Alternative to Morse-Novikov Theory for closed 1-form (I)

TL;DR

This paper generalizes an alternative approach to Morse-Novikov theory for closed 1-forms on compact spaces, introducing configurations that recover Novikov-Betti numbers and complex ranks under mild conditions.

Contribution

It extends the alternative to Morse-Novikov theory from maps to closed 1-forms on compact ANRs, defining configurations that encode topological invariants.

Findings

Configurations recover Novikov-Betti numbers.

Configurations determine ranks of Novikov complex boundary maps.

Properties like stability and Poincaré duality are formulated.

Abstract

This paper extends the Alternative to Morse-Novikov theory we have proposed in Burghelea (New topological invariants for real- and angle valued maps, World Scientific, Hackensack, 2018) from real- and angle-valued map to closed 1-forms. For a topological closed 1-form on a compact ANR (= absolute neighborhood retract), a concept generalizing closed differential 1-form on a compact manifold, under the mild hypothesis of tameness, a field and a non-negative integer we propose two configurations of points, the first on the real line the second on the positive real line, which recover Novikov-Betti numbers and the Novikov complex associated with a Morse closed 1-form with non-degenerated zeros. Precisely, the sum of the multiplicities of the points in the support of the first configuration which correspond to the integer r equals the r-th Novikov-Betti number and that of the points in the support of second configuration which corresponds to the integer r equals the rank of the boundary map in the Novikov complex. We formulate the basic properties of these configurations, the stability property and the Poincare duality property when the compact ANR is a closed orientable topological manifold, which in full generality will be proven in the second and third part of this work.

Equations

$$\sum _{t\in \mathbb{R}}{\mathbf{\delta }}_r^{\omega }(t)={\beta }_r^N(X;\xi (\omega ))$$

$$c_r(\omega )=\sum _{t\in \mathbb{R}}{\mathbf{\delta }}_r^{\omega }(t)+\sum _{t\in {\mathbb{R}}_{+}}{\mathbf{\gamma }}_r^{\omega }(t)+{\mathbf{\gamma }}_{r-1}^{\omega }(t)+\boxed{{\lambda }_{r-1}^{\omega }}$$

$$\mathrm{rank}{d}_r=\sum _{t\in {\mathbb{R}}_{>0}}{\mathbf{\gamma }}_r^{\omega }(t).$$

$${\mathcal{Z}}^1(X)={\bigsqcup }_{\xi \in H^1(X;\mathbb{R})}{\mathcal{Z}}^1(X;\xi ).$$

$$||\omega |{|}_{\mathcal{U}}:=\underset{\alpha \in A}{\sup }\underset{x\in {\overline{U}}_{\alpha }}{\sup }|f_{\alpha }^{\omega }(x)|,$$

$$D({\omega }_1,{\omega }_2{)}_{\mathcal{U}}:=||{\omega }_1-{\omega }_2|{|}_{\mathcal{U}}.$$

$$D({\omega }_1,{\omega }_2{)}_{\tilde{x}}:=\underset{y\in \tilde{X}}{\sup }|f_{\tilde{x}}^{{\omega }_1}(y)-f_{\tilde{x}}^{{\omega }_1}(y)|$$

$$X_t^f:=f^{-1}((-\infty ,t]),$$

$$X_f^t:=f^{-1}([t,\infty )),$$

$$H_r^{inv}(X;f):=\underset{t\to -\infty }{\underleftarrow{\lim }}{H}_r(X_t^f),H_r^{dir}(X;f):=\underset{t\to \infty }{\underleftarrow{\lim }}{H}_r(X_t^f)$$

$$i_{-\infty }^t{=}^f{i}_{-\infty }^t(r):H_r^{inv}(X;f)\to$$

$$i_t^{\infty }{=}^f{i}_t^{\infty }(r):H_r(X_t^f)\to$$

$${\mathbb{J}}_r^f:{=}^f{i}_{-\infty }^{\infty }(r):H_r^{inv}(X;f)\to$$

$$H_r(X_t^{f_1})\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$H_r^{inv}(X;f_1)=$$

$$H_r^{dir}(X;f_1)=$$

$${}^{f_1}{i}_{-\infty }^{\infty }{=}^{f_2}{i}_{-\infty }^{\infty }=$$

$$H^{inv}(X;\xi )$$

$$H^{dir}(X;\xi )$$

$${\mathbb{J}}_r^{\xi }$$

$${\mathbb{J}}_r^{\xi }:H^{inv}(X;\xi )\to H^{dir}(X;\xi ).$$

$$R_a^f(r):=\dim H_r(X_a^f,X_{<a}^f)\in {\mathbb{Z}}_{\ge 0}\bigsqcup \infty ,R_f^a(r):=\dim H_r(X_f^a,X_f^{>a})\in {\mathbb{Z}}_{\ge 0}\bigsqcup \infty .$$

$$H_r^{dir}(X;f)=H_r(X),$$

$$\mathcal{U}(\delta ;U_1,\cdots U_k,V):=\{{\delta }^{\prime }\in Conf(Y\setminus K)\mid \{\begin{array}{l}\mathrm{s}upp{\delta }^{\prime }\subset ({\bigcup }_{i=1,2,\cdots k}{U}_i)\cup V\\ {\sum }_{y\in U_i}{\delta }^{\prime }(y)=\delta (y_i)\end{array}\}$$

$$\mathcal{U}(\delta ,U_1,\cdots U_k):=\mathcal{U}(\delta ,U_1,\cdots U_k),\varnothing ).$$

$$\cdots {\tilde{X}}_t^f\subset$$

$$\cdots {\tilde{X}}_f^t\supset$$

$$\cdots {\mathbb{I}}_t^f(r)\subseteq$$

$$\cdots {\mathbb{I}}_f^t(r)\supseteq$$

$${\mathbb{I}}_t^f(r):=\mathrm{i}mg(H_r({\tilde{X}}_t)\to H_r(\tilde{X}))\mathrm{and}{\mathbb{I}}_{\mathrm{f}}^{\mathrm{t}}(\mathrm{r}):=\mathrm{img}({\mathrm{H}}_{\mathrm{r}}({\tilde{\mathrm{X}}}^{\mathrm{t}})\to {\mathrm{H}}_{\mathrm{r}}(\tilde{\mathrm{X}}))$$

$$(\sum _{i=1,\cdots k}{a}_{g_i}{g}_i)\cdot x:=\sum _{i=1,\cdots k}{a}_{g_i}(⟨g_i⟩(x))$$

$$(\sum _{i=1,\cdots k}{a}_{g_i}{g}_i)\cdot x\in {\mathbb{I}}_{t+\max g_i}^f(r),resp.(\sum _{\mathrm{i}=1,\cdots \mathrm{k}}{\mathrm{a}}_{{\mathrm{g}}_{\mathrm{i}}}{\mathrm{g}}_{\mathrm{i}})\cdot \mathrm{x}\in {\mathbb{I}}_{\mathrm{f}}^{\mathrm{t}+\min {\mathrm{g}}_{\mathrm{i}}}(\mathrm{r}).$$

$$(a_0{g}_0+a_1{g}_1+\cdots a_k{g}_k)\cdot x=0$$

$$a_{k}x=-(a_0(g_0-g_k)+a_1(g_1-g_k)+\cdots a_{k-1}(g_{k-1}-g_k))\cdot x,$$

$$0=\sum _{1\le i\le K}{P}_{r_i}\cdot (⟨g_{r_i}⟩(x))=(\sum _{1\le i\le K}{P}_{r_i}{g}_{r_i})\cdot x.$$

Full text

Dedicated to the memory of Stefan Papadima

1. 1 Introduction
2. 2 Topological closed one forms and tameness
3. 3 Topology
4. 4 The maps $\delta^{f}_{r}$ and $\gamma^{f}_{r}$
5. 5 The configurations $\boldsymbol{\delta}^{\omega}_{r}$ and $\boldsymbol{\gamma}^{\omega}_{r}$
6. 6 Appendix.1

1 Introduction

In this paper 111 The present version of this this paper is prompted by the discovery of a mistake in the proof of Theorem 3.2 page 723 of the printed version of this work in EJM 2020 no 6. Fortunately all results but Theorem 3.2 as formulated remain true by essentially the same arguments with minor improvements once an additional requirement ( hypothesis H, cf section 2 for definition), be added to the tameness of the TC1-form. This requirement is conjecturally superfluous and indeed superfluous for all TC1-forms whose cohomology class $\xi$ belongs to $img(H^{1}(M;\mathbb{Z})\to H^{1}(M;\mathbb{R})$ and for many other cases of pairs $(M;\xi\in H^{1}(M;\mathbb{R}))$ as discussed in subsection 2.2 we define the configurations $\delta^{\omega}$ and $\gamma^{\omega}$ for a tame topological closed one form $\omega$ on a compact ANR. They are analogues of the configurations $\delta^{f}_{r}$ and $\gamma^{f}_{r},$ previously defined in [3] for a tame real- or angle-valued maps $f.$ As a consequence we extend our Alternative to Morse-Novikov Theory (cf. [1]) from tame angle-valued maps to tame topological closed 1-form on a compact ANR.The concept topological closed 1-form, abbreviated TC1-form, is a generalization of closed differential 1-form on a smooth manifold. As in the case of real- or angle-valued maps (which corresponds to the case of closed 1-form of degree of irrationality zero or one) we will analyze the real-valued map $f^{\omega}:\tilde{X}\to\mathbb{R},$ a lift of $\omega,$ defined on the total space of the associated principal $\Gamma=\mathbb{Z}^{k}-$covering $\pi:\tilde{X}\to X$ associated to the cohomology class $[\omega]\in H^{1}(X;\mathbb{R})$ with $k$ the degree of irrationality of $\omega.$

It might appear that this should be a routine extension of the case $k=1$ but this is not quite so because:

1. the map $f^{\omega}$ is never proper when degree of irrationality $k$ is greater than $1$, so the homology vector spaces which are involved in, finite dimensional in the case $k\leq 1,$ might be infinite dimensional in the case $k\geq 2,$

2. the set of critical values of $f^{\omega}$ is not discrete when $k>1$ but the opposite, always dense if not empty; the approach of Zig-Zag persistence based on graph representations, cf. [2], is apparently not applicable.

However, the tameness of the lift $f^{\omega}$ of $\omega,$ ,see secytion 2 for definition) and the fact that the group $\Gamma-$ defined by the form $\omega$ appears as a subgroup of $\mathbb{R}$ make the approach described in sections 6 and 7 of [1] 222initiated in [3] applicable. Ultimately this leads to the finite configuration $\boldsymbol{\delta}^{\omega}_{r}$ and $\boldsymbol{\gamma}^{\omega}_{r}$ of points in $\mathbb{R}$ and $\mathbb{R}_{+}$ derived however from the $\mathbb{Z}_{\geq 0}-$valued maps $\boldsymbol{\delta}^{f^{\omega}}_{r}$ and $\boldsymbol{\gamma}^{f^{\omega}}_{r}$ which are not configurations.

To prove our result we consider in section 3 an apparently new definition of Novikov-Betti numbers based on the lifts of $\omega$ and verify in section 5.2 that this definition is equivalent to the standard ones (cf. [7] for definitions). We are unaware if it already exists in literature.

For the configurations $\boldsymbol{\delta}_{r}^{\omega}$ and $\boldsymbol{\gamma}^{\omega}_{r}$ one can prove a stability property and a Poincaré duality property similar to Theorems 1.3 and 1.5 in [3] and, in view of the stability property of the assignment $\omega\rightsquigarrow\boldsymbol{\gamma}^{\omega}_{r}$ (cf. Theorem 1.3), show that the configurations $\boldsymbol{\delta}^{\omega}_{r}$ can be actually defined for any TC1-form, not necessary tame. It also can be refined in the spirit of [5]and [6] to an assignment with values $\kappa[\Gamma]-$modules but we are not interested in this aspect nor in its implications at this time.

The main results about the configurations $\boldsymbol{\delta}^{\omega}_{r}$ and $\boldsymbol{\gamma}^{\omega}_{r}$ are stated in Theorems 1.1 Theorems 1.2 and 1.3. In this paper (part I) only Theorem 1.1 is proven entirely, the other two will be established in part II and part III of this work.

To formulate them we note :

For a fixed field $\kappa,$ $X$ a compact ANR, $\xi\in H^{1}(X;\mathbb{R}),$ denote by $\beta^{N}_{r}(X;\xi)$ the $r-$th Novikov-Betti number, $r=0,1,2,\cdots$ and by $\mathcal{Z}^{1}(X;\xi)$ the set of topological closed 1-form on $X$ in the cohomology class $\xi$ (cf. definition section 2 ) equipped with the compact open topology.

For a closed TC-1 form $\omega$ denote by $CR(\omega):=\{t=c^{\prime}-c^{\prime\prime}\mid c^{\prime},c"\in CR(f^{\omega})\}$ with $CR(f^{\omega})$ the set of critical values of $f^{\omega}$ a lift of $\omega.$ Clearly this set of real numbers is independent of the lift $f^{\omega}$.

As described in Section 2 the cohomology class $\xi\in H^{1}(X;\mathbb{R})$ defines two $\kappa[\Gamma]-$modules, $H^{inv}(X;\xi)$ and $H^{dir}(X;\xi)$ and a $\kappa[\Gamma]-$linear map $J^{\xi}_{r}:H^{inv}(X;\xi)\to H^{dir}(X;\xi)$ conjecturally always injective and verified to be injective for any $\xi$ in the image of $H^{1}(X;\mathbb{Q})\to H^{1}(X;\mathbb{R}),$ as shown in Section 2.

For a space $Y$ and closed subset $K\subset Y$ one denotes by $Conf_{N}(Y)$ the space of configurations of total cardinality $N$ equipped with the collision topology and by $Conf(Y\setminus K)$ the space of configuration of points in $Y\setminus K$ equipped with the bottleneck topology, all these topologies described in sections 2 and 3 below. In this paper $Y=\mathbb{R}$ and $K=(-\infty,0].$

Theorem 1.1

If $\omega$ is a weakly tame topological closed 1-form on $X$ (cf. section 2 for definition) then

[TABLE]

where $\xi(\omega)=[\omega]$ denotes the cohomology class determined by $\omega.$

The support of $\boldsymbol{\delta}_{r}^{\omega}$ and $\boldsymbol{\gamma}_{r}^{\omega}$ are real numbers in $CR(\omega).$ 2. 2.

If $X=M^{n}$ is a closed smooth manifold of dimension $n,$ $\omega$ is a tame closed differential 1-form (cf definition in section 2) in the cohomology class $\xi$ with all zeros of Morse type and $c_{r}(\omega)$ denotes the number of zeros of Morse index $r$ then

[TABLE]

with $\lambda_{r-1}^{\omega}=0$ if $J_{r-1}^{\xi}$ is injective. The definition of $\lambda^{\omega}_{r}$ is given in section 4 (11 ).

Item 1. explains in what sense $\boldsymbol{\delta}^{\omega}_{r}$ refines the Novikov-Betti numbers.

Since a chain complex of finite dimensional vector spaces is, up to a non canonical isomorphism, completely determined by the dimensions of its homology vector spaces and the dimensions of its components Item 2. explains to what extent the configurations $\boldsymbol{\delta}^{\omega}_{r}$ and $\boldsymbol{\gamma}^{\omega}_{r}$ supplemented by the numbers $\lambda^{\omega}_{r}$ if nonzero) provide together a refinement of the Novikov complex when considered over the Novikov field. Note that the formula in Item 2. gives the rank of $d_{r}:C_{r}\to C_{r-1},$ the boundary map in the Novikov complex. Precisely

[TABLE]

The extension of this result from smooth Morse closed one forms on a closed manifold to tame TC1-form on a compact ANR should be regarded as a considerable weakening of the hypothesis ” all zeros of $\omega$ are of Morse type” in order to recover all informations provided by the Novikov complex.

