˜ x m ⟩ . Note that if 0 is already contained in A , then A = A . ere { ˜ x m } denotes a finite set of preferred lifts as in the proof of Proposition 2.34. DefineΨ ∶ CM ● ( ˜ h ) ⊗ Z [ Γ A ] Nov (A∣ B ) Ð→ CM ● ( h, Nov (A∣ B )) , Ψ ( ˜ x m ⊗ λ ) = λ ⟨ x m ⟩ . By the above observation Ψ is a well defined Z [ Γ A ] -linear map. It is clear that Ψ issurjective. For injectivity we observe thatΨ ( ˜ x m ⊗ λ ) = Ψ ( ˜ x n ⊗ µ ) ⇐⇒ λ ⟨ x m ⟩ = µ ⟨ x n ⟩ . Therefore we must have λ = µ and x m = x n , hence ˜ x m = ˜ x n – recall that we are workingwith a preferred set of critical points in each fiber. This proves injectivity.Next we show that Ψ is Nov (A∣ B ) -linear. Pick λ, µ ∈ Nov (A∣ B ) and observe:Ψ ( µ ⋅ ( ˜ x m ⊗ λ )) = Ψ ( ˜ x m ⊗ ( µ ⋅ λ )) = ( µ ⋅ λ ) ⟨ x m ⟩ = µ ⋅ ( λ ⟨ x m ⟩) = µ ⋅ Ψ ( ˜ x m ⊗ λ ) . Hence Ψ is a Novikov-linear isomorphism. We are only left to show ∂ ○ Ψ = Ψ ○ ( ∂ M ⊗ id ) .Recall that ∂ M is defined by counting Morse trajectories of ˜ h on ̃ M A . In particular, theboundary operator ∂ on CN ● ( dh, A∣ B ) agrees with ∂ M on Crit ( ˜ h ) . Let us adopt thenotation of the proof of Proposition 2.34 and write ∂ M ˜ x m = ∂ ˜ x m = ∑ n λ m,n ˜ y n , λ m,n ∈ Z [ Γ A ] . ThereforeΨ ○ ( ∂ M ⊗ id ) ˜ x m ⊗ λ = Ψ ( ∑ n λ m,n ˜ y n ⊗ λ ) = Ψ ( ˜ x m ⊗ ∑ n λ m,n ⋅ λ ) = ( ∑ n λ m,n ⋅ λ ) ⟨ x m ⟩ . On the other hand Proposition 2.34 says that up to identifying ˜ x m and x m we have ∂ ˜ x m = ∂ x m , thus ∂ ○ Ψ ( ˜ x m ⊗ λ ) = ∂ ( λ ⟨ x m ⟩) = λ ∂ x m = λ ⋅ ∑ n λ m,n y n = ( λ ⋅ ∑ n λ m,n ) ⟨ y n ⟩ . But Nov (A∣ B ) is a commutative ring, hence we conclude the chain property of Ψ and thusthat Ψ defines a Novikov-linear chain isomorphism. Using the results developed in the Section 2 and the 0-vertex trick (cf. Lemma 3.1)we are going to prove: We are implicitly identifying Λ A∣ B = Nov ( A∣ B ) , see Proposition 2.27. Sanity check: λ m,n = ∑ A ∈ Γ A alg M( ˜ x m , A ˜ y n ; ˜ h ) A , but ⋃ A ∈ Γ A M( ˜ x m , A ˜ y n ; ˜ h ) ≅ M( x m , y n ; h ) and the latter isfinite. heorem 3.3. Let f be a Morse function and a ∈ H ( M ) a cohomolgoy class. Thenfor every Morse representative α ∈ a there exists a chain homotopy equivalence CN ● ( α ) ≃ CM ● ( f, Nov ( a )) of Novikov-modules. One can prove that the twisted Morse homology computes singular homology withcoefficients Nov ( a ) , see [2, Theorem 4.1]. Combining this with Theorem 3.3 and takingthe homology then shows:
Corollary 3.4. (Twisted Novikov Homology Theorem) For any cohomology class a ∈ H ( M ) there exists an isomorphism HN ● ( a ) ≅ H ● ( M, Nov ( a )) of Novikov-modules. Remark 3.5.
This is a slightly different incarnation of the classical Novikov Morse Homol-ogy Theorem as Corollary 3.4 relates the Novikov homology to twisted singular homologyrather than equivariant singular homology. Moreover, as the proof will show, we do not invoke the Eilenberg Theorem (or its Morse analogue from the previous subsection) andinstead produce a direct connection between the Novikov complex and the twisted Morsecomplex via the 0-vertex trick – this chain of arguments appears to be novel.
Proof of Theorem 3.3.
