On the conical Novikov homology

Abstract

Let $\omega$ be a Morse form on a closed connected manifold M. Let $p:{\widehat{M}}\rightarrow M$ be a regular covering with structure group G such that $p^*([\omega ])=0$. The period homomorphism $\pi _1(M)\rightarrow {{\mathbb {R}}}$ corresponding to $\omega$ factors through a homomorphism $\xi :G\rightarrow {{\mathbb {R}}}$. The rank of $\mathrm{Im}\, \xi$ is called the irrationality degree of $\xi$. Denote by $\Lambda$ the group ring ${{\mathbb {Z}}}G$ and let ${{\widehat{\Lambda }}}_\xi$ be its Novikov completion. Choose a transverse $\omega$-gradient v. The classical construction of counting the flow lines of v defines the Novikov complex freely generated over ${{\widehat{\Lambda }}}_\xi$ by the set of zeroes of $\omega$. We introduce a refinement of this construction. We define a subring $\widehat{\Lambda }_\Gamma$ of ${{\widehat{\Lambda }}}_\xi$ (depending on an auxiliary parameter $\Gamma$ which is a certain cone in the vector space $H^1(G,{{\mathbb {R}}})$) and show that the Novikov complex is defined actually over $\widehat{\Lambda }_\Gamma$ and computes the homology of the chain complex $C_*({\widehat{M}})\underset{\scriptscriptstyle \Lambda }{\otimes }\widehat{\Lambda }_\Gamma$. In the particular case when $G\approx {{\mathbb {Z}}}^2$, and the irrationality degree of $\xi$ equals 2, the ring $\widehat{\Lambda }_\Gamma$ is isomorphic to the ring of series in two variables x, y of the form $\sum _{r\in {{\mathbb {N}}}} a_r x^{n_r}y^{m_r}$ where $a_r, n_r, m_r\in {{\mathbb {Z}}}$ and both $n_r, m_r$ converge to $\infty$ when $r\rightarrow \infty$. The algebraic part of the proof is based on a suitable generalization of the classical algorithm of approximating irrational numbers by rationals. The geometric part is a straightforward generalization of the author’s proof of the particular case of this theorem concerning the circle-valued Morse maps (Pazhitnov in Ann Fac Sci Toulouse Math 4(2):297–338, 1995). As a byproduct we obtain a simple proof of the properties of the Novikov complex for the case of Morse forms of irrationality degree $>1$. The paper contains two appendices. In Appendix 1 we give an overview of Pitcher’s work on circle-valued Morse theory (1939). We show that Pitcher’s lower bounds for the number of critical points of a circle-valued Morse map coincide with the torsion-free part of the Novikov inequalities (1982). In Appendix 2 we construct a circle-valued Morse map and its gradient such that its unique Novikov incidence coefficient is a power series in one variable with an arbitrarily small convergence radius.

Appendices

Appendix 1. On the Pitcher inequalities for circle-valued Morse maps

In the paper [16] Pitcher obtained a lower bound for the number of critical points of a circle-valued Morse map. His remarkable work, dating back to 1939, is probably the first development in the circle-valued Morse theory. In this appendix we give an exposition of Pitcher’s work and relate it to the Novikov homology. We show in particular that the numbers $Q_k$ introduced by Pitcher equal Novikov Betti numbers, so that Pitcher’s inequalities are equivalent to the torsion-free part of the Novikov inequalities.

1.1 Pitcher inequalities

We will use the terminology of Pitcher in order to stay as close as possible to his setup. Let L be a closed manifold, and $\theta :L\rightarrow S^1={{\mathbb {R}}}/2\pi {{\mathbb {Z}}}$ a Morse map. The image $\theta _*(H_1(L))\subset H_1(S^1)={{\mathbb {Z}}}$ is a subgroup $\alpha {{\mathbb {Z}}}$ of ${{\mathbb {Z}}}$, where $\alpha$ is a positive integer. Let us assume for simplicity of exposition that $\alpha =1$. Let $K\rightarrow L$ be the corresponding infinite cyclic covering, and $F^*:K\rightarrow {{\mathbb {R}}}$ be a Morse function making the following diagram commutative:

Denote by T the generator of the structure group of the covering such that $F^*(Tx)=2\pi +F^*(x)$. Choose a regular value A of $F^*$ and let $B=A+2\pi$. The set is a fundamental domain for the action of the group ${{\mathbb {Z}}}$ on K. To give the definition of Pitcher’s invariant $Q_k$, let us introduce some more terminology. The closure of the fundamental domain above will be denoted by W; it is a cobordism whose boundary is a disjoint union $(F^*)^{-1}(A) \sqcup (F^*)^{-1}(B)$ of two regular level surfaces of $F^*$. Let $t=T^{-1}$, denote $(F^*)^{-1}(B)$ by V, then $\partial W= V\sqcup tV$. Let $)$ then . Pitcher defines two numerical homological invariants of this configuration (see the two paragraphs before Theorem I on p. 430 of [16]).

• The count of the new k-cycles. In our notation this is the dimension of the quotient $H_k(V^-)/tH_k(V^-)$ (all homology groups are with rational coefficients). Let us denote this number by $R_k$.

• The count of newly bounding k-cycles. This is the dimension of . Let us denote this number by $S_k$. The invariant $Q_k$ introduced by Pitcher is by definition the count of the new k-cycles less the count of newly bounding k-cycles, that is,

  $$\begin{aligned} Q_k=R_k-S_k. \end{aligned}$$

  (15)

  We shall see a bit later, that $Q_k$ is a positive integer. It follows immediately from the exact sequence of the pair $(V^-, tV^-)$ that

  

  (16)

  (where denotes the Betti number in degree k of the pair ).

Denote by $M_k$ the number of critical points of $\theta$ of index k. The classical Morse inequalities applied to the function $F^*$ on the cobordism W imply the inequalities

One deduces the inequalities including the alternated sums of the above invariants.

Theorem 8.1

([16, Theorem I]) For every k we have

$$\begin{aligned} M_k-M_{k-1}+M_{k-2} +\cdots \geqslant Q_k-Q_{k-1}+Q_{k-2} +\cdots \end{aligned}$$

Proof

Let us abbreviate to $\beta _k$. Using the definition (15) of the Pitcher numbers $Q_k$ and the formula (16) it is easy to see that

$$\begin{aligned} \beta _k-\beta _{k-1}+\beta _{k-2} +\cdots = S_k + Q_k-Q_{k-1}+Q_{k-2} +\cdots \end{aligned}$$

The classical Morse inequalities say

$$\begin{aligned} M_k-M_{k-1}+M_{k-2} +\cdots \geqslant \beta _k-\beta _{k-1}+\beta _{k-2} +\cdots \end{aligned}$$

and the theorem follows. $\square$

1.2 Pitcher numbers and Novikov Betti numbers

Let . The numbers $Q_k$ have a simple interpretation in terms of the P-module structure of the homology $H_k(V^-)$. The canonical decomposition of this module writes as follows:

where $n_i\in {{\mathbb {N}}}, 0<n_i\leqslant n_{i+1}$ and are non-constant polynomials with non-zero free term, . We then have

$$\begin{aligned} R_k = a_k+c_k, \quad S_k= c_k. \end{aligned}$$

Therefore $a_k=Q_k$. Let . Then , and the rank of the $\Lambda$-module $H_k(K)$ equals $a_k=Q_k$. We deduce therefore the following results.

Theorem 8.2

([16, Theorem IV]) The numbers $Q_k$ are homotopy invariants of L and the homotopy class of $\theta$.

Theorem 8.3

The number $Q_k$ equals the Novikov Betti number ${\widehat{b}}_k(L, [\theta ])$.

Appendix 2. An example: Novikov complex whose incidence coefficient has arbitrarily small convergence radius

Let q be any integer $\geqslant 3$. In this appendix we construct a circle-valued Morse function f on a 3-manifold, and its transverse gradient u with the following properties:

• (i) The function f has exactly two critical points with indices 2 and 1.