Theorem 1.2

Suppose $M$ is a closed topological manifold and $\omega\in\mathcal{Z}^{1}_{t}(X;\xi),$

$\boldsymbol{\delta}^{\omega}_{r}(t)=\boldsymbol{\delta}^{\omega}_{n-r}(-t),$** 2. 2.

$\boldsymbol{\gamma}^{\omega}_{r}(t)=\boldsymbol{\gamma}^{-\omega}_{n-r-1}(t).$**

Let $\mathcal{Z}^{1}_{t}(X;\xi)\subset\mathcal{Z}(X;\xi)$ denote the space of tame topological closed 1-forms with the topology induced from the compact open topology on $\mathcal{Z}^{1}(X;\xi),$ 333 $\mathcal{Z}^{1}_{t}(X;\xi)$ is dense in $\mathcal{Z}(X;\xi)$ defined in Section 2. The topology on $Conf_{\beta^{N}_{r}(X:\xi)}(\mathbb{R})$ is the collision topology and the topology on $Conf(\mathbb{R}_{+}),$ with $\mathbb{R}_{+}$ viewed as $Y\setminus K$ for $Y=\mathbb{R}$ and $K=(-\infty,0],$ is the bottleneck topology described in subsection (3.1).

Theorem 1.3

The assignment $\boldsymbol{\delta}_{r}:\mathcal{Z}^{1}_{t}(X;\xi)\rightsquigarrow Conf_{\beta^{N}_{r}(X:\xi)}(\mathbb{R})$ is continuous and extends to a continuous assignment on the entire $\mathcal{Z}^{1}(X;\xi).$ 2. 2.

The assignment $\boldsymbol{\gamma}_{r}:\mathcal{Z}^{1}_{t}(X;\xi)\rightsquigarrow Conf(\mathbb{R}_{+})$ is continuous.

To understand the relations between this paper and the previous works, [2], [3] and [1], the following observations are useful:

1. When $X$ is connected, a topological closed 1-form $\omega$ can be represented by a real valued map $f:\tilde{X}\to\mathbb{R},$ called lift of $\omega$ cf. subsection 2.1, $\tilde{X}$ the total space of the principal covering associated to $\omega.$ This lift is determined up to an additive constant, cf. section 2 ( i.e. $f_{1}$ and $f_{2}$ lifts of the form $\omega$ implies $f_{1}=f_{2}+t,t\in\mathbb{R}$). The configurations $\boldsymbol{\delta}_{r}^{\omega}$ and $\boldsymbol{\gamma}_{r}^{\omega}$ are derived from the integer-valued maps $\delta_{r}^{f}$ resp. $\gamma_{r}^{f}$ which have as supports points $(a,b)\in\mathbb{R}^{2}$ resp. $\mathbb{R}^{2}_{+}=\{(x,y)\mid x<y\}$ with $a,b$ critical values of $f.$ Since the support of $\delta^{f+t}$ resp. $\gamma^{f+t}$ is the $t-$diagonal translate of the support $\delta^{f}$ resp. $\gamma^{f},$ in order to get independence on the representative $f,$ one passes to the quotient spaces $\mathbb{R}^{2}/\mathbb{R}=\mathbb{R}$ resp. $\mathbb{R}^{2}_{+}/\mathbb{R}=\mathbb{R}_{+},$ where the quotient is taken w.r. to the diagonal action action $\mu_{\Delta}(t,(a,b))=(a+t,b+t).$ The support of $\boldsymbol{\delta}_{r}^{\omega}$ and $\boldsymbol{\gamma}_{r}^{\omega}$ is the image by $p:\mathbb{R}^{2}\to\mathbb{R},\ p(a,b)=b-a,$ of the support of $\delta_{r}^{f}$ and $\gamma_{r}^{f}.$

2. In this paper the notation $\gamma^{f}_{r}$ refers to the restriction to $\mathbb{R}^{2}_{+}=\{(x,y)\in\mathbb{R}^{2}\mid y-x>0\}$ of the map denoted by the same letter $\gamma^{f}_{r}$ in [1]. Note that such restriction to $\mathbb{R}^{2}_{+}$ collects information on the so called closed-open bar codes of $f,$ the ones of relevance in the Morse-Novikov theory, while the restriction to $\mathbb{R}^{2}_{-}=\{(x,y)\in\mathbb{R}^{2}\mid y-x<0\}$ collects information on the open-closed bar codes of $f.$ Note also that the open-closed bar codes of $f$ correspond to the closed-open bar codes of $-f$ via the correspondence $(a,b)\to(-b,-a).$

3. An interesting example of a tame closed topological 1-form is provided by a simplicial 1-cocycle on a finite simplicial complex. An algorithm to derive the bar codes (i.e. points in the support of $\boldsymbol{\delta}^{\omega}$ and $\boldsymbol{\gamma}^{\omega}$ with their multiplicity is desirable. This is possible and will be the topic of subsequent work.

In this paper we write ”$=$” for equality or canonical isomorphism and $\simeq$ for isomorphism, not necessary canonical.

An alternative treatment via persistent homology of Floer-Novikov theory was proposed by Usher and Zhang cf. [8]. Their work has challenged us to extend the results presented in [1] from angle-valued maps to topological closed one forms.

2 Topological closed one forms and tameness

2.1 Topological closed 1-form

A topological closed 1-form, abbreviated TC1-form, extends the concept of closed differential 1-form on a smooth manifold $M$ to an arbitrary topological space $X.$ One way to obtain this is to view it as an equivalence class of multivalued maps (first definition), an other way is to view it as an equivalence class of equivariant maps on the associated principal $\mathbb{Z}^{k}-$covering (second definition).

First definition

A multi-valued map is a systems $\{U_{\alpha},f_{\alpha}:U_{\alpha}\to\mathbb{R},\alpha\in A\}$ s.t.

1. (a)

$U_{\alpha}$ are open sets with $X=\cup U_{\alpha},$ 2. (b)

$f_{\alpha}$ are continuous maps s.t $f_{\alpha}-f_{\beta}:U_{\alpha}\cap U_{\beta}\to\mathbb{R}$ is locally constant. 2. 2.

Two multi-valued maps are equivalent if put together remain a multi-valued map.

Definition 2.1

A TC1-form is an equivalence class of multi-valued maps.

A TC1-form $\omega$ determines a cohomology class $\xi=\xi(\omega)\in H^{1}(X;\mathbb{R}).$ It suffices to show that a representative $\{U_{\alpha},f_{\alpha}:U_{\alpha}\to\mathbb{R}\}$ of $\omega$ defines for any continuous path $\gamma:[a,b]\to X$ the number $\int_{\gamma}\omega\in\mathbb{R},$ independent of the homotopy class rel. boundary of $\gamma$ and additive w.r. to juxtaposition of paths. Indeed, if $\gamma[a,b]\subset U_{\alpha}$ for some $\alpha,$ then $\int_{\gamma}\omega=f_{\alpha}(b)-f_{\alpha}(a);$ if not, one choses a subdivision of $[a,b],$ $a=t_{0}<t_{1}<\cdots t_{r}=b,$ such that $\gamma_{i}:=\gamma|_{[t_{i},t_{i+1}]}$ lie in some open set $U_{\alpha}$ and define $\int_{\gamma}\omega:=\sum\int_{\gamma_{i}}\omega.$ This assignment defines an homomorphism $\xi(\omega):H_{1}(X;\mathbb{Z})\to\mathbb{R},$ equivalently a cohomology class $\xi(\omega).$ One denotes by $\mathcal{Z}^{1}(X)$ the set of all TC1-forms and by $\mathcal{Z}^{1}(X;\xi):=\{\omega\in\mathcal{Z}^{1}(X)\mid\xi(\omega)=\xi\}.$ Clearly $\mathcal{Z}^{1}(X)$ is an $\mathbb{R}-$ vector space and

[TABLE]

In view of this definition, for any $X$ compact ANR one can find open covers of $X$, $\{U_{\alpha},\alpha\in A\}$ with the properties that $A$ is a finite set and $\overline{U}_{\alpha}$ is compact, connected and simply-connected. Such cover is called good cover. A choice $x_{\alpha}\in U_{\alpha}$ makes $\omega$ uniquely represented by a multivalued map $\{f_{\alpha}^{\omega}:U_{\alpha}\to\mathbb{R}\}$ with $f_{\alpha}^{\omega}(x_{\alpha})=0.$ One calls the system $\mathcal{U}:=\{U_{\alpha},x_{\alpha},\alpha\in A\}$ with $\{U_{\alpha},\alpha\in A\}$ good cover a base-pointed good cover of $X.$

The choice of a base pointed good cover $\mathcal{U}$ defines for the vector space $\mathcal{Z}^{1}(X)$ a complete norm,

[TABLE]

and then a distance in $\mathcal{Z}^{1}(X)$ and implicitly in $\mathcal{Z}^{1}(X;\xi),$

[TABLE]

Different base-pointed good covers lead to equivalent norms. The induced topology on $\mathcal{Z}^{1}(X)$ is referred to as the compact open topology. The subsets $\mathcal{Z}^{1}(X;\xi)$ are the connected components of $\mathcal{Z}^{1}(X).$ Examples:

A closed differential 1-form, $\omega\in\Omega^{1}(M)$ $d\omega=0$, defines a TC1-form.

Indeed, in view of Poincaré Lemma, for any $x\in M$ one chooses an open neighborhood $U_{x}\ni x$ and $f_{x}:U_{x}\to\mathbb{R}$ a smooth map s.t. $\omega_{x}|_{U_{x}}=df_{x}.$ The system $\{U_{x},f_{x}:U_{x}\to\mathbb{R}\}$ provides a representative of a TC1-form. 2. 2.

A simplicial 1- cocycle on the simplicial complex $X$ defines a TC1-form.

If $X$ is a simplicial complex, $\mathcal{X}_{0}$ the collection of vertices and $S\subset\mathcal{X}_{0}\times\mathcal{X}_{0}$ the collection of pairs $(x,y),$ $x,y\in\mathcal{X}_{0}$ s.t. $x,y$ are boundaries of a $1-$simplex, then a simplicial 1-cocycle is a map $\delta:S\to\mathbb{R}$ with the properties $\delta(x,y)=-\delta(y,x)$ and for any three vertices $x,y,z$ with $(x,y),(y,z),(x,z)\in S$ one has $\delta(x,y)+\delta(y,z)+\delta(z,x)=0.$ The collection of open sets $U_{x},$ $U_{x}$ the open star of the vertex $x\in\mathcal{X}_{0},$ and the maps $f_{x}:U_{x}\to\mathbb{R},$ the linear extensions to open simplexes of $U_{x}$ of the map given on the vertices in $U_{x}$ by $f_{x}(y):=\delta(x,y)$ and $f_{x}(x)=0$ provides a representative of a TC1-form.

Second definition. Let $\xi\in H^{1}(X;\mathbb{R})=Hom(H_{1}(X;\mathbb{Z}),\mathbb{R})$ and $\Gamma=\Gamma(\xi):=\mathrm{i}mg(\xi)\subset\mathbb{R}.$ If $X$ is a compact ANR then $\Gamma\simeq\mathbb{Z}^{k}$ with $k$ called the degree of irrationality of $\xi.$

The surjective homomorphism $\xi:H_{1}(X;\mathbb{Z})\to\Gamma$ defines the associated $\Gamma-$principal covering, $\pi:\tilde{X}_{\xi}=\tilde{X}\to X$, i.e. a free action $\mu:\Gamma\times\tilde{X}\to\tilde{X}$ with $\pi$ the quotient map $\tilde{X}\to\tilde{X}/\Gamma=X.$ This principal covering is unique up to isomorphism. When $X$ is connected so is $\tilde{X}.$

A continuous map $f:\tilde{X}\to\mathbb{R}$ is $\Gamma-$equivariant if ${f(\mu(g,x))=f(x)+g}.$

Definition 2.2

A TC1-form $\omega$ of cohomology class $\xi$ is an equivalency class of continuous $\Gamma-$equivariant real-valued maps $f:\tilde{X}\to\mathbb{R}$ where $f_{1}$ is equivalent to $f_{2}$ iff $f_{1}-f_{2}$ is locally constant.

One refers to any representative $f$ in this class as a lift of $\omega.$ Clearly any $\Gamma-$ equivariant map on a $\Gamma-$ principal covering $\tilde{X}\to X$ defines a cohomology class in $H^{1}(X;\mathbb{R})=Hom(H_{1}(X;\mathbb{Z}),\mathbb{R}),$ the same for equivalent equivariant maps. This because for any continuous path $\gamma:[0,1]\to X$ and $\tilde{x}\in\tilde{X}$ with $\pi(\tilde{x})=\gamma(0)$ there is a unique $\tilde{\gamma}:[0,1]\to\tilde{X}$ with $\tilde{\gamma}(0)=\tilde{x}$ and $\gamma=\pi\cdot\tilde{\gamma}$ and, by taking $\int_{\gamma}\omega:=f(\tilde{\gamma}(1))-f(\tilde{x})),$ one obtains an homomorphism $H_{1}(X;\mathbb{Z})\to\mathbb{R}.$ Denote by $\mathcal{Z}^{1}(X;\xi)$ the set of TC1-forms in the cohomology class $\xi.$ In view of this definition the choice of the base point $\tilde{x}$ in $\tilde{X}$ (actually one in each connected component if $\tilde{X}$ is not connected) provides a unique lift $f^{\omega}_{\tilde{x}}:\tilde{X}\to\mathbb{R}$ of $\omega$ with $f^{\omega}_{\tilde{x}}(\tilde{x})=0.$ When $X$ is compact one defines the distance $D(\omega_{1},\omega_{2})_{\tilde{x}}$ by

[TABLE]

which, in view of the compacity of $X$ and of the $\Gamma-$equivariance of the lifts, is a complete metric. It is not hard to see that different choices of the base point $\tilde{x}$ lead to equivalent distances and therefore to the same induced topology with the same collection of Cauchy sequences. It is not hard to show that the two definitions of $\mathcal{Z}^{1}(X)$ viewed as vector spaces equipped with complete metrics are equivalent. To see this one chooses a good cover $\{U_{\alpha},\alpha\in A\}$ of $X.$

Indeed, a multivalued map $\{f_{\alpha}:U_{\alpha}\to\mathbb{R}\}$ representing $\omega$ (cf. the first definition) can be modified to an equivalent multivalued map $\{f^{\prime}_{\alpha}:U_{\alpha}\to\mathbb{R}\}$ by adding appropriate constants on each open set $U_{\alpha}$ so that $f^{\prime}_{\alpha}\cdot\pi_{\alpha}:\pi^{-1}(U_{\alpha})\to\mathbb{R}$ defines a $\Gamma-$equivariant map on $\tilde{X},$ hence a representative of a TC1-form (cf. the second definition) in the same cohomology class.

Conversely, a $\Gamma-$equivariant map representing $\omega$ (in the second definition) and a collection of continuous section $s_{\alpha}:U_{\alpha}\to\pi^{-1}(U_{\alpha})$ (i.e. $\pi\cdot s_{\alpha}=id_{U_{\alpha}}$) give a multivalued map $\{f\cdot s_{\alpha}:U_{\alpha}\to\mathbb{R}\}$ representing a TC1-form (first definition) in the same cohomology class.

It is not hard to check that the identifications above make the distances defined by different choices of base points in $U_{\alpha}$ and $\tilde{X}$ equivalent and consequently with the same Cauchy sequences.