Pick 0 , a in H ( M ) , set A ∶ = ⟨ , a ⟩ , and consider any section θ ∶ A → Ω ( M ) . Up to perturbing θ we may assume that θ is a section ϑ (note that here A = A ) asin Lemma 3.1. In particular, setting B = ⟨ a ⟩ and invoking Lemma 3.1 we get a Novikovchain homotopy equivalenceCN ● ( θ a , A∣ a ) ≃ CM ● ( h, Nov (A∣ a )) . From Example 2.33 we deduce that Nov (A∣ a ) = Nov ( a ) , therefore we obtainCN ● ( θ a ) ≃ CM ● ( h, Nov ( a )) as Novikov-modules. Twisted Morse homology, just as ordinary Morse homology, doesnot depend on the choice of Morse function. Indeed, using continuation methods one canprove CN ● ( h, Nov ( a )) ≃ CM ● ( f, Nov ( a )) as Novikov-modules. This provesCN ● ( θ a ) ≃ CM ● ( f, Nov ( a )) as Novikov-modules. Observing θ a ∈ a and taking the homology on both sides completesthe proof. The authors mention in the proof of [2, Lemma 6.30] that the isomorphism in [2, Theorem 4.1] is anisomorphism in the category of the underlying local coefficient system, thus a Novikov-module isomorphism inour case. See the proof of [2, Theorem 3.9] for more details. .3 A polytope Novikov Principle Combining the two lemmata from Subsection 3.1 and the Main Theorem 2.24 we willprove a polytope Novikov Principle (see Theorem 3.6). As a corollary we recover theordinary Novikov Principle (cf. Corollary 3.8). In fact, the results here also cover thosein the previous Subsection 3.2. We opted to keep them apart to emphasise the novelty ofthe “twisted” approach to the Novikov Homology Theorem (see Remark 3.5).The polytope Novikov Principle yields a new proof (in the abelian case) to a recentresult due to Pajitnov [18, Theorem 5.1].
Theorem 3.6 (Polytope Novikov Principle) . Let θ ∶ A → Ω ( M ) be any section and B ⊆ A a subpolytope. Then there exists a perturbed section ϑ ∶ A → Ω ( M ) such that CN ● ( ϑ a , A∣ B ) ≃ C ● (̃ M A ) ⊗ Z [ Γ A ] Nov (A∣ B ) , ∀ a ∈ A , (26) as Novikov-modules. Here C ● denotes the singular chain complex with Z -coefficients. Proof of Theorem 3.6 .
Let θ ∶ A → Ω ( M ) be a section that extends θ with A thepolytope generated by the vertices of A and 0. By the 0-vertex trick, i.e. Lemma 3.1,there exists a perturbed section ϑ ∶ A → Ω ( M ) such thatCN ● ( ϑ a , A∣ B ) ≃ CM ● ( h, Nov (A∣ B )) , ∀ a ∈ A as Novikov-modules. Combining this with Lemma 3.2 we obtainCN ● ( ϑ a , A∣ B ) ≃ CM ● ( ˜ h ) ⊗ Z [ Γ A ] Nov (A∣ B ) , ∀ a ∈ A as Novikov-modules. From standard Morse theory we know that the Morse chain complexCM ● ( ˜ h ) is chain homotopy equivalent over Z [ Γ A ] to the singular chain complex C ● (̃ M A ) ,see for instance [17, Page 415] and [16, Appendix]. Denote by i ∶ CM ● ( ˜ h ) Ð→ C ● ( ̃ M A ) , j ∶ C ● ( ̃ M A ) Ð→ CM ● ( ˜ h ) such a chain homotopy equivalence. Then one can easily check that i ⊗ id Nov ( A ∣ B ) and j ⊗ id Nov ( A ∣ B ) define a Novikov-linear chain homotopy equivalenceCM ● ( ˜ h ) ⊗ Z [ Γ A ] Nov (A∣ B ) ≃ C ● ( ̃ M A ) ⊗ Z [ Γ A ] Nov (A∣ B ) . HenceCN ● ( ϑ a , A∣ B ) ≃ CM ● ( ˜ h ) ⊗ Z [ Γ A ] Nov (A∣ B ) ≃ C ● ( ̃ M A ) ⊗ Z [ Γ A ] Nov (A∣ B ) , ∀ a ∈ A . Setting ϑ ∶ = ϑ ∣ A finishes the proof. One can also deduce from category theory by observing that the functor F ∶ mod Z [ A ] → mod Nov ( A ∣ B ) , F ( O ) ∶ = O ⊗ Z [ Γ A ] Nov (A∣ B ) , F ( i ) = i ⊗ id , O, P ∈ obj ( mod Z [ Γ A ] ) , i ∈ hom ( O, P ) is additive. emark 3.7. If one is interested in a particular Morse form ω , then the following im-provement can be made: let ( ω, g ) be Morse-Smale and assume that A is a polytopearound [ ω ] that admits a section θ ∶ A → Ω ( M ) sufficiently close to ( ω, g ) in the senseof Corollary 2.17. Denote by ϑ ω ∶ A → Ω ( M ) the associated perturbation with referencepair ( ω, g ) . The ϑ = ϑ ∣ A section in the proof above might come from an exact referencepair since 0 could a priori be far away from θ . However, Corollary 2.23 assertsCN ( ϑ ωa , A∣ B ) ≃ CN ( ϑ a , A∣ B ) , ∀ a ∈ A . Combining this with (26) for a = [ ω ] givesCN ● ( ω, A∣ B ) ≃ C ● (̃ M A ) ⊗ Z [ Γ A ] Nov (A∣ B ) since ϑ ω [ ω ] = ω .In the special case of ordinary Novikov theory, i.e. A = ⟨ a ⟩ and A = ⟨ , a ⟩ , Theorem3.6 reduces to the ordinary Novikov Principle: Corollary 3.8. (Ordinary Novikov Principle) Let ( α, g ) be Morse-Smale. Then CN ● ( α ) ≃ C ● ( ̃ M a ) ⊗ Z [ Γ a ] Nov ( a ) . Proof.