• (ii) The unique Novikov incidence coefficient is a power series of the form $\sum _k a_k t^k$ where , where $C \ne 0$ (see formula (17)).

• (iii) This incidence coefficient is stable with respect to $C^0$-small perturbations of the gradient.

The property (ii) above implies that the convergence radius of the Novikov incidence coefficient equals 1 / q, thus it converges to 0 as $q\rightarrow \infty$. The construction generalizes the example from the author’s work [12, Section 3]. The proof uses the author’s theory of cellular gradients [13, 15], and we begin by a brief outline of this theory. The example itself is constructed in Sect. 9.2, and the reader can start reading this section consulting the introductory Sect. 9.1 when necessary.⁷

1.1 Cellular gradients and rationality theorem

1.1.1 Cellular gradients of Morse functions on cobordisms

Let $f:W\rightarrow [a,b]$ be a Morse function on a compact cobordism W, put $\partial _1 W= f^{-1}(b)$, $\partial _0 W= f^{-1}(a)$. Pick an f-gradient V. For we denote by D(a, v) the descending disc of a, that is, the stable manifold of a with respect to flow induced by v. We denote by D(v) the union of all descending discs and by $D({\scriptstyle {\text { ind}\leqslant {k}}}; v)$ the union of all descending discs of critical points of indices $\leqslant k$. For we denote by ${(-v)}^{\rightsquigarrow }(x)$ the point where the $(-v)$-trajectory $\gamma (x,t;-v)$ starting at x intersects $\partial _0 W$. The correspondence $x\mapsto {(-v)}^{\rightsquigarrow }(x)$ is then a diffeomorphism of onto . If this map is not extensible to a continuous map of $\partial _1 W$ to $\partial _0 W$. However we have shown in [15], see also [13, Part 3], that for a $C^0$-generic gradient this map can be endowed with a structure that closely resembles a cellular map of a CW complex.

Definition 9.1

• Let $\phi :N\rightarrow {{\mathbb {R}}}$ be a self-indexing Morse function on a closed manifold N (that is, ). Put . The filtration

  $$\begin{aligned} \varnothing = N_{-1}\subset N_0\subset \cdots \subset N_r =N \end{aligned}$$

  where $r=\dim N$ is called the Morse–Smale filtration associated to $\phi$ (or MS-filtration for brevity).

• For a given MS-filtration $\{N_i\}$ of N, the filtration by submanifolds is also an MS-filtration, called the dual MS-filtration of the filtration $\{N_i\}$.

Remark 9.2

The term $N_s$ of an MS-filtration is the result of attaching to $N_{-1}$ of handles of indices $\leqslant s$; it is a manifold with boundary homotopy equivalent to an s-dimensional CW complex.

Definition 9.3

An f-gradient v is called almost transverse if $D(p,v) \pitchfork D(q, -v)$ whenever $\mathrm{ind}\,p \leqslant \mathrm{ind}\,q$. The set of all f-gradients is denoted by G(f), the set of all almost transverse f-gradients is denoted by $G_{\mathrm{A}}(f)$, the set of all transverse f-gradients is denoted by $G_{\mathrm{T}}(f)$.

Definition 9.4

Let $f:W\rightarrow [a,b]$ be a Morse function on a cobordism W, v an f-gradient, and v an almost transverse f-gradient. We say that v satisfies the condition $({\mathfrak {C}})$ if there is a Morse–Smale filtration $\{\partial _1 W^k\}$ of $\partial _1 W$ and a Morse–Smale filtration $\{\partial _0 W^k\}$ of $\partial _0 W$ such that for every k,

The gradients satisfying the condition $({\mathfrak {C}})$ will be also called cellular gradients, or ${\mathfrak {C}}$-gradients. The set of all cellular gradients of f will be denoted by $G_{\mathrm{C}}(f)$.

The following theorem is one of the main results of [15], we cite it here using the terminology of [13, Part 3].

Theorem 9.5

The subset $G_{\mathrm{C}}(f)\subset G_{\mathrm{A}}(f)$ is open and dense in $C^0$-topology.