2.2 Weakly tame and tame real-valued maps and topological closed 1-form

Fix a field $\kappa$. The homology considered is always with coefficients in the fixed field $\kappa,$ (for simplicity in writing omitted from the notation) hence a $\kappa-$vector space. For a continuous map $f:X\to\mathbb{R}$ denote by:

[TABLE]

Consider the direct system $\{H_{r}(X^{f}_{t}):^{f}i_{t}^{t^{\prime}}(r):H_{r}(X^{f}_{t})\to H_{r}(X^{f}_{t^{\prime}}),\ t\leq t"\}$ with ${}^{f}i_{t}^{t^{\prime}}(r)$ the inclusion induced linear maps and denote by

[TABLE]

and the induced linear maps

[TABLE]

Often when the decorations $f$ or $r$ are implicit from the context they will be dropped off the notation.

If $f_{1},f_{2}:X\to\mathbb{R}$ are two continuous maps with $|f_{1}-f_{2}|<C$ then the inclusions $X^{f_{1}}_{t}\subseteq X^{f_{2}}_{t+C}\subseteq X^{f_{1}}_{t+2C}\subseteq X^{f_{2}}_{t+3C}$ induce

[TABLE]

which, by passing to homology, implies

[TABLE]

Suppose that $f^{\omega}:\tilde{X}_{\xi}\to\mathbb{R}$ is the lift of TC-1 form $\omega$ of cohomology class $\xi$. The $\Gamma-$equivariance of $f^{\omega}$ induces a $\Gamma-$action on the system $\{H_{r}(X^{f}_{t}),^{f}i_{t}^{t^{\prime}}\}$ which provides a structure of $\kappa[\Gamma]-$modules on $H_{r}^{inv}(\tilde{X};f^{\omega})$ and $H_{r}^{dir}(\tilde{X};f^{\omega})$ and make $\mathbb{J}^{f^{\omega}}_{r}$ a $\kappa[\Gamma]-$linear map. If $f^{\omega_{1}},f^{\omega_{2}}:\tilde{X}_{\xi}\to\mathbb{R}$ are two lifts of the TC-1 forms $\omega_{1},\omega_{2}\in\mathcal{Z}^{1}(X;\xi),$ then $|f^{\omega_{1}}(x)-f^{\omega_{2}}(x)|\leq C=\sup_{x\in K}|f^{\omega_{1}}(x)-f^{\omega_{2}}(x)|,$ where $K$ is the closure of a fundamental domain of the free action $\mu.$

Then one can also write

[TABLE]

and

[TABLE]

The map $\mathbb{J}^{\xi}_{r}$ is a $\kappa[\Gamma]-$linear map of $\kappa[\Gamma]-$modules.

Hypothesis H : $\mathbb{J}^{\xi}_{r}:H^{inv}(X;\xi)\to H^{dir}(X;\xi)$ is injective, When $X$ is a compact ANR this hypothesis is satisfied fo all $\xi$ of degree of irrationality $\leq 1$ and satisfied for many pars $(X,\xi)$ with $\xi$ of any degree of irrationality cf. Corollary 3.4. We conjecture that this hypothesis is true for any $\xi$ provided $X$ is a compact ANR.

For $a\in\mathbb{R}$ let

[TABLE]

The value $a\in\mathbb{R}$ is called regular value (w.r. to $\kappa$ ) if $R^{f}_{a}(r)+R_{f}^{a}(r)=R^{f}_{a}(r)+R^{-f}_{-a}(r)=0$ for any $r$ and critical value if not regular. value Denote by $CR_{r}(f)\subset\mathbb{R}$ the set of critical values of $f$ with the property $R^{f}_{a}(r)+R_{f}^{a}(r)\neq 0$ and by $CR(f):=\cup_{r}CR_{r}(f).$

Definition 2.3

A continuous map $f:X\to\mathbb{R}$ is called tame if :

for any closed interval $I\subseteq\mathbb{R}$ the subspace $f^{-1}(I)$ is an ANR, in particular $X$ is an ANR. 2. 2.

for any $a\in\mathbb{R}$ and any $r,$ $R^{f}_{a}(r)+R^{a}_{f}(r)<\infty.$ 3. 3.

the set $CR(f)$ is countable.

Since $\tilde{X}=\cup\tilde{X}^{f}_{t}$ item 1. implies that

[TABLE]

and when $X$ is locally compact (and separable) items 1. and 2. imply item 3.

Let $\omega$ be a TC1- form on a connected space $X$ and let $f:\tilde{X}\to\mathbb{R}$ be a lift of $\omega.$ The sets $CR_{r}(f)$ and $CR(f)$ are $\Gamma-$invariant with respect to the action of $\Gamma$ on $\mathbb{R}$ by translation (recall $\Gamma\subset\mathbb{R}$) and the action is free. The set of orbits $CR_{r}(f)/\Gamma$ and $CR(f)/\Gamma$ will be denoted by $\mathcal{O}_{r}(f)$ and $\mathcal{O}(f)$ respectively. If $f_{1}$ and $f_{2}$ are two lifts of $\omega$ then $f_{2}=f_{1}+t.$ The translation by $t$ provides a canonical bijective map $T_{t}:\mathcal{O}(f_{1})\to\mathcal{O}(f_{2})$ which preserves the $r-$component $\mathcal{O}_{r}(f).$ Denote then $\mathcal{O}(\omega):=\bigcup_{f\in\omega}\mathcal{O}(f)/\sim$ with $o_{1}\sim o_{2}$ ($o_{i}\in\mathcal{O}(f_{i})$) iff $T_{t}(o_{1})=o_{2}.$

Definition 2.4

A TC1-form $\omega\in\mathcal{Z}^{1}(X),$ $X$ a compact ANR, is weakly tame if one lift $f^{\omega}$ and then any other is tame. 2. 2.

A TC1-form $\omega\in\mathcal{Z}^{1}(X),$ $X$ compact ANR, is tame if one lift $f^{\omega}$ (and then any other) is tame and the set $\mathcal{O}(\omega)$ is finite.

When $X$ is not connected $\omega$ is weakly tame resp. tame if its restriction to each component is weakly tame resp. tame. Examples of tame TC1-forms

A locally polynomial 444 locally polynomial means that locally there exists coordinates s.t. the coefficients of the form are polynomial functions closed differential 1-form with all zeros isolated on a closed smooth manifold is tame. 2. 2.

A generic simplicial 1- cocycle on a finite simplicial complex defines a TC1-form which is tame. Here generic means that the 1- cocycle takes nonzero values on all 1- simplexes 555The tameness remains true without the hypothesis all zeros are isolated in case 1. and generic in case 2. but via more elaborated arguments. By similar arguments the restriction of a differential closed 1-form on a manifold $M$ to a compact Thom-Mather stratified subset $X\subset M$ defines a tame TC1-form on $X$ .

Let us check case 1. for closed manifolds. The arguments provided remain true when the manifold is compact and the restriction of $\omega$ to the boundary has no zeros.

Let $\pi:\tilde{M}\to M$ be the associated $\Gamma-$principal covering, $f:\tilde{M}\to\mathbb{R}$ a lift of $\omega,$ $\mathcal{X}$ the set of zeros of $\omega$ and $\tilde{\mathcal{X}}=\pi^{-1}(\mathcal{X})$ the set of critical points of $f.$ Note that $\mathcal{X}$ is finite. Let $\tilde{\mathcal{X}}(t):=\tilde{\mathcal{X}}\cap f^{-1}(t).$

Observe that $\Gamma$ acts freely on the set $\tilde{X}$ and the set of orbits of this action is in bijective correspondence to the set $\mathcal{X},$ hence is finite. Observe also that the restriction of $\pi$ to $\tilde{\mathcal{X}}(t)$ is injective.

If $t\in\mathbb{R}$ is a regular value then $f^{-1}(t)$ is a codimension one smooth submanifold and if $t$ is a critical value then $f^{-1}(t)$ is a codimension one submanifold with finitely many conic singularities, as many as the cardinality of $\tilde{\mathcal{X}}(t).$

For either case, $t$ regular or critical value, $f^{-1}(t)$ is a closed subset which is an ANR and then so is $f^{-1}(I)$ for any closed interval $I.$ This verifies requirement (1) in Definition 2.3.

Note that if $t$ is a regular value $\tilde{M}_{t}$ is a manifold with boundary with interior $\tilde{M}_{<t}$ hence $H_{r}(\tilde{M}_{t},\tilde{M}_{<t})=0.$ If $t$ is a critical value then $\tilde{M}_{t}\setminus\tilde{\mathcal{X}}(t)$ is a manifold with boundary with interior $\tilde{M}_{<t},$ hence $H_{r}(\tilde{M}_{t},\tilde{M}_{<t})=H_{r}(\tilde{M}_{t},\tilde{M}_{t}\setminus\tilde{\mathcal{X}}(t))=H_{r}(D_{t},D_{t}\setminus\tilde{\mathcal{X}}(t))$ with $D_{t}=\tilde{M}_{t}\cap D,$ where $D$ is a disjoint union of closed small discs embedded in $\tilde{M}$, whose interior is a neighborhood of $\tilde{\mathcal{X}}(t).$ The hypothesis ”local polynomial” permits to choose such small discs that makes $D_{t}$ and $S_{t}=(\partial D)\cap\tilde{M}_{t}$ compact ANRs and $D_{t}\setminus\tilde{\mathcal{X}}(t)$ retractible by deformation to $S_{t}.$ Since $(D_{t},S_{t})$ is a pair of compact ANRs then $H_{r}(D_{t},D_{t}\setminus\tilde{\mathcal{X}}(t))=H_{r}(D_{t},S_{t})$ is a vector space of finite dimension. This verifies the finite dimensionality of $H_{r}(\tilde{M}_{t},\tilde{M}_{<t}).$ The same arguments verify the finite dimensionality of $H_{r}(\tilde{M}^{t},\tilde{M}^{>t}),$ hence the requirement (2) in Definition 2.3 for $t$ a critical value. The requirement 3. is obvious in view of the compacity of $M.$

In case 2. the arguments are similar. Note that if $t$ is a simplicial regular value for the lift $f$ then $f^{-1}(t)$ has a collar neighborhood inside the simplicial complex $\tilde{X}$ in which case $(\tilde{X}_{t},\tilde{X}_{<t})$ can be treated homologically as $(\tilde{M}_{t},\tilde{M}_{<t})$ above. If $t$ is a simplicial critical value, in view of the genericity, except for a finite set of points $\mathcal{V}_{t}=\{x_{1},\cdots,x_{k}\}\subset f^{-1}(f),$ $f^{-1}(t)\setminus\mathcal{V}_{t}$ has a collar neighborhood inside $\tilde{X}\setminus{\mathcal{V}_{t}}.$ With these observations the homological arguments in the smooth case can be repeated. In the above examples $\mathcal{O}(\omega)$ is a finite set.

3 Topology

3.1 Configurations of points, collision topology, bottleneck topology

Consider a pair $(Y,K)$, $Y$ a locally compact space and $K\subset Y$ a closed subset.

A configuration of points in $Y$ is a map $\delta:Y\to\mathbb{Z}_{\geq 0}$ with finite support. The total cardinality of the support is the non negative integer $\sum_{y\in Y}\delta(y).$ Denote by $Conf(Y)$ the set of all configurations of points in $Y$ and by $Conf_{N}(Y)$ the subset of configurations whose support have total cardinality $N.$

For a configuration $\delta\in Conf(Y\setminus K)$ with support $\mathrm{s}upp\ \delta:=\{y_{1},y_{2},\cdots y_{k}\}$ and a collection of disjoint open sets $U_{1},U_{2},\cdots U_{k},V$ with $x_{i}\in U_{i},K\subset V$ denote by

[TABLE]

and for $\delta\in Conf(Y),$ and $K=\emptyset$ write

[TABLE]

On the set $Conf_{N}(Y)$ consider the topology generated by the collections of neighborhoods

$\{\mathcal{U}(\delta,U_{1},\cdots U_{k})\}$ of each $\delta\in Conf_{N}(Y)$ and refer to it as the collision topology. As a topological space $Conf_{N}(Y)$ identifies to $Y^{N}/\Sigma_{N},$ the quotient of the $N-$fold cartesian product of $Y$ by the group of permutations of $N$ elements.

On the set $Conf(Y\setminus K)$ consider the topology generated by the collections of neighborhoods

$\{\mathcal{U}(\delta,U_{1},\cdots U_{k},V)\}$ of each $\delta\in Conf(Y\setminus K)$ and refer to it as the bottleneck topology. Note that if $K=\emptyset$ the bottleneck topology and collision topology are the same.

In this paper we will consider only the case $Y=\mathbb{R}$ and $K=(-\infty,0]$ hence $Y\setminus K=\mathbb{R}_{+}.$

3.2 Some algebraic topology of a pair $(X,\omega)$

Let $\kappa$ be a field, $X$ a compact ANR, $\omega$ a tame TC1-form in the cohomology class $\xi$ of degree of irrationality $k$ and $\Gamma=\Gamma(\xi)\subset\mathbb{R}$ the group defined by $\xi,$ cf. subsection 2.1. Note that if $k\geq 2$ then $\Gamma$ is dense in $\mathbb{R}$.

Let $\tilde{X}\to X$ be the associated principal $\Gamma-$covering with the free action $\mu:\Gamma\times\tilde{X}\to\tilde{X}$ and $f:\tilde{X}\to\mathbb{R}$ be a lift of $\omega.$ For any $g\in\Gamma$ the homeomorphism $\mu(g,\cdots):\tilde{X}\to\tilde{X}$ induces the isomorphism $\langle g\rangle:H_{r}(\tilde{X})\to H_{r}(\tilde{X})$.

The map $f$ provides two filtrations of $\tilde{X}$ indexed by $t\in\mathbb{R},$ for $t<t^{\prime}<t^{\prime\prime},$

[TABLE]

which induce in homology the filtrations

[TABLE]

with

[TABLE]

Clearly $\langle g\rangle(\mathbb{I}^{f}_{t}(r))=\mathbb{I}^{f}_{t+g}(r)$ and $\langle g\rangle(\mathbb{I}_{f}^{t}(r))=\mathbb{I}_{f}^{t+g}(r).$

Note that:

The $\kappa-$vector space $H_{r}(\tilde{X})$ is actually a f.g. $\kappa[\Gamma]-$module (since $X$ is a compact ANR) actually a Noetherian module, 2. 2.

$\mathbb{I}^{f}_{-\infty}(r):=\cap_{t\in\mathbb{R}}\mathbb{I}^{f}_{t}(r)$ and $\mathbb{I}_{f}^{\infty}(r):=\cap_{t\in\mathbb{R}}\mathbb{I}_{f}^{t}(r)$ are $\kappa[\Gamma]-$submodules, 3. 3.

$H_{r}(\tilde{X})=\cup_{t}\ \mathbb{I}^{f}_{t}(r)=\cup_{t}\ \mathbb{I}_{f}^{t}(r),$ 4. 4.

$H^{N}_{r}(X;\xi):=H_{r}(\tilde{X})/TorH_{r}(\tilde{X})$ is a f.g. torsion free $\kappa[G]-$ module of rank $\beta_{alg,r}^{N}(X;\xi)$ (i.e. the rank of a maximal free submodule), number referred to as the algebraic Novikov-Betti number or simply as Novikov-Betti number.

Note that when $\kappa=\mathbb{R}$ or $\mathbb{C}$ then $\beta^{N}_{r}(X;\xi)$ equals the $L_{2}-$Betti number $\beta^{L_{2}}_{r}(\tilde{X})$ of $\tilde{X}$ (cf. [9] , Lemma 1.34).

Proposition 3.1

$TorH_{r}(\tilde{X})=\mathbb{I}^{f}_{-\infty}(r).$** 2. 2.

$TorH_{r}(\tilde{X})=\mathbb{I}_{f}^{\infty}(r).$**

Proof.

