Polytope Novikov homology

Let M be a closed manifold and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}} \subseteq H^1_{\mathrm {dR}}(M)$$\end{document}A⊆HdR1(M) a polytope. For each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a \in {\mathcal {A}}$$\end{document}a∈A, we define a Novikov chain complex with a multiple finiteness condition encoded by the polytope \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}$$\end{document}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 chain homotopy equivalent polytope Novikov complexes over a Novikov ring associated with said polytope. As applications, we present a novel approach to the (twisted) Novikov Morse Homology Theorem and prove a new polytope Novikov Principle. The latter generalizes the ordinary Novikov Principle and a recent result of Pajitnov in the abelian case.


Introduction
Given a closed manifold M and a cohomology class a ∈ H 1 dR (M ), one can define the so-called Novikov homology HN • (a), introduced by Novikov [11,12]. Roughly speaking, HN • (a) is defined by picking a Morse representative α ∈ a and a cover on which α pulls back to an exact form df , and then mimicking the definition of Morse homology usingf as the underlying Morse function. ) and their respective twisted coefficient systems 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 what we call polytope Novikov homology HN • (a, A) by including multiple finiteness conditions imposed by a polytope A = a 0 , . . . , a k ⊆ H 1 dR (M ) containing a. These polytope Novikov homology groups HN • (a, A) retain the three features of HN(a) mentioned above, modulo replacing Nov(a) by a "smaller" Novikov ring Nov(A). The Main Theorem in Sect. 2 gives a dynamical relation between HN • (a, A) and HN • (b, A), i.e., by staying in the realm of Novikov homology and not resorting to the algebraic counterpart of twisted singular homology. The statement of the Main Theorem might be known to some experts in the field, but lacks a proof in the literature. Similar variants of the Main Theorem have been proved in different settings; most noteworthy are [5,7,13,25]. For example, in [13], Ono considers Novikov-Floer homology on a closed symplectic manifold 3 and proves the following.
Theorem. (Ono [13]) If two symplectic isotopies 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 common 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's Theorem, but with less assumptions: the nearby assumption of the fluxes in Ono's result would translate to a smallness assumption on A, which is not needed here. Let us mention that the formulation and setup of the Main Theorem comes closest to a recent result due to Groman and Merry [5,Theorem 5.1].
At the end of the paper, we present two applications of the Main Theorem. In the first application, we recover the aforementioned Novikov Morse Homology Theorem: 4 The proof, modulo details, goes as follows: taking A = 0, a , setting B = a , invoking the lower isomorphism in the Main The proof idea is similar to the sketch above-one relates the polytope Novikov complex to a twisted Morse complex by including the 0-vertex in the polytope A and using the Main Theorem. We call this the 0-vertex trick (cf. Lemma 3.1). To get from the twisted Morse complex to the equivariant singular chain complex, we use a Morse-Eilenberg type result (cf. Lemma Immediate consequences of the polytope Novikov Principle include the ordinary Novikov Principle (cf. Corollary 3.8) and a recent "conical" Novikov Principle [17,Theorem 5.1] 6 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 [18], we combine ideas of Ono's Theorem, the magnetic case [5], and of the present paper to construct a polytope Novikov symplectic homology, which is related to Ritter's twisted symplectic homology [20]. The analogue of the Main Theorem remains true. Applications include Novikov number-type bounds on the number of fixed points of symplectomorphisms with prescribed flux on the boundary, and the study of symplectic isotopies of such maps.

Definition and properties of ordinary Novikov homology
In this subsection, we quickly recall the (ordinary) definition of the Novikov chain complex and its homology, together with some well known properties. The main purpose is to fix the notation for the remainder of the section. For a thorough treatment of Novikov homology, we recommend [4,19,22] and the recently published [1]. For more details on the construction of the Novikov ring, see for instance [6,Chapter 4]. Fix once and for all a closed smooth oriented and connected finitedimensional manifold M . For any Morse-Smale pair (α, g) one can define the Novikov chain complex whose homology is called Novikov homology of (α, g) It is a standard fact that two Morse-Smale pairs with cohomologous Morse forms induce isomorphic Novikov homologies, and thus, we shall write HN • (a) with a = [α] to denote the Novikov 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, while the respective lowercase Greek letters are representatives in the corresponding cohomology classes, e.g., α ∈ a, β ∈ b.