Let v be a cellular f-gradient for a Morse function f on a cobordism W. Consider the compact topological space $\partial _1 W{}^k/\partial _1 W{}^{k-1}$ obtained by shrinking the subspace $\partial _1 W^{k-1}$ to a point denoted $r_{k-1}$. The image of a point $y\in \partial _1 W^k$ in the space $\partial _1 W^k/\partial _1 W^{k-1}$ will be denoted by ${\overline{y}}$. Similar notation will be used for $\partial _0 W$, the shrunk subspace $\partial _0 W^{k-1}$ will be denoted by $s_{k-1}$. The next theorem describes the cellular-like structure on the map ${(-v)}^{\rightsquigarrow }$ (see [13, p. 234]).

Theorem 9.6

If v is a cellular f-gradient then for every k there is a continuous map

$$\begin{aligned} {(-v)}^{\displaystyle \twoheadrightarrow }:\partial _1 W^k/\partial _1 W^{k-1} \rightarrow \partial _0 W^k/\partial _0 W^{k-1} \end{aligned}$$

such that ${(-v)}^{\displaystyle \twoheadrightarrow }(r_{k-1}) = s_{k-1}$ and

$$\begin{aligned}&{(-v)}^{\displaystyle \twoheadrightarrow }(\overline{x}) = s_{k-1}&\quad \text {if}\;\; x\in D(-v), \\&{(-v)}^{\displaystyle \twoheadrightarrow } (\overline{x}) =\overline{{(-v)}^{\rightsquigarrow }(x)}&\quad \text {if}\;\; x\notin D(-v). \end{aligned}$$

Definition 9.7

The map induced by ${(-v)}^{\displaystyle \twoheadrightarrow }$ in homology is denoted by

and called the homological gradient descent.

This homomorphism is stable with respect to $C^0$-small perturbations of the gradient, as shown in [13, p. 282]:

Proposition 9.8

Let v be a cellular gradient of a Morse function $f:W\rightarrow [a,b]$. There is $\delta >0$ such that for every f-gradient w with $\Vert w-v\Vert <\delta$ the homomorphisms

are equal.

1.1.2 Cellular gradients for circle-valued Morse functions and their Novikov complexes

Let $f:M\rightarrow S^1={{\mathbb {R}}}/{{\mathbb {Z}}}$ be a Morse function, we will assume that its class [f] in $H^1(M,{{\mathbb {Z}}})$ is indivisible. Let $\overline{M} \rightarrow M$ be the corresponding infinite cyclic covering; lift the function f to a real-valued Morse function $F:\overline{M}\rightarrow {{\mathbb {R}}}$. Let $\lambda$ be a regular value of F, put $V=F^{-1}(\lambda )$. We have a cobordism $W=F^{-1}([\lambda -1,\lambda ])$ and a Morse function . Let t be a generator of the structure group $\approx {{\mathbb {Z}}}$ of the covering, such that . The map $t^{-1}$ determines a diffeomorphism $\partial _0 W\rightarrow \partial _1 W$ which will be denoted by I. An f-gradient v induces an F-gradient, denoted by the same symbol v.

Definition 9.9

• An f-gradient v is called cellular with respect to $\lambda$ if the induced F-gradient on W is cellular with respect to some MS-filtration $\{N_i\}$ on $\partial _0 W$ and the MS-filtration $\{I(N_i)\}$ on $\partial _1 W$.

• An f-gradient v is called cellular if it is cellular with respect to $\lambda$ for some regular value $\lambda$ of f.

• The set of all cellular gradients of f is denoted $G_{\mathrm{C}}(f)$.

The following theorem is one of the main results of [15], concerning circle-valued Morse functions; we cite it here in the terminology of [13, Chapter 12].

Theorem 9.10

• (i) The subset $G_{\mathrm{C}}(f)\subset G(f)$ is open and dense in G(f) with respect to $C^0$-topology.

• (ii) The subset $G_{\mathrm{C}}(f)\cap G_{\mathrm{T}}(f)\subset G_{\mathrm{T}}(f)$ is open and dense in $G_{\mathrm{T}}(f)$ with respect to $C^0$-topology.