The $\kappa[\Gamma]-$ module structure of $H_{r}(\tilde{X})$ is given by

[TABLE]

with $a_{g_{i}}\in\kappa,\ a_{g_{i}}\neq 0$ and then if $x\in\mathbb{I}^{f}_{t}(r)$ resp. $\mathbb{I}_{f}^{t}(r)$ one has

[TABLE]

To check the inclusion $TorH_{r}(\tilde{X})\subset\mathbb{I}^{f}_{-\infty}(r)$ in item 1. one starts with $x\in TorH_{r}(\tilde{X})$ which has to belong by (3.) above to some $\mathbb{I}^{f}_{t}(r).$ Suppose that

[TABLE]

with $g_{0}<g_{1}<\cdots<g_{k},\ a_{g_{i}}\neq 0.$ Then $x\in\mathbb{I}^{f}_{t}(r)$ implies

[TABLE]

hence $x\in\mathbb{I}^{f}_{t-(g_{k}-g_{k-1})}(r).$ By repeating the argument $x\in\mathbb{I}^{f}_{t-n(g_{k}-g_{k-1})}(r)$ for any $n,$ one derives $x\in\mathbb{I}^{f}_{-\infty}(r).$

Similarly, to check the inclusion $TorH_{r}(\tilde{X})\subset\mathbb{I}_{f}^{\infty}(r),$ one starts with $x\in\mathbb{I}_{f}^{t}(r),$ suppose that $g_{0}>g_{1}>\cdots>g_{k},\ a_{g_{i}}\neq 0$ and derives $x\in\mathbb{I}_{f}^{t+(-g_{k}+g_{k-1})}(r),$ hence $x\in\mathbb{I}_{f}^{t+n(-g_{k}+g_{k-1})}(r)$ for any $n,$ hence $x\in\mathbb{I}_{f}^{\infty}(r).$ To check that $I^{f}_{-\infty}(r)\subseteq TorH_{r}(\tilde{X})$ and $I_{f}^{\infty}(r)\subseteq TorH_{r}(\tilde{X})$ one uses the fact that $H_{r}(\tilde{X})$ is a f.g. $\kappa[\Gamma]-$module. If $x\in I^{f}_{-\infty}(r)$ then there exists an infinite collection of negative $g^{\prime}s$ in $\Gamma,$ say $\cdots<g_{r}<g_{r-1}<\cdots<g_{2}<g_{1},$ such that $\langle g_{r}\rangle(x)\in\mathbb{I}^{f}_{-\infty}(r)$ and, in view of the fact that $\mathbb{I}^{f}_{-\infty}(r)$ is f.g., a finite collection of elements $P_{r_{i}}\in\kappa[G],i=1,2,\cdots K$ s.t.

[TABLE]

Hence one obtains $x\in TorH_{r}(\tilde{X}).$ By a similar argument one concludes that $x\in\mathbb{I}_{f}^{\infty}(r)$ implies $x\in TorH_{r}(\tilde{X}).$ ∎

As an immediate consequence one has

[TABLE]

which in view of $\cup_{t\i\mathbb{R}}\mathbb{I}^{f}_{t}(r)=H_{r}(\tilde{X})$ implies

[TABLE]

For the purpose of this paper we are interested in deciding when (conjecturally always) for $\xi\in H^{1}(X;\mathbb{R}),$ the linear map $\mathbb{J}^{\xi}_{r}$ is injective . Proposition 3.3 below provides some partial answer to this question. To formulate them one needs some preliminary definitions and observations.

• •

For $p:X\to Y$ a continuous map,   $X,Y$ compact ANRs denote by $CR(p)$ the critical values set as

[TABLE]

• •

For $f$ a lift of the tame TC-1 form $\omega\in\mathcal{Z}(X;\xi),$ $\xi\in H^{1}(X;\mathbb{R}),$ $X$ compact ANR, (to avoid notational repetitions) it will be convenient to write also

[TABLE]

[TABLE]

use the abbreviations $i_{t}(r)$ and $\pi_{t}(r)$ for the canonical maps

[TABLE]

and the notaions $\iota_{t}(r)$ and $\iota^{t}(r)$ for the inclusion induced linear maps

[TABLE]

where $X^{f}(t)=f^{-1}(t).$

Then for any $0\neq x\in H_{r}(\tilde{X}_{t}),t\in(-\infty,\infty)$ define

[TABLE]

and put

[TABLE]

The omission of $t$ in this notation is justified by Item (1) in the following list of properties:

(1): $\tau_{t^{\prime}}(i_{t}^{t^{\prime}}(x))=\tau_{t}(x)$ provided $t^{\prime}<\tau_{t}(x).$

(2): $\tau(\lambda x)=\tau(x)$ for $\lambda\in\kappa\setminus 0,$

(3) $\tau(gx)=\tau(x)+g$ for $g\in\Gamma,$

(4) $\tau(x+y)\leq\sup\{\tau(x),\tau(y)\}.$

As a consequence of (2), (3) (4) above one has the following

Observation 3.2

a) If the elements of the collection $\{x_{\alpha}\in H_{r}(\tilde{X}_{t})\mid t\geq-\infty\}$ satisfy $\tau(x_{\alpha})\neq\tau(x_{\beta})$ for any $\alpha\neq\beta$ then they are $\kappa-$linearly independents.

b) If $x\in\mathbb{H}^{inv}_{r}(X;\xi)$ with $\tau(x)<\infty$ then the collection of elements $\{gx\in\mathbb{H}^{inv}_{r}(X,\xi)\mid g\in\Gamma\}$ are $\kappa-$linearly independent and for any $t$ the collection of elements $\{i^{t}_{-\infty}(gx)\in H_{r}(\tilde{X}_{t})\mid g+\tau(x)>t\}$ are $\kappa-$linearly independent.

c) Moreover for any $g$ with $g>t-\tau(x)$ and $x$ as in (b) there exists $y_{g}\in\ker(\iota_{t}(r):H_{r}(\tilde{X}(t))\to H_{r}(\tilde{X}))$ s.t. $\iota_{t}(r)(y_{g})=\pi_{t}(r)(gx)$ and the collection of elements $\{y_{g}\in H_{r}(\tilde{X}(t))\}$ are $\kappa-$linearly independent. Consequently, if $\dim(\ker(\iota_{t}(r):H_{r}(\tilde{X}(t))\to H_{r}(\tilde{X}_{t}))$ is finite for at least one $t,$ then $x$ has to be [math].

The proof of Item (c) needs in addition to (2), (3) and (4) the Mayer-Vietoris sequence

\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{r}(\tilde{X}^{f}(t))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{H_{r}(\tilde{X}^{f}_{t})\oplus H_{r}(\tilde{X}_{f}^{t})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{H_{r}(\tilde{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}

with $\iota=\iota_{t}(r)\oplus\iota^{t}(r),j=i_{t}(r)+i^{t}(r)$ where $i^{t}(r):H_{r}(\tilde{X}^{t})\to H_{r}(\tilde{X})$ is the linear map induced by the inclusion $\tilde{X}_{f}^{t}\subset\tilde{X}.$

Proposition 3.3

Suppose $p:X\to T^{k}$ is a continuous map, $X$ compact ANR, such that $CR(p)$ is contained in a finite collection of disjointly embedded $k-$dimensional disks $D_{1},D_{2},\cdots D_{N}\subset T^{k}$ and $\xi_{0}\in H^{1}(T^{k};\mathbb{R}).$ Then the linear map $\mathbb{J}_{r}^{\xi}$ with $\xi={p^{\ast}(\xi_{0})}$ is injective.

As a consequence one has

Corollary 3.4

1. (a)

For any $\xi\in H^{1}(X;\mathbb{R}),$ of degree of irrationality $k\leq 1,$ $X$ compact ANR, the linear map $\mathbb{J}_{r}^{\xi}$ is injective. 2. (b)

For any $\xi\in H^{1}(T^{k};\mathbb{R})$ the linear map $\mathbb{J}^{\xi}_{r}$ is injective. 3. (c)

For any $p:X\to T^{k}$ with homotopy theoretic fiber a compact ANR and $\xi_{0}\in H^{1}(T^{k};\mathbb{R})$ the linear map $\mathbb{J}^{\xi}$ with $\xi=p^{\ast}(\xi_{0})$ is injective. 4. (d)

the construction below provides for any$P$* a compact ANR (resp. closed manifold) and any $k$ positive integer a compact ANR $X$ (closed manifold) and $\xi\in H^{1}(X:\mathbb{R})$ with the linear map $\mathbb{J}^{\xi}_{r}$ injective.*

Proof of Proposition 3.3: Consider the pullback diagram with $\pi_{0}$ the canonical $\mathbb{Z}^{k}-$ principal covering

[TABLE]

and $\omega_{0}=\sum_{i=1}^{k}a_{i}d\theta_{i}$ with $\theta_{1},=e^{it_{1}},\cdots,\theta_{k}=e^{it_{k}}$ the angular coordinates on $T^{k},$ $\{t_{1},\cdots t_{k}\}$ the cartesian coordinates on $R^{k},$ and $\{a_{i}\in\mathbb{R}\}$ a collection of real numbers. With respect to these coordinates $\pi_{0}(t_{1},\cdots,t_{k})=(\theta_{1},\cdots,\theta_{k}).$ Note that any cohomology class $\xi_{0}\in H^{1}(T^{k};\mathbb{R})$ has a smooth closed one form of this type as representative.

The map $f_{0}(t_{1},\cdots t_{k})=\sum_{i=1}^{k}a_{i}t_{i}$ satisfies $df_{0}=\pi_{0}^{\ast}(\omega_{0})$ and if $CR(p)$ is a finite set then there exists $t$ s.t. $\pi_{0}(f^{-1}(t))\cap CR(p)=\emptyset.$ Then for $f=f_{0}\cdot\tilde{\pi}$ the level $f^{-1}(t)$ is a fibration over a hyperplane $f_{0}^{-1}(t)$ with compact fibers, hence of finite dimensional homology. By Observation 3.2 the injectivity of $\mathbb{J}^{\xi}_{r}$ follows.

If $CR(p)$ si contained in $\sqcup_{i=1.\cdots,N}D_{i}$ consider $u:T^{k}\to T^{k}$ a map homotopic to $id_{T^{k}}$ whose restriction to $T^{k}\setminus\sqcup D_{i}$ is a diffeomorphism on the image and shrinks each disc to its center. Clearly for $p^{\prime}=u\cdot p$  $CR(p^{\prime})$ is a finite set of points, and since $p^{\ast}(\xi_{0})={p^{\prime}}^{\ast}(\xi_{0})=\xi$ the injectivity of $\mathbb{J}^{\xi}_{r}$ holds..

Note that for any $\xi\in H^{1}(X;\mathbb{R})$ one can produce $p:X\to T^{k}$ and $\xi_{0}\in H^{1}(T^{k};\mathbb{R})$ s.t. $\xi=p^{\ast}(\xi_{0})$ and if $\xi$ is of degree of irrationality $r$ then one can choose any $k>r$ with $\xi_{0}$ of degree of irrationality $r.$

q.e.d

Poof of Corollary (3.4): A simple look to the diagram (4) shows that for $k=1$ the map $f_{0}$ and then $f$ is proper hence the homology of any level $f^{-1}(t)$ is finite dimensional. This verifies (a).

For the Item (b) , when $X=T^{k},$ the levels of $f_{0}$ are hyperplanes and then by Observation 3.2 the injectivity of $\mathbb{J}^{\xi}_{r}$ holds .

For Item (c), note first that if $CR(p)=\emptyset$ the statement holds because the levels of $f$ have finite dimensional homology. If $p$ is only fibration up to homotopy, then by crossing with a compact contractible space $Y$ and by replacing $p$ by a bundle map $p^{\prime}:X\times Y\to T^{k}$ homotopic to $p\cdot pr_{X},$ one derives the statement from the previous situation.

For Item (d), choose $P^{d}$ a closed $d-$dimensional manifold and let $N^{d+k}$ be a compact $(d+k)-$ dimensional compact manifold with boundary $\partial N^{d+k}=P^{d}\times S^{k-1}.$ Let $D^{k}\subset T^{k}$ be a $k-$ dimensional (closed) disk embedded in $T^{k}=$ the cartesian product of $k-$copies of $S^{1},$ $u":N^{d+k}\to D^{k}$ be a continuous map which restricted to $\partial N^{d+k}$ is the projection on $S^{k-1}$ and $u^{\prime}:(T^{k}\setminus\overset{\circ}{D^{k}})\times P^{d}\to(T^{k}\setminus\overset{\circ}{D^{k}})$ be the projection on $(T^{k}\setminus\overset{\circ}{D^{k}}).$ Here $\overset{\circ}{D^{k}}$ denotes the interior of $D^{k}.$ Let $M^{d+k}$ be the closed manifold

[TABLE]

obtained by identifying the boundaries of the compact manifolds $(T^{k}\setminus\overset{\circ}{D^{k}}\times P^{d}$ and $N^{d+k}$ and $u:=u^{\prime}\cup u^{\prime\prime}.$ The homotopy class of $u$ defines a cohomology class in $H^{1}(M;\mathbb{Z}^{k})$ which together with a chosen injective homomorphism $\mathbb{Z}^{k}\to\mathbb{R},$ provides $\xi\in H^{1}(M;\mathbb{R})$ of degree of irrationality $k.$ By Proposition 3.3 the injectivity of $\mathbb{J}^{\xi}_{r}$ holds. Indeed the composition \textstyle{{T^{k}\setminus\overset{\circ}{D^{k}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subset}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{T^{k}} induces an isomorphism in dimension one integral homology hence $u$ induces a surjective homomorphism in dimension one integral homology which makes the hypothesis of Proposition 3.3 satisfied.

3.3 Novikov-Betti numbers (a topological definition)

Recall that for $f:\tilde{X}\to\mathbb{R},$ a lift of a tame TC1-form $\omega,$ the vector space $\mathbb{I}^{f}_{t}(r)/\mathbb{I}^{f}_{<t}(r)$ is zero when $t$ is a regular value and of finite dimension when $t$ is a critical value, and the isomorphism $\langle g\rangle:H_{r}(\tilde{X})\to H_{r}(\tilde{X})$ induces an isomorphism

[TABLE]

Consider the $\kappa-$vector space

[TABLE]

and note that this sum

1. involves only components corresponding to $t$ critical values, hence of at most countable many,

2. is a $\kappa[\Gamma]-$ module whose multiplication by $g$ is provided by the component-wise isomorphism $\langle g\rangle_{t},$

3. is independent of the lift $f$ up to an isomorphism since

[TABLE]

4. is a free $\kappa[\Gamma]-$module canonically isomorphic to $V\otimes_{\kappa}\kappa[\Gamma]$ where $V$ is the finite dimensional vector space

[TABLE]

for any choice $a^{o}\in o\in\mathcal{O}_{r}(f).$ Different choices of $a^{o}$ lead to isomorphic vector spaces $V.$

One defines

[TABLE]

which will be shown in section (5.2) to be same as

[TABLE]

One can provide a similar definition using $\mathbb{I}_{f}^{t}$ instead of $\mathbb{I}^{f}_{t}$. Clearly $\mathbb{I}^{f}_{t}=\mathbb{I}_{-f}^{t}$ and $\mathbb{I}^{f}_{<t}=\mathbb{I}_{-f}^{>t}$ with $-f$ being a lift of the TC1-form $-\omega.$ From the algebraic perspective the first is based on the group $\Gamma$ the second on the group $\Gamma^{\prime}$ canonically isomorphic o $\Gamma$ by the isomorphism $g\to g^{\prime}=-g.$ This leads to the same numbers $\beta^{N}_{alg,r}$ and $\beta^{N}_{top,r}$.