Let us quickly recall the relevant definitions. Each cohomology class a determines a period homomorphism Φ a : π 1 (M ) → R defined by integrating any representative α ∈ a over loops γ in M . 7 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 1 (M ) ker(a) . Then, α pulls back to an exact form on M a , i.e., π * α = df α for somef α ∈ C ∞ ( M a ). Define where Crit i (f α ) denotes the critical points off α with Morse index i. The ith Novikov chain group CN i (α) can then be defined as the downward completion of V i (α) with respect tof α , which shall be denoted by Explicitly, elements ξ ∈ CN i (α) are infinite sums with a finiteness condition determined byf α : The boundary operator is defined by counting Novikov-Morse trajectories of f α Cohomologous one-forms induce the same period homomorphism by Stokes' Theorem. where is the usual moduli space withg = π * g the pullback metric. Denote M(x,ỹ;f α ) = M(x,ỹ;f α ) / R . Similarly, we denote by M(x, y; α) and M(x, y; α) the moduli spaces downstairs. The # alg indicates the algebraic count, i.e., counting the Novikov-Morse trajectories with signs determined by a choice of orientation of the underlying unstable manifolds. The Novikov ring Λ α associated with α ∈ a is defined as the upward completion of the group ring Z[Γ a ] with respect to the period homomorphism Φ a , and therefore The Novikov ring Λ α does not depend on the choice of representative α ∈ a, and thus, we shall write Λ a . Moreover, Λ a acts on CN • (α) in the obvious way. By fixing a preferred liftx j in each fiber of the finitely many zeros x j ∈ Z(α) := x ∈ M α(x) = 0 , one can view CN • (α) as a finitely generated Λ a -module Λ a x j as Novikov ring modules. ( Another standard fact asserts that the boundary operator ∂ is Λ a -linear, and consequently, the Novikov homology HN • (a) carries a Λ a -module structure. The latter is implicitly using the fact that isomorphism of Novikov homologies for cohomologous Morse forms, which is 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 2 in all the definitions above.

Novikov homology with polytopes
We are now ready to refine the Novikov chain complex using polytopes-this notion is key for the proofs of all incoming theorems. To any polytope A, we associate a regular cover Example 2.3. For the polytope A = a , the covering M A agrees with the abelian cover M a associated with a ∈ H 1 dR (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 trivial cohomology class, and so does every a ∈ A; see Lemma 2.7. We writef α ∈ C ∞ ( M A ) to denote some primitive of π * α for α a representative of a ∈ A. Now, we fix a smooth section of the projection of closed one-forms to their cohomology class. In other words, θ a is a representative of a. This enables us to talk about a "preferred" representative of each cohomology class in the polytope.
For every polytope a ∈ A, we define The subtle but crucial difference to V i (θ a ) is that M A does not necessarily coincide with the abelian cover M a .

Definition 2.4. Let
A be a polytope with section θ : A → Ω 1 (M ). Then, the (polytope) Novikov chain complex groups are defined as the intersections of the downward completions of V i (θ a , A) with respect to anyf β : M A → R for b ∈ A. In other words, with the notation of (1) Remark 2.5. Let β ∈ b be any representative. The choice of primitivef β of π * β is unique up to adding constants and hence does not affect the finiteness condition. Additionally, two primitivesf β andf β induce the same finiteness Unpacking Definition 2.4, we see where it does not matter which primitivesf β we use, cf. Remark 2.5. The right-hand side describes a finiteness condition that has to hold for all b ∈ A, and hence, we will refer to it as the multi finiteness condition. Notation. In view of Remark 2.5, we shall write In a similar fashion, we can define yet another completion of V i (θ a , A) by taking the completion with respect to less one-forms. Definition 2.6. Let B ⊆ A be a subpolytope, i.e., the convex hull of a subset of the vertices of A. Then, we define By definition, we get the inclusion The next lemma asserts that CN • (θ a , A) is uniquely determined by the vertices of A. In other words, one only needs to check the multi-finiteness condition for the finitely many vertices a l . This is a straightforward adaptation of [25,Lemma 7.3].
More generally, for every subpolytope B ⊆ A spanned by b j = a lj One can play a similar game with the Novikov rings: Definition 2.8. Define the (polytope) Novikov ring where Z[Γ A ] b denotes the upward completion of the group ring Z[Γ A ] with respect to the period homomorphism Φ b : Γ A → R. Analogously, for every subpolytope B ⊆ A, we define the restricted polytope Novikov ring The obvious analogue to Lemma 2.7 holds for Novikov rings as well. These rings enable us to view the polytope Novikov chain complexes as finite Novikov modules just as in the ordinary setting (2). Next, we try to equip the groups CN • (θ a , A) with a boundary operators that turns them into a genuine chain complex. The obvious candidate would be Note that the moduli space above actually also depends on a choice of metric g, and so does the boundary operator ∂ θa . When we want to keep track of the metric, we will write CN • (θ a , g, A). For restrictions A| B , we define the boundary operator analogously.