Let v be a cellular f-gradient. For every k we have an endomorphism

Proposition 9.8 implies the following corollary.

Corollary 9.11

Let v be a cellular f-gradient. There is $\delta >0$ such that for every f-gradient w with $\Vert v-w\Vert <\epsilon$ and every r we have .

It turns out that the Novikov complex $N_*(f,v)$ can be computed in terms of this homomorphism. Let . Choose the lifts ${\bar{p}}, {\bar{q}}$ of the points p, q to $\overline{M}$ in such a way that ${\bar{p}}, {\bar{q}} \in t^{-1}W$. Since v is cellular, the k-dimensional submanifold $T=D({\bar{p}},v)\cap \partial _1 W$ is in $\partial _1 W^k$, and the set is compact. Choose some orientations of descending discs D(p, v), D(q, v). We have then the fundamental class . Assume for simplicity of exposition that M is oriented. Similarly, the $(r-k)$-dimensional submanifold $S=D( t{\bar{q}}, -v)\cap \partial _1 W$ determines a class

$$\begin{aligned}{}[S]\in H_{r-k}\bigl (\widehat{\partial }_1 W{}^{r-k}, \widehat{\partial }_1 W{}^{r-k-1}\bigr ), \end{aligned}$$

where $r=\dim V=\dim M -1$. The intersection index $\langle [T], [S] \rangle \in {{\mathbb {Z}}}$ is defined. The next theorem (see [13, p. 379]) expresses the Novikov incidence coefficient in terms of the gradient descent homomorphism.

Theorem 9.12

We have

(Here $n_0(p,q;v)\in {{\mathbb {Z}}}$ is the incidence coefficient of the critical points p, q in the cobordism W; the brackets denote the intersection index.)

The next corollary is obtained by a standard argument from linear algebra.

Corollary 9.13

For any cellular gradient v the Novikov incidence coefficient N(p, q; v) is a rational function of the form where and $Q(0)=1$.

1.2 An example

Let ${{\mathbb {T}}}^2$ be the 2-dimensional torus, $\alpha$ be its parallel, $\beta$ its meridian. Consider two disjoint closed discs $D_1, D_2$ in ${{\mathbb {T}}}^2$ which do no intersect $\alpha \cup \beta$. Removing their interiors from ${{\mathbb {T}}}^2$ we obtain a surface S, whose boundary is the disjoint union of two circles $\partial _1 S$ and $\partial _2 S$ (see the upper image on Fig. 1). Attach a copy S(1, 1) of S to another copy S(1, 2) of S, identifying $\partial _2 S(1,1)$ with $\partial _1 S(1,2)$. We obtain a surface of genus 2 with two components of boundary: $\partial _1 S(1,1)$ and $\partial _2 S(1,2)$. Attaching the copies $D_1(1), D_2(1)$ of the discs $D_1, D_2$ to these components gives a closed surface N of genus 2. One more copy of this surface will be denoted by K (see the bottom of Fig. 1).

Fig. 1

Construction of the manifold N

Similarly we glue together three copies S(1 / 2, 1), S(1 / 2, 0), S(1 / 2, 2) of S and attach to it two discs $D_1(1/2), D_2(1/2)$ to obtain a closed surface L of genus 3 (depicted in the middle of the figure). Associate to every point in $D_1(1/2)\cup S(1/2,1) \cup S(1/2,0)$ its copy in $D_1(1)\cup S(1,1)\cup S(1,2)$; this determines a diffeomorphism which will be denoted by I(1 / 2, 1). Similarly, we construct a diffeomorphism I(1 / 2, 0) of the surface $S(1/2,0)\cup S(1/2,2)\cup D_1(1/2)$ onto $S(0,1)\cup S(0,2) \cup D_2(0)$. A surgery along the circle $\beta (1/2, 2)$ yields a surface naturally diffeomorphic to N. Attaching the corresponding handle of index 2 to gives a cobordism $W_1$ endowed with a Morse function $F_1:W_1\rightarrow [1/2, 1]$. This Morse function has one critical point $x_2$ of index 2. Pick a gradient $w_1$ for this function in such a way that:

• (A1) The ascending disc $D(x_2, -w_1)$ intersects the level surface $N=F^{-1}_1(1)$ by two points in the interior of $D_2(1)$.