Both numbers $\beta_{top,r}^{N}(X;\omega)$ and $\beta_{alg,r}^{N}(X;\xi(\omega))$ whose equality is verified in section 5.2 are referred to as the Novikov-Betti numbers.

4 The maps

$\delta^{f}_{r}$ and $\gamma^{f}_{r}$

In this section $f:X\to\mathbb{R}$ will be a tame map, cf. Definition 2.3. Recall from the previous section the notations:

$X^{f}_{a}:=f^{-1}((-\infty,a]),X^{f}_{<a}:=f^{-1}((-\infty,a)),X^{f}_{\infty}=X,$

$X_{f}^{a}:=f^{-1}([a,\infty)),$  $X_{f}^{>a}:=f^{-1}((a,\infty)),$ $X^{-\infty}_{f}=X,$

$\mathbb{I}^{f}_{a}(r):=\mathrm{i}mg(H_{r}(X^{f}_{a})\to H_{r}(X)),$  $\mathbb{I}^{f}_{<a}(r)=\mathrm{i}mg(H_{r}(X^{f}_{<a})\to H_{r}(X))=\cup_{\alpha<a}\mathbb{I}^{f}_{\alpha}(r),$

$\mathbb{I}_{f}^{a}(r):=\mathrm{i}mg(H_{r}(X_{f}^{a})\to H_{r}(X)),$  $\mathbb{I}_{f}^{>a}(r)=\mathrm{i}mg(H_{r}(X_{f}^{>a})\to H_{r}(X))=\cup_{\beta>a}\mathbb{I}_{f}^{\beta}(r).$

$\mathbb{I}^{f}_{-\infty}(r)=\cap_{a\in\mathbb{R}}\mathbb{I}^{f}_{a}(r),$  $\mathbb{I}_{f}^{\infty}(r)=\cap_{a\in\mathbb{R}}\mathbb{I}_{f}^{a}(r).$

For any $a,b\in\mathbb{R}$ define $\boxed{\mathbb{F}^{f}_{r}(a,b):=\mathbb{I}^{f}_{a}(r)\cap\mathbb{I}_{f}^{b}(r)}$ and when $a<b$ define

$\boxed{\mathbb{T}^{f}_{r}(a,b):=\ker(H_{r}(X^{f}_{a})\to H_{r}(X^{f}_{b})).}$

Extend the definitions of $\mathbb{T}^{f}_{r}(<a,b)$ to:

[TABLE]

Note that :

for $a<b<c\leq\infty$ the obvious linear map

[TABLE]

is an isomorphism, 2. 2.

for $a<b$ the homology exact sequence of the pair $(X^{f}_{b},X^{f}_{a})$ implies the short exact sequence

[TABLE] 3. 3.

the commutative diagram

[TABLE]

has all columns and the first three rows exact, which makes the forth row also exact, 4. 4.

By passing to direct limit when $a\to b$ the short exact sequence (7) and the last row of (8) imply the isomorphism

[TABLE]

The conjecturally trivial vector spaces $\hat{\lambda}^{f}_{r}$ and numbers $\lambda^{\omega}_{r}.$

For $a\in\mathbb{R}$ let

[TABLE]

be the canonical linear map induced from $i_{t}^{<a}(r):\mathbb{T}_{r}(t,a)\to\mathbb{T}_{r}(<a,a)$ by passing to the inverse limit when $t\to-\infty.$ Since $H_{r}(\tilde{X}_{a},\tilde{X}_{<a})$ is finite dimensional then so is

[TABLE]

Denote by

[TABLE]

and observe that

if $h$ is a different lift of $\omega,$ hence $h=f+t,$ then $\lambda^{h}_{r}(a)=\lambda^{f}_{r}(a+t)$ 2. 2.

$\lambda^{f}_{r}(a+g)=\lambda^{f}_{r}(a)$ for any $g\in\Gamma.$

Then one defines

[TABLE]

for $a\in o^{f}\in CR(f)/\Gamma,$ and then

[TABLE]

integer number independent of the lift of $\omega.$ In case $\omega\in\mathcal{Z}^{1}(X;\xi),$  $\mathbb{J}_{r}^{\xi}$ injective implies $\lambda^{\omega}_{r}=0.$

4.1 The assignments $\hat{\delta}_{r}^{f}$ and $\delta^{f}_{r}.$

Let $f:X\to\mathbb{R}$ be a fixed tame map. Since $f$ is fixed the decoration ”$f$” will not appear the notation below.

Call box a subset $B\subset\mathbb{R}^{2}$ of the form $B=(a^{\prime},a]\times[b,b^{\prime})$ for $-\infty\leq a^{\prime}<a,\quad b<b^{\prime}\leq\infty,$ and define

[TABLE]

Let

[TABLE]

be the canonical projection 666When implicit in the context, $r$ will be dropped off the notation.

For $B=B^{\prime}\sqcup B^{\prime\prime}$ with $B^{\prime}=(a^{\prime},a^{\prime\prime}]\times[b,b^{\prime})$ ,  $B^{\prime\prime}=(a^{\prime\prime},a]\times[b,b^{\prime}),$   $-\infty\leq a^{\prime}<a^{\prime\prime}<a,\ b<b^{\prime}<b^{\prime\prime}\leq\infty$ or with $B^{\prime}=(a^{\prime},a]\times[b^{\prime\prime},b^{\prime}),$   $B^{\prime\prime}=(a^{\prime},a]\times[b,b^{\prime\prime}),$    $-\infty\leq a^{\prime}<a,\ b<b^{\prime\prime}<b^{\prime}\leq\infty$ the inclusion $B^{\prime}\subseteq B$ induces the injective linear map

[TABLE]

and the inclusion $B^{\prime\prime}\subseteq B$ induces the surjective linear map

[TABLE]

For $-\infty\leq a^{\prime\prime}<a^{\prime}<a;\ b<b^{\prime}<b^{\prime\prime}\leq\infty$ one denotes by:

[TABLE]

and by

[TABLE]

b^{\prime\prime}$$b^{\prime}$$b$$a^{\prime\prime}$$a^{\prime}$$a$$B_{21}$$B_{11}$$B_{22}$$B_{12}Figure 1

In view of the definitions one has

Proposition 4.1

1. The sequence

[TABLE]

is exact.

2. The diagram

[TABLE]

is commutative with all rows and columns exact sequences.

One denotes by $\pi_{B}^{B_{22}}$ for the composition $\pi_{B}^{B_{22}}:=\pi^{B_{22}}_{B_{\cdot 2}}\cdot\pi^{B_{\cdot 2}}_{B}=\pi^{B_{22}}_{B_{2\cdot}}\cdot\pi^{B_{2\cdot}}_{B}$ and by $i^{B}_{B_{11}}$ the composition $i^{B}_{B_{11}}:=i^{B}_{B_{1\cdot}}\cdot i^{B_{1\cdot}}_{B_{11}}=i^{B}_{B_{\cdot 1}}\cdot i^{B_{\cdot 1}}_{B_{11}}$ and in general by

[TABLE]

when the box $B^{\prime\prime}$ is located in the lower-right corner of the box $B$ and by

[TABLE]

when the box $B^{\prime}$ is located in the upper-left corner of the box $B.$ The map $\pi_{B}^{B^{\prime}}$ is always surjective and $i_{B^{\prime}}^{B}$ always injective.

For $\epsilon>0$ one denotes by $B(a,b;\epsilon)$ the set $B(a,b;\epsilon):=(a-\epsilon]\times[b,b+\epsilon).$ Suppose $-\infty\leq a^{\prime}<a,\ \ b<b^{\prime}\leq\infty.$

Define

[TABLE]

w.r. to the surjective linear maps \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 30.68883pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-30.68883pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\mathbb{F}{r}(B(a,b;\epsilon))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 30.6274pt\raise 6.50694pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{\scriptstyle{\pi^{B(a,b;\epsilon^{\prime})}{B(a,b;\epsilon)}}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 54.68883pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 54.68883pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\mathbb{F}_{r}(B(a,b;\epsilon^{\prime}))}}}}}}}}\ignorespaces}}}}\ignorespaces, $\epsilon>\epsilon^{\prime},$ and denote by

[TABLE]

the composition \textstyle{\mathbb{F}_{r}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{B}^{B(a,b;\epsilon)}}$$\textstyle{\mathbb{F}_{r}(B(a,b;\epsilon))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{(a,b)}_{B(a,b;\epsilon)}}$$\textstyle{\hat{\delta}_{r}(a,b)} for $\epsilon<\inf\{(a^{\prime}-a),(b^{\prime}-b)\}$ and by

[TABLE]

the composition \textstyle{\mathbb{F}_{r}(a,b)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\{a,b\}}^{B(a,b;\epsilon)}}$$\textstyle{\mathbb{F}_{r}(B(a,b;\epsilon))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{(a,b)}_{B(a,b;\epsilon)}}$$\textstyle{\hat{\delta}_{r}(a,b).}

Both compositions are surjective maps independent on $\epsilon.$

One has

[TABLE]

Define

[TABLE]

w.r. to the surjective maps $\pi_{(a^{\prime},a]\times[b,b+\epsilon)}^{(a^{\prime},a]\times[b,b+\epsilon^{\prime})}(r):\mathbb{F}_{r}((a^{\prime},a]\times[b,b+\epsilon))\to\mathbb{F}_{r}((a^{\prime},a]\times[b,b+\epsilon^{\prime})),\epsilon>\epsilon^{\prime}$ and denote by

[TABLE]

and

[TABLE]

the canonical surjective maps induced by passing to limit when $\epsilon\to 0$.

One has

[TABLE]

Define

[TABLE]

w.r. to the surjective maps $\pi_{(a-\epsilon,a]\times[b,b^{\prime})}^{(a-\epsilon^{\prime},a]\times[b,b^{\prime})}(r):\mathbb{F}_{r}((a-\epsilon,a]\times[b,b^{\prime}))\to\mathbb{F}_{r}((a-\epsilon^{\prime},a]\times[b,b^{\prime})),\epsilon>\epsilon^{\prime},$ and denote by

[TABLE]

and

[TABLE]

the canonical surjective maps induced by passing to limit when $\epsilon\to 0.$

One has

[TABLE]

Define

[TABLE]

and

[TABLE]

with the inclusions

\textstyle{\mathbb{F}_{r}((-\infty,a^{\prime}]\times b)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$$\textstyle{\mathbb{F}_{r}((-\infty,a]\times b)} for $a^{\prime}<a$ and

\textstyle{\mathbb{F}_{r}(a\times[b^{\prime},\infty))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$$\textstyle{\mathbb{F}_{r}(a\times[b,\infty))} for $b^{\prime}>b$ induced by the linear injective maps $i_{B}^{\prime B}$ cf. Proposition 2 item 1.

Note that:

$\hat{\delta}_{r}(a,b)\neq 0$ implies a,b\in CR_{r}(f),\ 2. 2.

$\mathbb{F}_{r}(a\times\cdots)\neq 0$ implies a\in CR_{r}(f),\ 3. 3.

$\mathbb{F}_{r}(\cdots\times b)\neq 0$ implies $b\in CR_{r}(f).$ 4. 4.

$\varinjlim_{\epsilon\to 0}\mathbb{F}_{r}((a-\epsilon,a]\times b)=\hat{\delta}_{r}(a,b),$ 5. 5.

$\varinjlim_{\epsilon\to 0}\mathbb{F}_{r}(a\times[b,b+\epsilon))=\hat{\delta}_{r}(a,b),$ 6. 6.

$\varinjlim_{\epsilon,\epsilon^{\prime}\to 0}\mathbb{F}_{r}(a-\epsilon,a]\times[b,b+\epsilon^{\prime}))=\hat{\delta}_{r}(a,b)$.

The above definitions combined with Proposition 4.1 leads to the following proposition.

Proposition 4.2

For $-\infty\leq a^{\prime}<a^{\prime\prime}<a,\ b\in\mathbb{R}$ the sequence

[TABLE]

is exact and for $a\in\mathbb{R},b<b^{\prime\prime}<b^{\prime}\leq\infty$ the sequence

[TABLE]

is exact.

In both sequences $i$ and $\pi$ are the linear maps induced by the injective linear maps $i_{B^{\prime}}^{B}$ and the surjective linear maps $\pi_{B}^{B^{\prime\prime}}.$ 2. 2.

1. (a)

For any $a\in\mathbb{R}$ and $b<b^{\prime}\leq\infty$

[TABLE]

and when $a$ is a regular value $\dim\mathbb{F}_{r}(a\times[b,b^{\prime}))=dim\mathbb{F}_{r}(a\times\mathbb{R})=0.$ 2. (b)

For any $b\in\mathbb{R}$ and $-\infty\leq a^{\prime}<a$

[TABLE]

and when $b$ is a regular value $dim\mathbb{F}_{r}((a,a]\times b)=dim\mathbb{F}_{r}(\mathbb{R}\times b))=0.$

The inequalities above hold in view of (15 ) and (16) for $[b,b^{\prime})$ replaced by $[b,\infty)$ or $\mathbb{R}$ and $(a^{\prime},a]$ replaced by $(-\infty,a]$ or $\mathbb{R}$ respectively.

1. (a)

If either $a$ or $b$ are regular values then $\hat{\delta}_{r}(a,b)=0.$ 2. (b)

For any $a\in\mathbb{R}$ the set $\mathrm{s}upp\delta^{f}_{r}\cap(a\times\mathbb{R})$ is finite and if $a$ regular value is empty. 3. (c)

For any $b\in\mathbb{R}$ the set $\mathrm{s}upp\delta^{f}_{r}\cap(\mathbb{R}\times b)$ is finite and if $b$ regular value is empty.

The relation between the surjective linear maps $\pi_{\cdots}^{\cdots}$ is summarized by the following commutative diagram with $B=(a^{\prime},a]\times[b,b^{\prime}).$

[TABLE]

Definition 4.3

For $(a,b)\in CR_{r}(f)\times C_{r}(f)$ a splitting

[TABLE]

is a right inverse of the canonical projection $\pi_{\{a,b\}}^{(a,b)}(r):\mathbb{F}_{r}(a,b)\to\hat{\delta}^{f}_{r}(a,b),$  i.e. $\pi_{\{a,b\}}^{(a,b)}(r)\cdot i_{(a,b)}(r)=id.$

For $-\infty\leq a^{\prime}<a,\ b<b^{\prime}\leq\infty$ let $K$ be either one of the following sets:

a bounded or unbounded box $B=(a^{\prime},a]\times[b,b^{\prime}),$ 2. 2.

a bounded or unbounded horizontal open-closed interval $I=(a^{\prime},a]\times b,$ 3. 3.

a bounded or unbounded closed-open vertical interval $J=a\times[b,b^{\prime}),$ 4. 4.

$a\times\mathbb{R},$ 5. 5.

$\mathbb{R}\times b,$ 6. 6.

$\mathbb{R}^{2}.$

Call $(a,b)\in\mathbb{R}^{2}$ the relevant corner in case 1. and the relevant end in case 2. or 3. The interval $(a^{\prime},a^{\prime\prime}]\times b,$ when viewed as a subinterval of $(a^{\prime},a]\times b,$ is called left (open-closed) subinterval and $a\times[b^{\prime\prime},b^{\prime}),$ when viewed as a subinterval of $a\times[b,b^{\prime}),$ is called upper (closed-open) subinterval.

For $(a,b)\in K$ a splitting $i_{(a,b)}(r)$ provides the injective linear map

[TABLE]

defined as follows.

– If $(a,b)$ is the relevant corner or the relevant end then $i^{K}_{(a,b)}(r)$ is the composition

[TABLE]

with $\pi^{K}_{(a,b)}(r):\mathbb{F}_{r}(a,b)\to\mathbb{F}_{r}(K)$ the canonical projection.