Set a = [ α ] , pick A = ⟨ a ⟩ , B = A and θ ∶ A → Ω ( M ) the smooth section definedby θ a = α . The section θ is obviously close to ( α, g ) , thus Theorem 3.6 and Remark 3.7imply CN ● ( α ) ≃ C ● ( ̃ M a ) ⊗ Z [ Γ a ] Nov ( a ) as Novikov-modules.Even though it is hidden in the proof above, the main idea is still to use perturbations ϑ ∶ A → Ω ( M ) associated to an exact reference pair ( H, g ) . Recall that these sections ϑ are constructed by a “shift-and-scale” procedure so that each ϑ a is dominated by theexact term ε dH . This strategy to recover the ordinary Novikov Principle has been knownamong experts for quite awhile, see [15, Page 302] for a historical account, [14, Page 548]and [10, Theorem 3.5.2]. However, our approach is slightly different as it does not makeuse of gradient like vector fields.We conclude the present subsection by explaining how to recover [18, Theorem 5.1]from the polytope Novikov principle. For the reader’s convenience we briefly recall Pa-jitnov’s setting, keeping the notation as close as possible to [18]. Fix a Morse-Smale pair ( ω, g ) on M and let p ∶ ̂ M → M be a regular cover such that p ∗ [ ω ] =
0. Denote by r the rank of ω and define G = Deck ( ̂ M ) . Viewing the period homomorphism Φ ω on H ( M ; Z ) we get a splitting H ( M ; Z ) ≅ Z r ⊕ ker [ ω ] . The rank of a cohomology class a ∈ H ( M ) is defined as rank Z ( im Φ a ) . ajitnov calls a family of homomorhismΨ , . . . , Ψ r ∶ Z r → Z a Φ ω -regular family if • the Ψ i span hom Z ( Z r , Z ) and • the coordinates of Φ ω ∶ Z r → R in the basis Ψ i are strictly positive.We shall call Ψ = { Ψ i } a Φ ω - semi-regular family whenever the first bullet point above issatisfied. One should not be fooled by the length of name – the existence of a semi-regularfamily is obvious and merely an algebraic statement.To every (semi)-regular family Ψ = { Ψ , . . . , Ψ r } we associate the conical Novikov ring ̂ Λ Ψ = r ⋂ i = ̂ Z [ G ] Ψ i . The conical Novikov chain complex (N ● ( ω ) , ∂ ) is defined as N i ( ω, Ψ ) = N i ( ω ) = ⊕ x ∈ Z i ( ω ) ̂ Λ Ψ ⟨ x ⟩ , where the boundary operator ∂ is defined as expected: fix preferred lifts ˆ x of each x ∈ Z ( ω ) and define the y -component of ∂x by the (signed) count of ˆ f ω -Morse flow lines on the cover ̂ M from ˆ x to g ○ ˆ y for all g ∈ G . Pajitnov proves that there always exists a Φ ω -regularfamily Ψ so that (N ● ( ω ) , ∂ ) is a well defined ̂ Λ Ψ -module chain complex and shows: Theorem 3.9 (Pajitnov 2019, [18]) . For any Morse-Smale pair ( ω, g ) there exists a Φ ω -regular family Ψ such that N ● ( ω, Ψ ) = N ● ( ω ) is a well defined chain complex. Moreover,for any such Ψ it holds N ● ( ω ) ≃ C ● ( ̂ M ) ⊗ Z [ G ] ̂ Λ Ψ as ̂ Λ Ψ -modules. Using Theorem 3.6 we recover Theorem 3.9 in the abelian case.
Corollary 3.10.