• (A2) The diffeomorphism $\overset{\rightsquigarrow }{w_1}$ sends $D_2(1/2)$ to the interior of $D_2(1)$.

• (A3) The restriction of $\overset{\rightsquigarrow }{w_1}$ to $D_1(1/2)\cup S(1/2,1)\cup S(1/2,0)$ equals I(1 / 2, 1) everywhere except a small tubular neighbourhood $T_1$ of the circle $\partial D_1(1/2)$. Further, $\overset{\rightsquigarrow }{w_1} (T_1) = I(1/2, 1) (T_1)$ and the $\overset{\rightsquigarrow }{w_1}$-image of $D_1(1/2)$ contains $D_1(1)$ in its interior.

Similarly, we do a surgery along the circle $\beta (1/2,1)$ and obtain a surface naturally diffeomorphic to K. Attach the corresponding handle to , get a cobordism $W_0$ endowed with a Morse function $F_0:W_0\rightarrow [0,1/2]$ having one critical point $x_1$ of index 1. Pick an $F_0$-gradient $w_0$ such that:

• (B1) The descending disc $D(x_1, w_0)$ intersects the level surface $K=F^{-1}_0(0)$ by two points in the interior of $D_1(0)$.

• (B2) The diffeomorphism ${(-w_0)}^{\rightsquigarrow }$ sends $D_1(1/2)$ to the interior of $D_1(0)$.

• (B3) The restriction of ${(-w_0)}^{\rightsquigarrow }$ to $S(1/2,0)\cup S(1/2,2)\cup D_1(1/2)$ equals I(1 / 2, 0) everywhere except a small tubular neighbourhood $T_2$ of the circle $\partial D_2(1/2)$. Further, ${(-w_0)}^{\rightsquigarrow }(T_2) = I(1/2, 0) (T_2)$ and the ${(-w_0)}^{\rightsquigarrow }$-image of $D_2(1/2)$ contains $D_2(0)$ in its interior.

We have $\partial W_1 \approx N\sqcup L$, $\partial W_0 \approx L\sqcup K$; attaching $W_1$ to $W_0$ along the L-component of their boundaries we obtain a cobordism W with boundary $\partial W\approx N\sqcup K$, endowed with a Morse function $F:W\rightarrow [0,1]$, such that , . The gradients $w_0$ and $w_1$ can be glued together (modifying them appropriately nearby L if necessary) so that the resulting gradient w is cellular F-gradient (see Definition 9.4). To show this we introduce Morse–Smale filtrations on $\partial _0 W$ and $\partial _1 W$. Let

$$\begin{aligned} N_0=D_1(1), \quad N_1 = D_1(1) \cup S(1,1)\cup S(1,2), \quad N_2= N. \end{aligned}$$

The filtration $N_0\subset N_1\subset N_2$ is then an MS-filtration of N. The image of this filtration with respect to the natural diffeomorphism $J:N \rightarrow K$ is an MS-filtration $K_0\subset K_1 \subset K_2$ on K. Properties (A1)–(A3) and (B1)–(B3) imply the following:

$$\begin{aligned}&\mathrm{(D1)}&\qquad D(x_i, w) \cap \partial _0 W\subset \mathrm{Int}\,K_{i-1} \quad \text {for}\;\; i=1,2, \\&\mathrm{(D2)}&\qquad {(-w)}^{\rightsquigarrow } (N_i) \subset \mathrm{Int}\,K_i \quad \text {for}\;\; i=0,1,2. \end{aligned}$$