Formally, the definition of ∂ θa looks identical to the definition of ∂ on CN • (α), and morally it is. However, there are two major differences. First, the cover M A might differ from the abelian cover M a of a. Second, it is not clear whether ∂ = ∂ θa preserves the multi finiteness condition, i.e., whether ∂ξ lies in CN • (θ a , A). Luckily, we will achieve this by replacing the original section θ with a perturbed section ϑ : A → Ω 1 (M ) (cf. Theorem 2.14). Whenever the chain complex is defined, we make the following definition. Analogously, we define Remark 2.10. Analogously to ordinary Novikov homology, one can show that the Novikov homologies HN • (ϑ a , A) and HN • (ϑ a , A| B ) are both finitely generated modules over the Novikov 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 for the restricted case).

Technical results for Sect. 2.4
In this subsection, we state and prove all the technical auxiliary results needed for the proof of Theorem 2.14, which roughly speaking asserts the welldefinedness of the polytope chain complexes and their respective homologies after modifying the section θ : A → Ω 1 (M ) to a new section ϑ : A → Ω 1 (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 any smooth function H : M → R.
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 → 0, and (z k ) ⊂ M , such that By compactness of M , we can pass to a subsequence (z k ) converging to some z ∈ M . The above however implies ∇ g ρ(z) = 0, which is equivalent , which is a contradiction. This concludes the proof.
Notation. Such a constant C ρ > 0 is often referred to as a Palais-Smale constant (short: PS-constant). The main case of interest is the exact one, i.e., ρ = dH, for which we will abbreviate C dH = C H . Sometimes, we will also abbreviate C ρ = C. The next lemma builds the main technical tool of Sect. 2.4. The idea is to perturb one-forms α close to a given reference Morse-Smale pair (ρ, g), so that the pertubations, say α , maintain their cohomology classes of α, become Morse, have the same zeros as ρ, and are still relatively close to ρ. This is reminiscent of Zhang's arguments [25, Section 3]. Lemma 2.12. Let (ρ, g) be a Morse-Smale pair, δ > 0 so small that the balls B 2δ (x), with x ∈ Z(ρ), are geodesically convex 8 and lie in pairwise disjoint charts of M , and C = C ρ (δ, g) > 0 as in Proposition 2.11.
where · is the norm induced by g. Then, there exists a Morse-Smale pair Proof. Since ρ is a Morse form, there are only finitely many zeros x ∈ Z(ρ). Around each such x, we will perturb α without changing its cohomology We set . This means that for the inequality in the first bullet point, it suffices to argue why the bound holds inside each ball B 2δ (x i ). Inserting the definitions grants Recall that f i was chosen, such that f i (x i ) = 0. Due to the geodesic convexity of the balls B 2δ (x i ), we can apply the mean value inequality All in all, this implies This proves the first inequality in the first bullet point. From this, we will deduce that The reverse inclusion is obtained by observing that for y ∈ Z(α ), we have by assumption on ρ and the inequality above. Proposition 2.11 then implies that z has to 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 suffices to perturb g on an open set that intersects all the Novikov-Morse trajectories of (α , g); see [ The last assertion of the statement follows from the observation that, for α fixed, the map g → ∇ g α is continuous, and thus, for g close to g, we get using the above inequality, by the first bullet point, Taking ε ≤ C 4 and invoking Proposition 2.11 then conclude the proof. Lemma 2.12 can be applied to a whole section θ : A → Ω 1 (M ) nearby a reference Morse-Smale pair (ρ, g) and give rise to a perturbed section ϑ : A → Ω 1 (M ) that is still relatively close 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 > 0 as in Lemma 2.12 Then, there exists a section and a positive constant D = D(N, g) > 0 with the following significance: 10 • Moreover, for every ϑ a , there exists a Riemannian metric g ϑa close to g with where · ϑa is the operator norm induced by g ϑa .