– If $(a,b)$ is not relevant corner or end then:

a) in the case 1., 2., 3. one defines

[TABLE]

with $K^{\prime}\subset K,$ the only upper-left box, resp. left subinterval, resp. upper subinterval having $(a,b)$ as the relevant corner resp. end.

b) in the case 4., 5., 6., one defines $i_{(a,b)}^{K}(r)$ as the direct limit of $i_{(a,b)}^{K^{\prime}}(r)$ where $K^{\prime}$ runs among the subsets of $K$ of the same type and located as upper left box resp. left interval resp. upper interval which make $i_{K^{\prime}}^{K}(r)$ injective. Choose a collection of splittings $\mathcal{S}:=\{i_{(a,b)}(r)\mid(a,b)\in CR(f)\times CR(f),r\in\mathbb{Z}_{\geq 0}\}.$ Let $K$ be a set as in cases in 1., 2., 3., 4., 6. above. and $A\subseteq CR(f)\times CR(f).$ Denote

[TABLE]

[TABLE]

Proposition 4.4

For any choice of $\mathcal{S}$ the following holds:

The maps ${}^{\mathcal{S}}I(r)$ and ${}^{\mathcal{S}}I^{K}_{A\cap K}(r)$ are injective. 2. 2.

*If $A=CR(f)\times CR(f)$ and $K$ is in either case 2., 3.. 4., or 5. then *

${}^{\mathcal{S}}I^{K}_{K}(r):=^{\mathcal{S}}I^{K}_{A\cap K}(r)$* is an isomorphism.*

Proof.

Item 1: It suffices to verify the statement for $A$ a finite set. This is done by induction on cardinality of $A$ as follows. When $\sharp A=1$ this follows from the fact that $i_{r}(\alpha,\beta)$ is a splitting. When $\sharp(K\cap A)\geq 2$ one can write $K=K_{1}\sqcup K_{2}$ with $\sharp(K_{i}\cap A)<\sharp(K\cap A).$ In view of the definition of ${}^{\mathcal{S}}I^{\cdots}_{\cdots}$the following diagram is commutative.

[TABLE]

Since both raws are short exact sequence with the left and the right vertical arrows injective the mid vertical arrow is injective too.

Item 2: In view of the tameness property of $f,$ if $a\notin CR(f)$ then $\mathbb{F}_{r}(a\times\mathbb{R})=0.$ If $a\in CR(f)$ then $\mathbb{F}_{r}(a\times\mathbb{R})=\mathbb{I}_{a}(r)/\mathbb{I}_{<a}(r)$ in view of (16) and by Proposition 4.2 item 2. is of finite dimension. Denote by $\mathbb{S}(a,b)=\mathbb{I}_{a}\cap\mathbb{I}^{b}/\mathbb{I}_{<a}\cap\mathbb{I}^{b}$ and observe that

[TABLE]

for $b<b^{\prime}.$ The finite dimensionality of $\mathbb{F}(a\times\mathbb{R})$ implies the existence of a finite collection of critical values $b_{1}<b_{2}<\cdots b_{N}$ s.t. $\mathbb{S}(a,-\infty)=\mathbb{S}(a,b_{1})\supset\mathbb{S}(a,b_{1})\supset\cdots\supset\mathbb{S}(a,b_{N-1})\supset\mathbb{S}(a,b_{N})\supset 0$ and in view of the definition of $\mathbb{S}(a,b)$ one has $\mathbb{S}(a,b^{\prime})=\mathbb{S}(a,b_{i})$ provided $b^{\prime}\in(b_{i-1},b_{i}].$ This implies that $\hat{\delta}(a,b_{i})=\mathbb{S}(a,b_{i})/\mathbb{S}(a,b_{i+1})$ and therefore

[TABLE]

This implies that ${}^{\mathcal{S}}I^{K}_{A\cap K}(r)$is an isomorphism and

[TABLE]

If $b\notin CR(f)$ then $\mathbb{F}(\mathbb{R}\times b)=0$ and if $b\in CR(f)$ then by (15),

$\mathbb{F}_{r}(\mathbb{R}\times b)=\mathbb{I}^{b}(r)/\mathbb{I}^{>b}(r)$ and by Proposition 4.2 item 2. of finite dimension. Denote by $\mathbb{U}(a,b)=\mathbb{I}_{a}\cap\mathbb{I}^{b}/\mathbb{I}_{a}\cap\mathbb{I}^{>b}$ and observe that

[TABLE]

for $a^{\prime}>a.$ The finite dimensionality of $\mathbb{F}_{r}(\mathbb{R}\times b)$ implies the existence of a finite collection of critical values $a_{1}>a_{2}>\cdots>a_{N}$ s.t.  $\mathbb{U}(\infty,b)=\mathbb{U}(a_{1},b)\supset\mathbb{U}(a_{2},b)\supset\cdots\supset\mathbb{U}(a_{N-1},b)\supset\mathbb{U}(a_{N},b)\supset 0$ and in view of the definition of $\mathbb{U}(a,b)$ one has $\mathbb{U}(a^{\prime},b)=\mathbb{U}(a_{i},b)$ if $a^{\prime}\in[a_{i+1},a_{i}).$ Therefore

[TABLE]

This implies that ${}^{\mathcal{S}}I^{K}_{A\cap K}(r)$ is an isomorphism and

[TABLE]

∎

Define

[TABLE]

As a consequence of Proposition 4.4 for any $a\in\mathbb{R}$ the set $\mathrm{s}upp\ \delta^{f}_{r}\cap(a\times\mathbb{R})$ is finite and of total cardinality

[TABLE]

hence equal to zero when $a$ is a regular value. Similarly, for any $b\in\mathbb{R}$ the set $\mathrm{s}upp\ \delta^{f}_{r}\cap(\mathbb{R}\times b)$ is finite of total cardinality

[TABLE]

hence equal to zero when $b$ is a regular value.

4.2 The assignments $\hat{\gamma}_{r}^{f}$ and $\gamma^{f}_{r}.$

Call box above diagonal, abbreviated ad-box, a subset $B\subset\mathbb{R}^{2}_{+}=\{(x,y)\mid x<y\}$ of the form $B=(a^{\prime},a]\times(b^{\prime},b],$ with $a^{\prime}<a\leq b^{\prime}<b.$ As in the previous subsection let $f:X\to\mathbb{R}$ be a fixed tame map. Since $f$ is fixed the decoration ”$f$” will not appear the notations below.

For $a^{\prime}<a\leq b^{\prime}<b$ the inclusions $X^{f}_{a^{\prime}}\subset X^{f}_{a}\subseteq X^{f}_{b^{\prime}}\subset X^{f}_{b}$ induce in homology the commutative cartesian diagram

[TABLE]

with the property that $\mathrm{i}mg\ u(r)=\mathrm{i}mg\ i_{a^{\prime}}^{a}(r)\cap\mathbb{T}_{r}(a,b^{\prime}).$ Here $i_{a^{\prime}}^{a}(r):\mathbb{T}_{r}(a^{\prime},b)\to\mathbb{T}_{r}(a,b)$ is the restriction of $i_{a^{\prime}}^{a}(r):H_{r}(X_{a^{\prime}})\to H_{r}(X_{b}).$ In order to avoid heavy notation, when implicit from the context we will simply write $i(r)$ instead of $i_{a^{\prime}}^{a}(r).$ Define

[TABLE]

Suppose that $B_{1},B_{2},B$ are three ad-boxes with $B_{1}\sqcup B_{2}=B$ in either one of the two relative positions:

$B_{1}$ the left side ad-box and $B_{2}$ the right side ad-box (for example $B=B_{\cdot 2},B_{1}=B_{12},B_{2}=B_{22}$ in Figure 2),

$B_{1}$ the down side ad-box and $B_{2}$ the upper side ad-box (for example $B=B_{1\cdot},B_{1}=B_{11},B_{2}=B_{12}$ in Figure 2).

The inclusion $B_{1}\subseteq B$ induces the injective linear map $i_{B_{1}}^{B}(r):\mathbb{T}_{r}(B_{1})\to\mathbb{T}_{r}(B)$ and the inclusion $B_{2}\subset B$ the surjective linear map $\pi_{B}^{B_{2}}(r):\mathbb{T}_{r}(B)\to\mathbb{T}_{r}(B_{2}).$

One still call ad-box the set $(-\infty,a]\times(b^{\prime},b],\ a\leq b^{\prime},$ and define

[TABLE]

For $-\infty\leq a^{\prime\prime}\leq a^{\prime}<a;\ b^{\prime\prime}\leq b^{\prime}<b$ one considers the ad-boxes

[TABLE]

[TABLE]

b$$b^{\prime}$$b^{\prime\prime}$$a^{\prime\prime}$$a^{\prime}$$a$$B_{11}$$B_{12}$$B_{21}$$B_{22}Figure 2

By elementary but tedious arguments, for details see the Appendix below, one can show.

Proposition 4.5

1. If $B_{1},B_{2},B$ are ad boxes s.t. $B=B_{1}\sqcup B_{2}$ then the sequence

[TABLE]

is exact.

2. If $B_{11},B_{12},B_{21},B_{22}$ are ad-boxes as in Figure 2 then the diagram

[TABLE]

is commutative with all rows and columns exact sequences.

As a consequence, for any $-\infty\leq a^{\prime}<a\leq b^{\prime}<b\rm\ {and}\ \epsilon>\epsilon^{\prime}$ the inclusion induced linear maps:

[TABLE]

[TABLE]

and

[TABLE]

where $B(a,b;\epsilon):=(a-\epsilon]\times(b-\epsilon,b],$ are surjective.

Define

for $a<b$

[TABLE]

with respect to the maps $\pi^{B(a,b;\epsilon^{\prime})}_{B(a,b;\epsilon\ )}(r),$ 2. 2.

for $-\infty\leq a^{\prime}<a<b$

[TABLE]

with respect to the maps $\pi_{(a^{\prime},a]\times(b-\epsilon,\ b]}^{(a^{\prime},a]\times(b-\epsilon^{\prime},b]}(r)$ and then observe that $\varinjlim_{\epsilon\to 0}\mathbb{T}_{r}((a-\epsilon,a]\times b)=\hat{\gamma}_{r}(a,b).$ 3. 3.

for $a\leq b^{\prime}<b$

[TABLE]

with respect to the maps $\pi_{(a-\epsilon,\ a]\times(b^{\prime},b]}^{(a-\epsilon^{\prime},a]\times(b^{\prime},b]}(r)$ and then observe that $\varinjlim_{\epsilon\to 0}\mathbb{T}_{r}(a\times(b-\epsilon,b])=\hat{\gamma}_{r}(a,b).$

These maps induce for $-\infty\leq a^{\prime}<a^{\prime\prime}<a\leq b^{\prime}<b^{\prime\prime}<b<\infty$ the following exact sequences.

[TABLE]

with $I_{1}=(a^{\prime},a^{\prime\prime}]\times b,I=(a^{\prime},a^{\prime\prime}]\times b,I_{2}=(a^{\prime\prime},a]\times b$ and

[TABLE]

with $I_{1}=(a\times(b^{\prime},b^{\prime\prime}],I=a\times(b^{\prime},b],I_{2}=a\times(b^{\prime\prime},b].$

Note that the exactness of (22) for $a^{\prime}=-\infty$ can be derived by passing to the inverse limit in (22) when $a^{\prime}\to-\infty.$ The exactness is maintained in view of the surjectivity of $\mathbb{T}_{r}((a^{\prime}_{1},a]\times b)\to\mathbb{T}_{r}((a_{2}^{\prime},a]\times b)$ for $-\infty<a_{1}^{\prime}<a_{2}^{\prime}<a.$

Schematically (22) and (23) can be written as

[TABLE]

is an exact sequence. Extend the definitions above to

[TABLE]

with respect to the maps induced by inclusions $(a,b-\epsilon]\times b$ into $(a,b-\epsilon^{\prime}]\times b$ for $\epsilon>\epsilon^{\prime},$

[TABLE]

with respect to the maps induced by inclusions $a\times(a,b_{1}]$ into $a\times(a,b_{2}]$ for $a<b_{1}<b_{2}<b.$ and observe

[TABLE]

with respect to the surjective maps induced by inclusions $(a\times(a+\epsilon,b])$ into $(a\times(a+\epsilon^{\prime},b])$ for $\epsilon>\epsilon^{\prime}.$

The reader can also check that

[TABLE]

with respect to the linear maps $t_{B_{\epsilon}}^{B_{\epsilon^{\prime}}}=\iota^{B_{\epsilon^{\prime}}}_{(a,b-\epsilon]\times(b-\epsilon^{\prime},b]}\cdot\pi_{B_{\epsilon}}^{(a,b-\epsilon]\times(b-\epsilon^{\prime},b]}$ where $B_{\epsilon}=(a^{\prime},b-\epsilon]\times(b-\epsilon,b]$ and $\epsilon^{\prime}<\epsilon.$

One can verify that

[TABLE]

and that

[TABLE]

The description of the canonical projections $\pi_{\cdots}^{\cdots}$ summarized in the diagram below is implicit in (25).

[TABLE]

In view of the above definitions Proposition 4.5 leads to the following

Proposition 4.6

2. (a)

For $a<b^{\prime}<b$ one has

[TABLE] 2. (b)

For $a^{\prime}<a<b$ one has

[TABLE]

The same inequalities hold for $(b^{\prime},b]$ in (a) replaced by $(a,\infty)$ and $(a^{\prime},a]$ in (b) replaced by $(-\infty,b)$ 2. 2.

1. (a)

If either $a$ or $b$ are regular values then $\hat{\gamma}_{r}(a,b)=0,$ 2. (b)

For any $a,$ $\mathrm{s}upp\gamma^{f}_{r}\cap(a\times(a,\infty))$ is a finite set and when $a$ regular value is empty. 3. (c)

For any $b,$ $\mathrm{s}upp\gamma^{f}_{r}\cap((-\infty,b)\times b)$ is a finite set and when $b$ regular value is empty.

Proof.

Item 1.: To check part a) observe that from the commutative diagram (27) with all raws and columns exact

[TABLE]

one derives the injectivity of $\mathbb{T}_{r}(a,b)/i\mathbb{T}_{r}(<a,b))\to H_{r}(X_{a})/iH_{r}(X_{<}a).$ In the sequence

[TABLE]

the right to left arrow is surjective and both the left to right arrow and the left to down arrow are injective. Then, in view of the finite dimensionality of $H_{r}(X_{a},X_{<a}),$ the statement follows.

To check part b) one considers the commutative diagram (28) with all rows and columns exact

[TABLE]

and one derives the surjectivity of

[TABLE]

From the long exact sequence of the triple $(X_{a}\subset X_{<b}\subset X_{b})$ one derives the injectivity of

$H_{r+1}(X_{b},X_{a})/iH_{r+1}(X_{<b},X_{a})\to H_{r+1}(X_{b},X_{<b}).$ Then, in view of the finite dimensionality

of $H_{r+1}(X_{b},X_{<b}),$ the diagram

[TABLE]

implies the statement b). The extension to intervals $(a,\infty),(-\infty,b)$ follows in view of the definitions of $\mathbb{T}_{r}(a\times(a,\infty))$ and $\mathbb{T}_{r}((-\infty,b)\times b).$

Item 2.: follows from Item 1. ∎

Suppose that $K$ is one of the following type of sets.

a bounded or unbounded ad-box $B=(a^{\prime},a]\times(b^{\prime},b],$ $-\infty\leq a^{\prime}<a<b,\ \ b^{\prime}<b\leq\infty,$ 2. 2.

a horizontal open-closed interval $I=(a^{\prime},a]\times b,$ $-\infty\leq a^{\prime}<a,$ 3. 3.

a vertical open-closed interval $J=a\times(b^{\prime},b],$ $a\leq b^{\prime}<b,$ 4. 4.

$(-\infty,b)\times b,$ 5. 5.