Let ( ω, g ) be a Morse-Smale pair, p ∶ ̂ M → M be an abelian regular coverwith p ∗ [ ω ] = .Then there exists Φ ω -semi-regular family Ψ , a section θ ∶ A → Ω ( M ) around [ ω ] with ̃ M A = ̂ M , and a perturbation ϑ ∶ A → Ω ( M ) such that CN ● ( ϑ a , A∣ B ) ≃ C ● ( ̂ M ) ⊗ Z [ G ] ̂ Λ Ψ , ∀ a ∈ A as Novikov-modules with ϑ [ ω ] = ω . In particular, CN ● ( ω, A∣ B ) ≃ C ● (̂ M ) ⊗ Z [ G ] ̂ Λ Ψ . The definition in [18] is slightly more general, as they define the count with respect to any transverse ω -gradient. roof. First of all we construct the polytope A and a small section θ ∶ A → Ω ( M ) byadapting a rational approximation idea due to Pajitnov [16], see also [22, Section 4.2].Consider the splitting from before H ( M ; Z ) ≅ Z r ⊕ ker [ ω ] , where r is the rank of [ ω ] . Pick r -many generators γ , . . . , γ r of the first summand above.By de Rham’s Theorem there r -many pairwise distinct integral classes a , . . . , a r dual to γ , . . . , γ r such that ker ( a l ) ⊃ ker [ ω ] , ∀ l = , . . . , r. In particular, we can write Φ ω = r ∑ l = u l ⋅ Φ a l , for a unique vector u = ( u , . . . , u r ) ∈ R r , hence [ ω ] = ∑ rl = u l ⋅ a l . Thus, for our fixed Morserepresentative ω ∈ [ ω ] there exist α l ∈ a l with ω = r ∑ l = u l ⋅ α l and ker [ ω ] = r ⋂ l = ker ( a l ) . (27)For fixed ε > r -many rational closed one-forms β l that are ε -close to ω (inthe operator norm induced by g ). Pick a sufficiently small vector v = v ( ε ) ∈ R r suchthat for β = ω + r ∑ l = v l ⋅ α l we have ∥ ω − β ∥ ≤ r ∑ l = ∣ v l ∣ ⋅ ∥ α l ∥ < ε and b ∶ = [ β ] ∈ H ( M ; Q ) . This is possible since Q is dense in R . Define β = ω + r ∑ l = v l ⋅ α l , v ∈ R r , satisfying the same properties with v − v = ( v − v ´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶ ≠ , , . . . , ) . In particular, β − β = ( v − v ) ⋅ α , which implies b ≠ b . Proceeding inductively (e.g. v agreeing with v except for thesecond entry etc.) we end up with r -many rational one-forms β , . . . , β r satisfying • ∥ ω − β j ∥ ≤ ∑ rl = ∣ v jl ∣ ⋅ ∥ α l ∥ < ε for all j = , . . . , k , • b l ≠ b j for all l ≠ j . ker ( b l ) ⊃ ker [ ω ] for all l = , . . . , k , by (27).Since all b l are rational, there exists a positive integer q ∈ N so that every cohomologyclass q ⋅ b l is integral. From the bullet points above we thus conclude thatΨ ∶ = { Φ q ⋅ b l } defines a Φ ω -semi-regular family. Next set A − = ⟨[ ω ] , b , . . . , b r ⟩ and ε = C ω D ⋅ , cf. Corollary 2.17. It is clear that there exists a section θ − ∶ A − → Ω ( M ) sending [ ω ] to ω and b l to β l . In particular ∥ θ − a − ω ∥ < ε, ∀ a ∈ A − . Analogously to the construction of the β l ’s, one can define rational one-forms close to ω that do not vanish on ker [ ω ] . Including some of those cohomology classes allows us toextend A − to A so that ̃ M A = ̂ M with a section θ ∶ K → Ω ( M ) that extends θ − and is still ε -close to ω . By choice of ε we can invoke Corollary 2.17 and define a perturbed section ϑ ω ∶ A → Ω ( M ) with ( ω, g ) as the underlying reference pair. But now we are in a position to use Theorem3.6 and Remark 3.7 with B ∶ = ⟨ b , . . . , b r ⟩ to obtainCN ● ( ϑ ωa , A∣ B ) ≃ C ● ( ̂ M ) ⊗ Z [ G ] Nov (A∣ B ) = C ● ( ̂ M ) ⊗ Z [ G ] r ⋂ l = Nov (A∣ b l ) . The last equality follows from the Z -analogue of Lemma 2.7. Note that Nov (K∣ b l ) = ̂ Z [ G ] b l = ̂ Z [ G ] q ⋅ b l , therefore the Novikov ring on the RHS above does coincide with ̂ Λ Ψ .By definition of the perturbation ϑ ω we get ϑ ω [ ω ] = ω , which finally concludes the proof. References [1] A. Banyaga and D. Hurtubise.
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