Proof. Since the whole section θ : A → Ω 1 (M ) is C 8 -close to (ρ, g), we can take (ρ, g) as a reference pair and apply Lemma 2.12 to every θ a and denote ϑ a the corresponding perturbation. Recall from the proof of Lemma 2.12 that ϑ a is obtained by an exact perturbation of θ a around the zeros of ρ-a closer inspection reveals that this exact perturbation varies smoothly along θ a , in 10 The choice of D > 0 is independent of the assumption (7). particular that ϑ defines a smooth section. The first two bullet points follow immediately from Lemma 2.12. We choose g ϑa = g (θa) just as g in Lemma 2.12, i.e., by means of a small perturbation of g inside N . The argument in [19] shows that sufficiently small perturbations give rise to Riemannian metrics that are uniformly equivalent to the original g; in other words, we may choose g ϑa , such that (ϑ a , g ϑa ) is Morse-Smale and 1 with D > 0 a constant that only depends on N and g. Using this inequality and invoking, the first bullet point of Lemma 2.12 conclude the proof.

Section perturbations
We can finally state and prove Theorem 2.14 by applying the previous results in the special case of exact reference pairs: and a choice of Riemannian metrics g ϑa with the following significance: defined for every pair (ϑ a , g ϑa ) as above; • (Ray invariance) The chain complexes are equal upon scaling, i.e., The rough idea is to "shift-and-scale": we shift and scale the polytope A, so that it is 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 three bullet 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 in chain homotopy equivalent complexes. This is proven in the next subsection (cf. Theorem 2.19).
At the cost of imposing a smallness condition on the underlying section, we get the same results for perturbations associated with 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 This is still a section, but does not satisfy the bullet points above. Since θ is smooth, there exists with respect to · induced by g, C H = C H (δ, g) > 0 and D = D(N, g) > 0 chosen as in Proposition 2.13. Now, we can apply Proposition 2.13 to the section ε · a → ε · θ a + dH and obtain a new section 12 Finally, we scale back and redefine Thus, we have 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 the Morse-Smale property. This proves the first bullet point.
Indeed, assume for contradiction that there exists a Novikov chain ξ = This means that there are some ε · b ∈ ε · A, c ∈ R and sequencesx n with ξx n = 0,ỹ n pairwise distinct,γ n ∈ M x n ,ỹ n ;f ε·ϑa , and see Remark 2.5. Denote by γ n = π •γ n the Novikov-Morse trajectories downstairs. The energy expression can then be massaged as follows: 12 Explicitly, this section is of the form where f i depends smoothly on θa and satisfies df i = ε · θa, f i (x i ) = 0 around critical points x i of H, see Lemma 2.12, (5) and (6) applied to ε · θa + dH and ρ = dH.
Showing that the rightmost term is bounded by m · E(γ n ), m ∈ (0, 1) suffices to obtain a contradiction: admitting such a bound leads to In particular, c ≤f ϑ ε (ε·b) (x n ) for all n. However, ξ belongs to CN • (ε · ϑ a , ε · A) and ξx n = 0, and thus, the multi-finiteness condition implies that there are only finitely many distinctx n . Up to passing to a subsequence, we can therefore assumex n =x and alsoỹ n ∈ π −1 (y). 13 The corresponding Novikov-Morse trajectories have uniformly bounded energy therefore, γ n has a C ∞ loc -convergent subsequence. At the same time M (x, y; ϑ ε (ε · a)) is a 0-dimensional manifold, which means that the convergent subsequence γ n eventually does not depend on n. This contradicts our assumption that the endpointsỹ n upstairs are pairwise disjoint.
Therefore, we are only left to show the bound to conclude the Claim. For this purpose, we define 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 ε · a)).
Lemma 2.12 says that for s ∈ R \ S n , we get The Lebesgue measure μ(S n ) can be bounded using the energy And finally (8), This proves the Claim. Now, we observe that scaling ϑ a by r > 0 does not affect the zeros and that the moduli spaces associated with (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 > 0, the polytope chain complexes associated with (r · ϑ a , g a ) agree with each other. This proves the ray invariance. Setting r = 1 ε and using the Claim prove the remaining first bullet point.
Remark 2.15. Instead of running the argument for the sections ϑ ε : ε · A → Ω 1 (M ), we could also work with ϑ = 1 ε · ϑ ε : A → Ω 1 (M ) by directly by applying Proposition 2.13 to the section a → θ a + d ε −1 H and (ε −1 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 of Morse functions. Note that the analogous bound at the end of the proof of Theorem 2.14 holds upon replacing H by ε −1 H. This follows from the nice scaling behavior of the PS constants: 16. The skeptical reader might wonder whether ∂ 2 ϑa = 0 really holds. Viewing the chain group CN • (ϑ a , A) as a certain twisted chain group allows for a quick and simple proof-see Remark 2.35.
The key in the proof of Theorem 2.14 was to obtain control over the energy by perturbing the section θ : A → Ω 1 (M ) via Proposition 2.13. The  [25].