We have also the dual properties

$$\begin{aligned}&\mathrm{(U1)}&\qquad D(x_i, -w) \cap \partial _1 W\subset \mathrm{Int}\,{\widehat{N}}_{2-i} \quad \text {for} \;\; i=1,2, \\&\mathrm{(U2)}&\qquad \overset{\rightsquigarrow }{w} ({\widehat{K}}_i) \subset \mathrm{Int}\,{\widehat{N}}_i \quad \text {for} \; \;i=0,1,2 \end{aligned}$$

(recall that ${\widehat{N}}_i$ and ${\widehat{K}}_i$ denote the MS-filtrations dual to the filtrations $N_i$, resp. $K_i$). The conjunction of properties (D1), (D2) is just a reformulation of the condition ($\mathfrak {C}$1); similarly the conjunction of properties (U1), (U2) is equivalent to ($\mathfrak {C}$2). The F-gradient w is therefore cellular.

The 3-manifold M and a circle-valued function on it will be obtained by gluing N to K via a diffeomorphism that we will now describe. Put

$$\begin{aligned} a_1=[\alpha (1,1)], \quad a_2=[\alpha (1,2)], \quad b_1=[\beta (1,1)], \quad b_2=[\beta (1,2)]. \end{aligned}$$

The family is then a basis in $H_1(N)$. The same embedded circles determine the homology classes in $H_1(N_1/N_0)$, they will be denoted by the same letters by a certain abuse of notation. Similarly we obtain a base in $H_1(K)$ and $H_1(K_1/K_0)$. Denote by $J:N\xrightarrow { \, \approx \,} K$ the natural diffeomorphism, then we have . Consider the automorphism of $H_1(K)\approx {{\mathbb {Z}}}^4$ given in the base by the following matrix:

$$\begin{aligned} {{\mathfrak {S}}}=\left( \begin{matrix} 0 &{} \quad 2 &{} \quad 1 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{} \quad 1\\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ -1 &{}\quad q &{} \quad 0 &{} \quad -2\\ \end{matrix} \right) \end{aligned}$$

where q is any integer $\geqslant 3$. It is easy to check that ${{\mathfrak {S}}}$ preserves the intersection form on $H_1(N)$ therefore there is a diffeomorphism $\Phi :K\rightarrow K$ inducing ${{\mathfrak {S}}}$ in $H_1$. We can assume that $\Phi (x)=x$ for $x\in D_1(0)\cup D_2(0)$. Identifying each point $y\in K$ with $J\Phi (y)\in N$ we obtain a 3-manifold M; the Morse function F induces a map $f:M\rightarrow S^1$. The F-gradient w induces an f-gradient v. Observe that v is a cellular f-gradient with respect to regular level surface N and its MS-filtration $\{N_i\}$. The matrix of the endomorphism is easy to compute; it equals

$$\begin{aligned} {{\mathfrak {M}}}= \left( \begin{matrix} 0 &{}\quad 0 &{}\quad 0 &{}\quad 2\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{} \quad 1\\ 0 &{}\quad 0 &{}\quad -1 &{}\quad q \end{matrix} \right) . \end{aligned}$$

Pick a transverse f-gradient u sufficiently close to v in $C^{\infty }$ topology so that u is still a cellular f-gradient with respect to the level surface N and its MS-filtration $\{N_i\}$, and . The Novikov incidence coefficient $N( x_2, x_1; u)$ is now easy to compute. Let T be the $J^{-1}$-image in N of $D(x_2,u)\cap K$, and $\theta =[T]\in H_1(N_1/N_0)$. Then $\theta =b_1-2b_2$. Let $S=D(x_1,-u)\cap N$ then $[S]=b_1\in H_1(N_1/N_0)$. Applying Theorem 9.12 we obtain

$$\begin{aligned} n_{k+1}( x_2, x_1; u) = \big \langle {{\mathfrak {M}}}^k(\theta ), b_1\big \rangle . \end{aligned}$$

We have . Therefore $n_{k+1}( x_2, x_1; u)$ equals the first coordinate of the vector with respect to basis . Computing this first coordinate is a routine exercise in linear algebra which will be left to the reader. We give just the result:

(17)

The properties of f and v stated in the beginning of this appendix are now obvious.