$a\times(a,\infty),$

and observe that in view of Proposition 4.6, when $K$ is of type 2., 3., 4. or 5., the vector space $\mathbb{T}_{r}(K)$ has finite dimension.

Denote by $(CR(f)\times CR(f))_{+}:=\{(a,b)\mid a,b\in CR(f),a<b\}.$

Splittings:

For any $(a,b)\in(CR(f)\times CR(f))_{+}$ a splitting

[TABLE]

is a right inverse of the canonical projection $\pi^{(a,b)}_{\{a,b\}}(r):\mathbb{T}_{r}(a,b)\to\hat{\gamma}^{f}_{r}(a,b).$ 2. 2.

For any $K$ in either one of the situations above and $(a,b)\in K$ one defines

[TABLE]

first for the case the point $(a,b)$ is the relevant corner or vertex of $K$ then for an arbitrary point of $K$ as in the previous subsection. 3. 3.

For a collection of splittings $\mathcal{S}=\{i_{(a,b)}(r)\mid(a,b)\in CR(f)\times CR(f)_{+}\},$ set $A\subset(CR(f)\times CR(f))_{+}$ and $K$ in one of the cases 1. to 5. above one defines

[TABLE]

Proposition 4.6 can be refined to Proposition 4.7

Proposition 4.7

For any choice of $\mathcal{S}$

The maps ${}^{\mathcal{S}}I_{A}^{K}(r)$ are injective. 2. 2.

If $A=(CR(f)\times CR(f))_{+}$ and $K$ is either of type 2.(when $-\infty<a^{\prime}$), 3., 4. or 5., then ${}^{\mathcal{S}}I^{K}_{A\cap K}$ is an isomorphism. In particular

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

Proof.

Item 1. follows by similar arguments as in the proof of Proposition 4.4 item 1.

Item 2. We treat $K$ of type 2and 4 as case 1 and type 3 and 5 as case 2.

For case 1 let $b\in\mathbb{R}$ and

[TABLE]

For case 2 let $a\in\mathbb{R}$ and

[TABLE]

We will verify two facts,

Fact 1: $\mathbb{T}_{r}(K)=0$ iff $\sup\hat{\gamma}_{r}\cap K=\emptyset.$ Fact 2: $\mathbb{T}_{r}(K)\neq 0$ implies in case 1 the existence of a finite collection of real numbers $\{a_{1},a_{2},\cdots a_{M}\}$ s.t. $a^{\prime}<a_{M}<\cdots<a_{2}<a_{1}\leq a$ in case 2 the existence of a finite collection of real numbers $\{b_{1},b_{2},\cdots b_{N}\},b^{\prime}<b_{1}<b_{2}<\cdots b_{N}\leq b$ and in case 1 $\sup\hat{\gamma}^{f}_{r}\cap K:=\{a_{1}\times b,a_{2}\times b,\cdots a_{M}\times b\}$ in case 2 $\sup\hat{\gamma}^{f}_{r}\cap K:=\{a\times b_{1},a\times b_{2},\cdots,a\times b_{N}\}.$

The verifications use the finite dimensionality of $\mathbb{T}_{r}(K)$ and the exact sequence (22) in case 1 and (23) in case 2. Since the verifications are essentially the same we will do it in details only Case 1.

Observe that by Item 1. $\mathbb{T}_{r}(K)=0$ implies $\sup\hat{\gamma}_{r}\cap K=\emptyset.$ and want to conclude that $\mathbb{T}_{r}(K)\neq 0$ implies $\mathrm{s}upp\gamma^{f}_{r}\cap K\neq\emptyset.$ This will verify Fact 1.

It suffices to check this for $a^{\prime}\neq-\infty,$ because if $a^{\prime}=-\infty,$ by (22), there exists $\alpha,$ $-\infty<\alpha<a$ such that $\mathbb{T}_{r}(\alpha,a]\times b)\neq 0.$ If no such $\alpha$ exists then $\mathbb{T}_{r}((-\infty,a]\times b)=0,$ again by (22), impossible in view of the hypothesis .

If $a^{\prime}\neq-\infty$ one can produce two infinite sequences $\{\alpha^{\prime}_{i}\}$ and $\{\alpha_{i}\}$ $a^{\prime}\leq\alpha^{\prime}_{1}\leq\alpha^{\prime}_{2},\leq\cdots\leq\alpha^{\prime}_{n}\leq\alpha^{\prime}_{n+1}\leq\cdots$ and $a\geq\alpha_{1}\geq\alpha_{2}\geq\cdots\geq\alpha_{n}\geq\alpha_{n+1}\geq\cdots$ such that

(i) $\alpha^{\prime}_{n}\leq\alpha^{\prime}_{n+1}<\alpha_{n+1}\leq\alpha_{n},$

(ii) $|\alpha_{n+1}-\alpha_{n+1}|=1/2|\alpha^{\prime}_{n}-\alpha_{n}|,$

(iii) $\mathbb{T}_{r}((\alpha^{\prime}_{n},\alpha_{n}]\times b)\neq 0$ for every $n.$

Indeed, in view of 22) one can take the interval $(\alpha^{\prime}_{n+1},\alpha_{n+1}]$ to be one of the two intervals $I=(\alpha^{\prime}_{n},\frac{\alpha_{n}+\alpha^{\prime}_{n}}{2}]$ or $(\frac{\alpha_{n}+\alpha^{\prime}_{n}}{2},\alpha_{n}]$

which satisfies $\mathbb{T}_{r}(I)\neq 0$ and inductively construct the sequences starting with $\alpha^{\prime}_{1}=a^{\prime}$ and $\alpha_{1}=a.$ They will satisfy the properties (i), (ii), (iii). In view of (i) and (ii) both sequences are convergent to the real number $\alpha\in(a^{\prime},a.]$ Since $\dim\mathbb{T}_{r}((\alpha^{\prime}_{n},\alpha_{n}]\times b)$ provides a decreasing sequence of positive integers, hence constant for $n$ large enough, hence with the vector space $\mathbb{T}_{r}((\alpha^{\prime}_{n},\alpha_{n}]\times b)$ stabilising to a nontrivial vector space $\hat{\gamma}^{f}_{r}(\alpha,b).$ Hence $(\alpha,b)\in\mathrm{s}upp\gamma^{f}_{r}\cap K.$

Case 2 is treated similarly based on (23) , producing sequences $\{\beta^{\prime}_{n}\},$ $\{\beta_{n}\}$ and ultimately $\beta\in(b^{\prime},b]$ s.t $(a,\beta)\in\mathrm{s}upp\gamma^{f}\cap K.$

To check Fact 2 note that Proposition (4.6) produces a finite sequence $\{a_{M}<\cdots<a_{2}<a_{1}\}$ in case 1 and $\{b_{1}<b_{2}<\cdots b_{N}\}$ in case 2 with $a_{i}$ and $b_{i}$ exactly the points where the map $(a^{\prime},a]\ni t\rightsquigarrow\dim\mathbb{T}_{r}(((t,a]\times b)$ in case 1 and $(b^{\prime},b]\ni t\rightsquigarrow\dim\mathbb{T}_{r}(a\times(b^{\prime},t])$ in case 2 have discontinuities. Clearly both $a_{i}$ and $b_{i}$ in view of (22) and (E23) are critical values and there are no points $(\alpha,b),\alpha\in(a_{i+1},a_{i})$ in case 1 or $(a,\beta),\beta\in(b_{i},b_{i+1})$ in case 2 in $\mathrm{s}upp\gamma^{f}_{r}\cap K.$

Item 1 guaranties the injectivity of ${}^{\mathcal{S}}I_{K\cap\sup\hat{\gamma}_{r}}^{K}(r).$ A supposed lack of surjectivity is ruled out using Fact 1. The isomorphism ${}^{\mathcal{S}}I_{K\cap\sup\hat{\gamma}_{r}}^{K}(r)$ implies (31) and (32) in case 1 and (29) and (E25’) in case 2.

∎

Corollary 4.8

1. $\mathrm{c}oker(H_{r}(X_{<a})\to H_{r}(X_{a}))\simeq(\oplus_{t\in\mathbb{R}}\hat{\delta}_{r}(a,t))\bigoplus(\oplus_{t\in\mathbb{R}}\hat{\gamma}_{r}(a,t))$

2. $\ker(H_{r}(X_{<b})\to H_{r}(X_{b}))\simeq\mathbb{T}_{r}((-\infty,b)\times b)\simeq\oplus_{t\in(-\infty,a)}\hat{\gamma}^{f}_{r}(t,a)\oplus\hat{\lambda}^{f}_{r}(b)$

Proof.

Item 1.: In view of last raw of (8 ) one has

[TABLE]

In view of the fifth equality (25) and of the equality (29) one has

[TABLE]

In view of the equality (19)

[TABLE]

In view of the exact sequence last raw of (8)

[TABLE]

Combining these three isomorphisms item 1. follows. Item 2. : In view of the seventh equality (25) one has

[TABLE]

In view of Proposition 4.7 and formula (10)

[TABLE]

Putting together these isomorphisms one obtains Item 2.

∎

To summarize, the map $\gamma^{f}_{r}:\mathbb{R}^{2}\to\mathbb{Z}_{\geq 0}$ satisfies :

for any $a\in\mathbb{R},$ $\mathrm{s}upp\ \gamma^{f}_{r}\cap(a\times\mathbb{R})$ is a finite set of total multiplicity

[TABLE]

in particular for $a$ regular value is zero. 2. 2.

for any $b\in\mathbb{R}$ the set $\mathrm{s}upp\ \gamma^{f}_{r}\cap(\mathbb{R}\times b)$ is finite and of total multiplicity

[TABLE]

in particular for $b$ regular value is zero.

As a consequence of Corollary 4.8 one obtains.

[TABLE]

4.3 The case of TC1 form

Suppose that $f:\tilde{X}\to\mathbb{R}$ is the lift of a tame TC1-form $\omega.$ Then $f$ $\Gamma-$equivariant map, the sets $CR_{r}(f)$ and $CR_{r}(f)$ are $\Gamma-$invariant w.r. to the translations by the elements of $\Gamma,$ the set $\mathcal{O}_{r}(f)=CR_{r}(f)/\Gamma$ is finite and the maps $\delta^{f}_{r}$ and $\gamma^{f}_{r}$ are $\Gamma-$ invariant i.e.

[TABLE]

In view of (19) and (37), one has

Corollary 4.9

For $f$ a lift of a tame TC1- form $\omega$ and for any choice $a^{o}\in o\in\mathcal{O}_{r}(f)$

[TABLE] 2. 2.

[TABLE]

In case $\omega$ is a Morse closed differential 1-form on a closed manifold $M$ and $f:\tilde{M}\to\mathbb{R}$ is a lift of $\omega$ then $\sum_{o\in\mathcal{O}(f)}\dim H_{r}(\tilde{X}_{a^{o}},\tilde{X}_{<a^{o}})$ is exactly the number of zeros of $\omega$ of Morse index $r.$ Indeed, let $\tilde{\mathcal{X}}_{r}$ be the set of critical points of index $r$ and $\mathcal{X}_{r}$ be the finite set of zeros of $\omega$ of Morse index $r.$ The group $\Gamma$ acts freely on $\tilde{\mathcal{X}}_{r}$ and the orbits of this action identify to $\mathcal{X}_{r}.$ Let $\pi:\tilde{\mathcal{X}}_{r}\to\mathcal{X}_{r}$ be the quotient map and let $\tilde{\mathcal{X}}_{r,a}:=\tilde{\mathcal{X}}_{r}\cap f^{-1}(a)$ . Note that the restriction of $\pi$ to $\tilde{\mathcal{X}}_{r,a}$ is injective.

Choose for any $o\in\mathcal{O}(f)$ a critical value $a^{o}$ and observe that $\cup_{o\in\mathcal{O}(f)}\tilde{\mathcal{X}}_{r,a^{o}}$ identifies by $\pi$ to $\mathcal{X}_{r}.$

Classical Morse theory identifies the $\kappa-$vector space generated by $\tilde{\mathcal{X}}_{r,a}$ with the vector space $H_{r}(\tilde{M}_{a},\tilde{M}_{<a})$ and therefore the cardinality of set $\mathcal{X}_{r}$ is the dimension of the vector space $\oplus_{o\in\mathcal{O}(f)}H_{r}(\tilde{M}_{a^{o}},\tilde{M}_{<a^{o}})$ calculated by Corollary 4.9 item 2., equivalently the rank of the free $\kappa[\Gamma]-$ module $\oplus_{a\in CR(f)}H_{r}(\tilde{M}_{a},\tilde{M}_{<a}).$

5 The configurations $\boldsymbol{\delta}^{\omega}_{r}$ and $\boldsymbol{\gamma}^{\omega}_{r}$

5.1 The supports of $\delta^{f}_{r}$ and $\gamma^{f}_{r}$

In view of Propositions 4.2 and 4.6, the support of $\delta^{f}_{r}$ and $\gamma^{f}_{r}$ are located on finitely many diagonal $\Delta^{\delta}_{s}$ and $\Delta^{\gamma}_{s},$ for a finite collection of values of $s$ ( $s>0$ in the case of $\gamma^{f}_{r}$). Here we denote by $\Delta_{s}:=\{(x,y)\in\mathbb{R}^{2}\mid y-x=s\}.$ One way to produce these diagonals, goes as follows. Since the procedure is the same for $\delta^{f}_{r}$ and $\gamma^{f}_{r}$ we describe in details only the case of $\delta^{f}_{r}.$

Choose one $a^{o}\in o\in\mathcal{O}_{r}(f)$ for each orbit $o.$ Each point in the set

[TABLE]

defines a diagonal corresponding to $s=b^{i}_{a^{o}}-a^{o},$ so ultimately one obtains a collection of at most $\sum_{o\in\mathcal{O}_{r}(f)}n(o)$ such diagonals. The index $s$ which appears is always a differences of critical values of the lift $f.$ In view of the equality $\delta^{f}_{r}(a,b)=\delta^{f}_{r}(a+g,b+g)$ the integer $n(o)$ is independent on the choice of $a^{o}.$ Note that different orbits $o^{\prime}$ and $o^{\prime\prime}$ lead to the same diagonal once the equality $b^{i}_{a^{o^{\prime}}}-a^{o^{\prime}}=b^{j}_{a^{o^{\prime\prime}}}-a^{o^{\prime\prime}}$ holds, so the same diagonal can appear multiple times, different choices of $a^{o}$ lead to the same diagonals and different lifts of $\omega$ also lead to the same diagonals with the same number of occurrences. Similarly one can produce diagonals by choosing $b_{o}\in o$ and considering the diagonals corresponding to the points $(b_{i}^{b_{o}},b_{o}),1\leq i\leq m(o)$ in the set $\mathrm{s}upp\delta_{r}^{f}\cap\mathbb{R}\times b_{o}$, hence with $s=(b_{o}-a_{i}^{b_{o}}).$ Again the integer $m(o)$ depends only on the orbit $o.$ The outcome by both procedures is expected to be the same as argued below.

To better explain the number of diagonals and the ”correct multiplicity” associates with each diagonal the following definitions are of help. Note that both cases $\delta^{f}_{r}$ and $\gamma^{f}_{r}$ are similar so one treats fin details only the case of $\delta^{f}_{r}$ and one points out the minor notational differences when the case.