The question remains why we used an (exact) reference pair (H, g) instead of a more general Morse-Smale pair (ρ, g) in Theorem 2.14. The answer is simple: the given argument already breaks down in the very first line-the corresponding ϑ is not a section anymore, since the ρ-shift changes the cohomology class. However, whenever the section θ : A → Ω 1 (M ) is already sufficiently close to (ρ, g) in terms of the corresponding PS-constant C ρ > 0, we do not need to shift and scale θ, and can perturb θ directly: with · the operator norm induced by g. Then, there exists a perturbed section and g ϑa , 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 the same as the one of Theorem 2.14.

Independence of the data
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 any valid perturbation ϑ i = ϑ i (θ i , H i , g i , ε i ) in the sense of Theorem 2.14 gives rise to chain homotopy equivalent chain complexes. The same is true for perturbations coming from Corollary 2.17, and at the end of the subsection, we will show that both perturbations lead to chain homotopy equivalent Novikov complexes.
Remark 2.18. All the chain maps and chain homotopy equivalences constructed from here on are Novikov-module morphisms, i.e., linear over the Novikov ring. We will not explicitly state this every time for better readability.
in the sense of Theorem 2.14, induce chain homotopy equivalent polytope complexes Vol Proof of Theorem 2. 19. We may assume ε 1 ≥ ε 0 . Denote by the respective sections on A as in the first part of the proof of Theorem 2.14. Let be a positive smooth function with h ≡ 0 on (−∞, e) and h ≡ 1 on (1 − e, +∞), for some small e > 0, and set Fix a ∈ A and pick g i := g ϑ i a , i = 0, 1 two metrics as in Theorem 2.14. Let g s = g s (a) be a homotopy of Riemannian metrics connecting g 0 to g 1 and assume that (ϑ s a , g s ) is regular-this is rectified by Remark 2.20. Note that here g s actually depends on a.
To this regular homotopy, we can now associate a chain continuation Analogously to the case of the boundary operator in Theorem 2.14, proving that Ψ 10 defines a well-defined Novikov chain map essentially boils down to proving that it respects the multi-finiteness condition-the rest follows by standard Novikov-Morse techniques. Thus, proceeding as in Theorem 2.14 reveals that it suffices 14 to bound by either a multiple m ∈ (0, 1) of the energy E(γ n ), where b is some cohomology class in A, or a uniform bound 15 altogether. Set This time around we need to divide by ε i as we are running the continuation directly on the original polytope A instead of the scaled polytope (see Remark 2.15). As in the previous proof of Theorem 2.14, the s ∈ R ≤0 \ S 0 n and 14 This is also implicitly using thatf ϑ s b =f ϑ 0 b + hs • π for a smooth family hs ∈ C ∞ (M ), since ϑ s b are cohomologous for all s. Hence for some uniform constant C. 15 Uniform in b and n ∈ N, that is.
where we have used Proposition 2.13 as in Theorem 2.14. We are left to bound (15) for s ∈ [0, 1]. For this, we compute via Cauchy-Schwarz where F > 0 is a uniform constant in s ∈ [0, 1] and b ∈ A-recall that g s depends on a, but that does not matter. In particular, this proves A case distinction now does the job: for any n ∈ N, we either have 1 5 In the first case, we can bound the norm of (15) by 2 5 · E(γ n ), whereas in the second case, we get 1 5 · E(γ n ) 1 2 < F , and thus, we may bound the norm of (15) by 10F 2 . This proves As explained before, this suffices to conclude that Ψ 10 defines a well-defined Novikov chain map, which defines the desired chain homotopy equivalence (see proof of Theorem 2.14 for more details). This concludes the proof.
Remark 2.20. The (linear) homotopy (ϑ s a , g s ) chosen in the proof of Theorem 2.19 might be non-regular. One can replace (ϑ s a , g s ) with an arbitrarily close regular homotopy ((ϑ s a ) , g s ) connecting the same data. The only bit where this affects the previous argument in Theorem 2.19 is when trying to bound max s∈[0,1] ϑ s b − (ϑ s a ) s . Using that ϑ s a is smooth in s and close to (ϑ s a ) , we still get the desired uniform bound b.
As a consequence of Theorems 2.14 and 2.19, we obtain the analogue results for restrictions to subpolytopes B ⊆ A: satisfying all the bullet points of Theorem 2.14. Any other choice ϑ = ϑ (θ , H , g ) does not affect the chain complexes up to chain homotopy equivalence. Moreover, the inclusion defines a Novikov-linear chain map for all a ∈ A.
Proof. The proof of the first part is literally the same as in Theorems 2.14 and 2. 19. To see that the inclusion defines a chain map, it suffices to observe that both boundary 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 using perturbed sections with respect to non-exact reference pairs (cf. Corollary 2.17). While this 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.
Moreover, both (18) and (19) continue to hold in the restricted case B ⊆ A.
Proof. The proof idea is again arguing via continuations as in Theorem 2.19 above-we will use the latter as carbon copy and adapt the same notation. Define By the same logic as in Theorem 2.19, it suffices to control the expression for all b ∈ A, to get the desired continuation chain map to conclude (18). For this purpose, we define Here, g 0 and g 1 (abusively) denote Riemannian metrics g ϑ ρ a and g ϑ H a coming from Proposition 2.13.
Observe that by assumption and choice of (ϑ ρ a , g 0 ), we have This suffices to obtain the desired control over (20) and proves (18); see proof of Theorem 2.19 for more details. Last but not least, (19) follows by applying (18) twice:  , A), ∀a ∈ A. Corollary 2.23 is the analogue to the independence of Morse-Smale pairs (α, g) in the case of ordinary Novikov homology. The latter can also be recovered from the former by taking the trivial polytope A = a . Nevertheless, keeping track of the section θ, or rather its perturbations, will prove useful, especially when establishing the commutative diagram in the Main Theorem 2.24.

Non-exact deformations and proof of the main theorem
The power of the polytope machinery will become evident in this subsectionroughly speaking, the notion of polytopes allows us to compare the Novikov homologies coming from two different cohomology classes, see Main Theorem 2.24. In Sect. 3, we present some applications of the Main Theorem 2.24.
where the horizontal maps are Novikov-linear chain homotopy equivalences. In particular  (13) and define a homotopy Pick g s a smooth homotopy connecting the two metrics g ϑa and g ϑ b and assume that (ϑ s ab , g s ) is regular (see Remark 2.20). The idea now is to show that the chain continuation map associated with the regular homotopy (ϑ s ab , g s ) is well defined and makes the desired diagram commute. The argument that Ψ ba is a well-defined Novikov chain map is the same as in Theorem 2.19 and follows by controlling terms of the form: Note that this time around we do not need to put an s-dependence on ϑ c (this corresponds to ϑ b in the proof of Theorem 2.19), since the endpoints of ϑ s ab 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 cohomologous assumption in Theorem 2.19, and indeed, the remaining part of the proof is verbatim the same and is thus omitted.
We use the very same homotopy to define a chain continuation From this, we obtain the following commutative diagram on the chain level: Here, the ι B denote the inclusions (17), which are chain maps (cf. Corollary 2.21). By symmetry and the standard argument, we get continuations Ψ ab and Ψ ab | B in the opposite direction by reversing the underlying regular homotopy. It is also easy to see that continuation maps are linear over the underlying Novikov ring. This proves that the two horizontal chain maps above define the desired chain homotopy equivalences. In particular, the chain diagram above induces the desired diagram in homology and thus concludes the proof.

ιB ιB
The upper and lower chain homotopy equivalences, however, do come from compositions of chain continuations rather than genuine chain continuations.

Twisted Novikov complex
Throughout this subsection, we shall assume that θ : A → Ω 1 (M ) has already been perturbed as in Theorem 2.14. 16 We present an alternative description of CN • (θ a , A) by means of local coefficients. For an extensive treatment of local coefficients, we recommend [23] and [1,Chapter 2] in the case of Morse homology. as the ring 17 consisting of formal sums g∈G n g t g with n g ∈ Z, satisfying the finiteness condition ∀c ∈ R : g n g = 0, g < c is finite.
Whenever G is the image of a period homomorphism Φ a : π 1 (M ) → R, we write Nov(a) := Nov(G), G = im(Φ a ).
It turns out that Nov(a) is isomorphic to Λ a , where the isomorphism is given by sending a deck transformation A ∈ Γ a to t Φa(A) -both finiteness conditions match and we obtain: In view of Proposition 2.27, we will also refer to Nov(a) as Novikov ring of a. Inspired by the definition of Nov(a), we will now define yet another ring Nov(A), which will be isomorphic to Λ A almost by definition. A := a 0 , . . . , a k

Definition 2.28. Let
with a multi-finiteness condition Similarly, for any subpolytope B ⊆ A, we define a (potentially) larger group Let us mention that these Novikov rings are commutative rings, since Γ A is abelian. Indeed, the commutator subgroup of π 1 (M ) is contained in every kernel ker(a l ); in particular Each polytope A comes with a representation Sometimes, we will write out Φ a l (η) = η a l . The importance of the minus sign will become clear in Definition 2.31. To any such representation, one can associate a local coefficient system Taking a closer look at (22) reveals that the Novikov ring isomorphism Nov(A)(γ) is given by multiplication with hence, we may also view it as a Z[Γ A ]-module isomorphism.