Two points $(x,y)\in\mathbb{R}^{2}$ and $(x^{\prime},y^{\prime})\in\mathbb{R}^{2}$ are $\Gamma-$equivalent, written $(x,y)\sim(x^{\prime},y^{\prime})$ iff there exists $g\in\Gamma$ s.t. $x=g+x^{\prime},y=g+y^{\prime}.$

Definition 5.1

A subset $\mathcal{B}^{f},$ $\mathcal{B}^{f}\subset\mathrm{s}upp\delta^{f}_{r}\subset CR(f)\times CR(f)$ is called a BASE for $\mathrm{s}upp\ \delta^{f}_{r}$ if the following holds:

for any $(\alpha,\beta)\in\mathrm{s}upp\delta^{f}_{r}$ there exists $g\in\Gamma$ and $(a,b)\in B^{f}$ such that $(\alpha,\beta)=(g+a,g+b),$ 2. 2.

If $(a,b)$ and $(a^{\prime},b^{\prime})$ are in $\mathcal{B}^{f}$ then $(a,b)\sim(a^{\prime},b^{\prime})$ implies $(a,b)=(a^{\prime},b^{\prime}).$

In view of 2. the pair $(a,b)$ and $g$ claimed by 1. are unique.

Observe that the following holds:

If $\mathcal{B}^{f_{1}}_{1}$ and $\mathcal{B}^{f_{2}}_{2}$ are two bases for the support of $\delta_{r}^{f_{1}}$ and $\delta^{f_{2}}_{r},$ $f_{1}$ and $f_{2}$ lifts of $\omega,$ then there exists a canonical bijective correspondence $\theta:\mathcal{B}_{1}\to\mathcal{B}_{2}$ with the property that $\delta^{f_{2}}(\theta(a,b))=\delta^{f_{1}}(a,b),$ 2. 2.

If for any $o\in\mathcal{O}_{r}(f)$ one chooses $a^{o}\in o$ and let $b_{a^{o}}^{1},b_{a^{o}}^{2},\cdots b_{a^{o}}^{n(o)}$ be all critical values 777finitely many in view of Proposition 4.2 s.t $\mathrm{s}upp\delta^{f}_{r}\cap a^{o}\times R=\{(a^{o},b_{a^{o}}^{1}),(a^{o},b_{a^{o}}^{2}),\cdots(a^{o},b_{a^{o}}^{n(o)})\}$ then the finite collection of points

[TABLE]

provides a base for the support of $\delta^{f}_{r}.$ Denote this base by $\mathcal{B}^{f}(\{a^{o}\})$ with $\{a^{o}\}$ the collection of elements $a^{o}.$ 3. 3.

If for any $o\in\mathcal{O}_{r}(f)$ one chooses $b_{o}\in o$ and let $a^{b_{o}}_{1},a^{b_{o}}_{2},\cdots a^{b_{o}}_{m(o)}$ be all critical values s.t $\mathrm{s}upp\delta^{f}_{r}\cap\mathbb{R}\times b_{o}=\{(a^{b_{o}}_{1},b_{o}),(a^{b_{o}}_{2},b_{o})\cdots(a^{b_{o}}_{m(o)},b_{o})\}$ then the collection of points

[TABLE]

provides a base for the support of $\delta^{f}_{r}.$ Denote this base by $\mathcal{B}^{f}(\{b_{o}\}).$ 4. 4.

Each element $(a,b)\in\mathcal{B}^{f}$ of a base provides a diagonal $\Delta_{s}$ with $s=b-a$ and each such diagonal $\Delta_{s}$ appear as many time as the number of pairs $\{(a,b)\in\mathcal{B}\mid b-a=s\}.$ It is convenient to assign to $\Delta_{s}$ the number

[TABLE]

which, by item 1. above, is independent of the base $\mathcal{B}^{f}.$

The same definition can be made in case of $\gamma^{f}$ and provide base $\mathcal{B}^{f}$ for $\mathrm{s}upp\ \gamma^{f}_{r}$. The same observations, 1., 2., 3., 4.remain valid when $\mathrm{s}upp\delta^{f}_{r}\cap a\times\mathbb{R}$ and $\mathrm{s}upp\ \delta^{f}_{r}\cap\mathbb{R}\times b$ are replaced by $\mathrm{s}upp\ \gamma^{f}_{r}\cap a\times(a,\infty)$ and $\mathrm{s}upp\ \gamma^{f}_{r}\cap(-\infty,b)\times b$ respectively.

The number assigned to $\Delta_{s}$ in the case of $\gamma^{f}_{r}$ is

[TABLE]

which, by item 1. is independent on the base $\mathcal{B}^{f}$ and the lift $f.$

Given any lift $f$ of a tame $\omega,$ $\boldsymbol{\delta}_{r}^{\omega}(s)$ and $\boldsymbol{\gamma}_{r}^{\omega}(s)$ can be calculated using either a base of the type $\mathcal{B}(\{a^{o}\})$ or of type $\mathcal{B}(\{b_{o}\})$ and one obtains for any choice of a lift $f$ and any choice $a^{o}\in o$ or $b_{o}\in o,$ $o\in\mathcal{O}_{r}(f)$ the following formulae.

[TABLE]

[TABLE]

The items 1. and 2. in Corollary 4.9 become

[TABLE]

[TABLE]

5.2 Derivation of the main results

Proof of Theorem (1.1):

Item 1. follows from equality (41) combined with the equality of $\beta^{N}_{alg,r}(X;\xi(\omega))=\beta^{N}_{top,r}(X;\omega)$ established by Theorem 5.2 below.

Item 2. follows from equality (42).

Theorem 5.2

if $\omega$ is a tame TC1-form then $\beta^{N}_{top,r}(X;\omega)=\beta^{N}_{alg,r}(X;\xi(\omega)).$

Observe that in view of Proposition 3.1 for $x\in H_{r}(\tilde{X})$ either there exists $a=\alpha(x)\in\mathbb{R}$ s.t. $x\in\mathbb{I}_{a}(r)\setminus\mathbb{I}_{<a}(r)$ or $x\in\cap_{t\in\mathbb{R}}\mathbb{I}_{t}(r)=TorH_{r}(\tilde{X}).$ Clearly $x\in\mathbb{I}_{a}(r)\setminus\mathbb{I}_{<a}(r)$ implies $\hat{x},$ the image of $x$ in $\mathbb{I}_{a}(r)/\mathbb{I}_{<a}(r),$ is not zero.

Observe that:

$\alpha(x)\in CR(f)_{r},$ 2. 2.

$\alpha(g\cdot x)\equiv\alpha(\langle g\rangle(x))=g+\alpha(x),$ 3. 3.

$\alpha(x+y)=max\{\alpha(x),\alpha(y)\}$ if $\alpha(x)\neq\alpha(y)$ or if $\alpha(x)=\alpha(y)$ and $\hat{x}+\hat{y}\neq 0,$ 4. 4.

$\alpha(x+y)<\alpha(x)=\alpha(y)$ if $\alpha(x)=\alpha(y)$ and $\hat{x}+\hat{y}=0$

Suppose that $e_{1},e_{2},\cdots e_{N}$ is a base of a free $\kappa[\Gamma]-$module $\mathcal{E}$ submodule of $H_{r}(\tilde{X}).$ Note that the multiplication with elements in $\Gamma$ of any of $e_{i}^{\prime}s$ does not change their status of remaining together a base for $\mathcal{E}$, but modifies $\alpha(e_{i})$ as indicated in (2.) above. In view of the above properties of $\alpha(x)$ one can modify this base into a base of $\mathcal{E}$ consisting of

[TABLE]

with the following properties:

$N=n_{1}+n_{2}+\cdots n_{r},$ 2. 2.

$\alpha(e_{i,j})=a_{i}\in CR_{r}(f),$ 3. 3.

$a_{1}>a_{1}>\cdots>a_{r}$ with $a_{i}\in o_{i}$ different orbits of $CR_{r}(f)/\Gamma=\mathcal{O}_{r}(f),$ i.e. $\Gamma-$independent.

First one observes that for any $i=1,2,\cdots r$ the set $\hat{e}_{i,1},\hat{e}_{i,2},\cdots\hat{e}_{i,n_{i}}$ are $\kappa-$linearly independent elements in $\mathbb{I}_{a_{i}}/\mathbb{I}_{<a_{i}}.$

Indeed if for a fixed $i$ one has $\sum_{j}\lambda_{j}\hat{e}_{i,j}=0\ \lambda_{i}\in\kappa,$ then $\alpha(\sum_{j}\lambda_{j}e_{i,j})<a_{i}$ by (4), and then $\sum_{j}\lambda_{j}e_{i,j}=\sum_{j}Q_{j}e_{i,j}+\sum_{j,s\neq i}P_{s,j}e_{s,j}$ where $Q_{j}\in\kappa[\Gamma]$ contains only negative elements in $\Gamma$ (i.e. in $\Gamma\cap(-\infty,0)).$ Then one obtains $\sum_{j}\lambda_{j}(1-Q_{j})e_{i,j}-\sum_{j,s\neq i}P_{s,j}e_{s,j}=0,$ hence $\lambda_{j}(1-Q_{j})=0,$ and because $(1-Q_{j})\neq 0$ $\lambda_{j}=0.$

Second, In view of $\Gamma-$independence of $a_{i}$ the entire collection $\{\hat{e}_{i,j}\}$ consists of elements $\kappa[\Gamma]-$linearly independent in the $\kappa[\Gamma]-$module $\oplus_{a\in\mathbb{R}}\mathbb{I}_{a}/\mathbb{I}_{<a},$ cf subsection 3.3. This implies $N\leq\beta^{N}_{top,r}(X;\omega)$ hence $\beta^{N}_{alg,r}(X;\xi(\omega))\leq\beta^{N}_{top,r}(X;\omega).$

The inequality $\beta^{N}_{top,r}(X;\omega)\leq\beta^{N}_{alg,r}(X;\xi(\omega)$ follows from the injectivity of ${}^{\mathcal{S}}I(r)$ established in Proposition 4.4. provided that a $\Gamma-$compatible collection of splittings is chosen and such collection exists. A collection of splittings $i_{a,b}(r):\hat{\delta}^{f}_{r}(a,b)\to\mathbb{F}_{r}(a,b)\subseteq H_{r}(\tilde{X})$ is $\Gamma-$ compatible if for any $g\in\Gamma$ one has

[TABLE]

where the isomorphism $\langle g\rangle_{a,b}:\hat{\delta}^{f}_{r}(a,b)\to\hat{\delta}^{f}_{r}(a+g,b+g)$ is induced by the isomorphism $\langle g\rangle.$ The choice of an arbitrary collection of splittings $i_{a,b}(r)$ for $(a,b)\in B^{f}$ a base for $\mathrm{s}upp\ \delta^{f}_{r},$ which obviously exists, defines via formula (43) a family of compatible $\Gamma-$splittings. If the collection $\mathcal{S}$ is $\Gamma-$ compatible then the $\kappa-$linear map ${}^{\mathcal{S}}I(r)$ is actually an injective $\kappa[\Gamma]-$ linear map from the free $\kappa[\Gamma]$ module of rank $\dim\sum_{a,b\in B^{f}}\delta^{f}_{r}(a,b)=\beta^{N}_{top,r}(X;\omega)$ to $H_{r}(\tilde{X})$, hence $\beta^{N}_{top,r}(X;\omega)\leq\beta^{N}_{alg,r}(X,\xi(\omega)).$

Theorems 1.2 and 1.3 in the generality stated will be proven in part II and part III of this work. However in case $\omega$ is of degree of irrationality $0,$ hence $\Gamma=0$ and then $\tilde{X}=X,$ they follow from Theorems 5.2 and 5.3 in [1] in view of the fact that $\boldsymbol{\delta}^{\omega}(t)=\sum_{a}\delta^{f}_{r}(a,a+t).$ In case $\omega$ is of degree of irrationality $1$ and the TC1-form determined by an angle valued map $f:X\to\mathbb{S}^{1}=\mathbb{R}/2\pi\mathbb{Z}$ then they follow from [1] Theorems 5.5, 5.6, 6.3 and 6.4 by observing that $\boldsymbol{\delta}^{\omega}_{r}(t)=\sum_{z\in\mathbb{C}\setminus 0\mid\ln|z|=t}\delta^{f}_{r}(z)$ and $\boldsymbol{\gamma}^{\omega}_{r}(t)=\sum_{z\in\mathbb{C}\setminus 0|\ln|z|=t,|z|>1}\delta^{f}_{r}(z).$ If $\omega$ is of degree if irrationality one then $\Gamma\simeq Z$ with the positive generator, a real number $l\in\mathbb{R}_{+}.$ One repeats the arguments in [1] with $2\pi$ replaced by $l$ and one derives the two results from the same theorems in [1]. Reference [1] actually reproduces results in [3].

6 Appendix.1

Proof of Proposition 4.5

Consider commutative diagrams of $\kappa-$vector spaces

[TABLE]

which satisfies the following three properties

$j_{E}$ and $j_{F}$ are injective 2. 2.

$j_{E}:\ker i_{1}\to\ker i_{2}$ is an isomorphism 3. 3.

$\mathrm{i}mg(u)=\mathrm{i}mgi_{2}\cap\mathrm{i}mgj_{F}=\mathrm{i}mgu$

and define $T(\mathcal{B}):=F_{2}/i_{2}(E_{2})+j_{F}(F_{1})$

Consider the diagram

[TABLE]

and observe that

$\bf O1:$

If each of the three diagrams $\mathbb{B}_{1},\mathbb{B}_{2},\mathbb{B},$ associated with (44),

$\mathbb{B}_{1}$ with vertices $A_{1},A_{2},B_{1},B_{2}$,

$\mathbb{B}_{2}$ with vertices $B_{1},B_{2},C_{1},C_{2}$, and

$\mathbb{B}$ with vertices $A_{1},A_{2},C_{1},C_{2}$

satisfy the properties (1) (2) (3) above then (44) induces the exact sequence

[TABLE]

with $i$ induced by $i^{B}_{2},$ (well defined because $\mathrm{i}mg(i^{B}_{2}\cdot j_{B})\subseteq\mathrm{i}mgj_{C}$) $p$ induced by the inclusion $(i_{2}(A_{2})+j_{C}(C_{1}))\subseteq(i^{B}_{2}(B_{2})+j_{C}(C_{1})).$

Clearly $p$ is surjective and $p\cdot i=0.$ Property (3) implies $i$ injective. Properties (1), (2) (3) imply that the sequence is exact.

Similarly consider the diagram

[TABLE]

and observe by the same arguments that

$\bf O2:$

If each of the three diagrams $\mathbb{B}_{1},\mathbb{B}_{2},\mathbb{B},$ associated with (45),

$\mathbb{B}_{1}$ with vertices $A_{2},A_{3},B_{2},B_{3}$,

$\mathbb{B}_{2}$ with vertices $A_{1},A_{2},B_{1},B_{2}$, and

$\mathbb{B}$ with vertices $A_{1},A_{3},B_{1},B_{2}$

satisfy the properties (1) (2) (3) of the diagram $\mathbb{D}$ Then (45) induces the exact sequence

[TABLE]

Note that any ad-box $B=(a^{\prime}a]\times(b^{\prime},b]$ $<a^{\prime}<a<b\leq b^{\prime}<b$ defines a diagram $D$ as above with $E_{2}=\mathbb{T}_{r}(a^{\prime},b),F_{2}=\mathbb{T}_{r}(a,b),E_{1}=\mathbb{T}_{r}(a^{\prime},b^{\prime}),F_{1}=\mathbb{T}_{r}(a,b^{\prime})$ and $i_{1},i_{2},j_{E},j_{F}$ the induced linear maps. The ad-boxes $B_{12},B_{22},B{\cdot 2}$, $B_{11},B_{21},B_{\cdot 1}$ and $B_{1,\cdot},B_{2,\cdot},B$ are in the situation provided by $\bf O1$, and the boxes $B_{11},B_{12},B{1,\cdot}$, $B_{21},B_{22},B_{2,\cdot}$ and $B_{\cdot 1},B_{\cdot 2},B$ in the situation provided by $\bf O2$. Consequently Proposition 5 follows.