Remark 2.30. Usually, local coefficients are considered to take values in the category of abelian 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 with bundle of abelian groups would merely grant group isomorphisms.
With this, we define the anticipated twisted Novikov complexes. The twisted boundary operator ∂ = ∂ θa is defined by The twisted chain complexes CN • (θ a , Nov (A| B )) , ∂ θa , ∀a ∈ A are defined analogously. The corresponding twisted Novikov homologies are denoted by HN • (θ a , Nov(A)) and HN • (θ a , Nov (A| B )) , ∀a ∈ A, B ⊆ A. 19 This stems from the construction of a category equivalence between the fundamental groupoid and the fundamental group of a sufficiently nice topological space. This procedure allows to switch back and forth between local coefficients and representations, see [10,Page 17], [1,Chapter 2] and [3] for more details. 20 Note that Γ(γ * η) = Γ(γ) • Γ(η). 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 behavior of γ. Indeed, if γ 0 , γ 1 are two paths from x to y with unique liftsγ 1 (0) =γ 0 (0) =x, theñ This will be key in the next subsection.
As the next examples show, we recover the twisted Morse chain complex and its homology as a special case.

Comparing twisted and polytope complexes
In general, the same issues as in Sect. 2.2 arises when trying to prove that the twisted Novikov complexes are well defined: it is not clear whether ∂ maps into the desired chain complex. This is a non-issue in the special case of θ 0 = dh, i.e., twisted Morse homology-the reason is that the 0-dimensional moduli spaces M(x, y; h) are compact, hence finite. Compare this to [20]. In the following however, we will see that the twisted chain groups can always be identified with CN • (θ a , A), so that ∂ and ∂ agree, which then resolves the well-definedness issue by Theorem 2.14. In other words, the twisted chain complex is an equivalent description of the polytope chain complex.  Proof. First of all recall that we can view the i-th polytope Novikov chain groups as finitely generated Novikov modules by fixing a finite set of preferred liftsx m ∈ π −1 (x m ), for each zero x m of θ a of index i Λ A x m , as Novikov ring modules.
Since Λ A ∼ = Nov(A) (cf. Proposition 2.29), we end up with Both boundary operators ∂ and ∂ are Λ A -and Nov(A)-linear, and thus, it suffices to compare ∂x m and ∂x m . On the one hand, we have and on the other hand In the previous subsection, we have seen that the Novikov twist t γ a0 0 · · · t γ a k k of γ uniquely determines the lifting behavior of γ. Thus, ifγ denotes the unique lift which starts atx n and ends at someỹ, we get . This proves that (24) and (25) agree up to identifying the respective isomorphic Novikov rings. The same proof also shows that the restricted complexes associated with A| B agree.
With the identification of twisted and polytope complexes at hand, we can invoke Theorem 2.14 (recall that we already assumed that θ is perturbed accordingly) and deduce that the twisted chain complex is well defined. By the first part, it follows that the corresponding homologies agree. This finishes the proof.
Remark 2.35. The twisted complex can be used to deduce properties of the polytope complex and vice versa. For instance, trying to prove that ∂ 2 = 0 is equivalent to proving ∂ 2 = 0, which has a far more pleasant proof-the reason is that the Novikov twists are nicely behaved with respect to the compactification of the moduli spaces, see for instance [  In particular, there exists a perturbed section ϑ 0 : as chain complexes, where dh = ϑ 0 0 . Proof. Since ker(0) = π 1 (M ), adding 0 as vertex does not affect the underlying abelian cover, i.e., M A = M A 0 , also see Example 2.3. Thus, the deck transformation groups Γ A and Γ A 0 are equal and so are the respective group rings. The finiteness conditions for both Nov(A| B ) and Nov(A 0 | B ) are determined by the subpolytope B ⊆ A, and thus, by the group ring equality, we also deduce Nov(A| B ) = Nov(A 0 | B ).
The equality as local coefficient systems then also follows by observing that the period homomorphism Φ 0 is identically zero, and hence, t Φ0(A) = 1 for all A ∈ Γ A 0 . Pick ϑ 0 as in Theorem 2.14 (or Corollary 2.17). The first chain polytope equality follows from the Novikov rings being equal and the chain homotopy equivalence stems from the Main Theorem 2.24. The last equality follows from Example 2.33 and the equality of local coefficient systems above. as chain complexes over Nov(A| B ), whereh = h • π. 21 Note that if 0 is already contained in A, then A 0 = A.
Here, {x m } denotes a finite set of preferred lifts as in the proof of Proposition 2.34. Define By the above observation, Ψ is a well defined Z[Γ A ]-linear map. It is clear that Ψ is surjective. For injectivity, we observe that Therefore, we must have λ = μ and x m = x n , and hence,x m =x n -recall that we are working with a preferred set of critical points in each fiber. This proves injectivity.
However, Nov(A| B ) is a commutative ring, and hence, we conclude the chain property of Ψ and thus that Ψ defines a Novikov-linear chain isomorphism.

The twisted Novikov Morse Homology Theorem
Using the results developed in Sect. 2 and the 0-vertex trick (cf. Lemma 3.1), we are going to prove: of Novikov modules.
One can prove that the twisted Morse homology computes singular homology with coefficients Nov(a); see [1,Theorem 4.1]. 24 Combining this with Theorem 3.3 and taking the homology then shows: Remark 3.5. This is a slightly different incarnation of the classical Novikov Morse Homology Theorem as Corollary 3.4 relates the Novikov homology to twisted singular homology rather 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) and instead produce a direct connection between the Novikov complex and the twisted Morse complex via the 0-vertex trick-this chain of arguments appears to be novel. Up to perturbing θ, we may assume that θ is a section ϑ 0 (note that here A 0 = A) as in Lemma 3.1. In particular, setting B = a and invoking Lemma 3.1, we get a Novikov chain homotopy equivalence CN • (θ a , A| a ) CM • (h, Nov(A| a )) .  Setting ϑ := ϑ 0 | A finishes the proof.
Remark 3.7. If one is interested in a particular Morse form ω, then the following improvement can be made: let (ω, g) be Morse-Smale and assume that A is a polytope around [ω] that admits a section θ : A → Ω 1 (M ) sufficiently close to (ω, g) in the sense of Corollary 2.17. Denote by ϑ ω : A → Ω 1 (M ) the associated perturbation with reference pair (ω, g). The section ϑ = ϑ 0 | A in the proof above might come from an exact reference pair, since 0 could a priori be far away from θ. However, Corollary 2.23 asserts CN(ϑ ω a , A| B ) CN(ϑ a , A| B ), ∀a ∈ A. Combining this with (26) for a = [ω] gives In the special case of ordinary Novikov theory, i.e., A = a and A 0 = 0, a , Theorem 3.6 reduces to the ordinary Novikov Principle. Even though it is hidden in the proof above, the main idea is still to use perturbations ϑ 0 : A 0 → Ω 1 (M ) associated with an exact reference pair (H, g). Recall that these sections ϑ 0 are constructed by a "shift-and-scale" procedure, so that each ϑ 0 a is dominated by the exact term 1 ε dH. This strategy to recover the ordinary Novikov Principle has been known among experts for quite awhile; see [14, Page 302] for a historical account, [13, Page 548] and [9, Theorem 3.5.2]. However, our approach is slightly different as it does not make use of gradient like vector fields.
We conclude the present subsection by explaining how to recover [17, Theorem 5.1] from the polytope Novikov principle. For the reader's convenience, we briefly recall Pajitnov's setting, keeping the notation as close as 26 One can also deduce from category theory by observing that the functor F : Pajitnov calls a family of homomorphism Ψ 1 , . . . , 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 is satisfied. One should not be fooled by the length of namethe existence of a semi-regular family is obvious and merely an algebraic statement.
To every (semi)-regular family Ψ = {Ψ 1 , . . . , Ψ r }, we associate the conical Novikov ring The conical Novikov chain complex (N • (ω), ∂) is defined as where the boundary operator ∂ is defined as expected: fix preferred liftsx of each x ∈ Z(ω) and define the y-component of ∂x by the (signed) count of f ω -Morse flow lines on the cover M fromx to g •ŷ for all g ∈ G. 28 Pajitnov proves that there always exists a Φ ω -regular family Ψ, so that (N • (ω), ∂) is a well-defined Λ Ψ -module chain complex and shows: Theorem 3.9. (Pajitnov 2019, [17]) 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 Using Theorem 3.6, we recover Theorem 3.9 in the abelian case.