# Topology of angle valued maps, bar codes and Jordan blocks.

Dan Burghelea  Dept. of Mathematics, The Ohio State University, 231 West 18th
Avenue, Columbus, OH 43210, USA<EMAIL_ADDRESS>and  Stefan
Haller  Department of Mathematics, University of Vienna, Nordbergstraße 15,
A-1090 Vienna, Austria<EMAIL_ADDRESS>
###### Abstract.

In this paper one presents a collection of results relating the “bar codes”
and “Jordan blocks”, a new class of invariants for a tame angle valued map,
with the topology of underlying space (and map). As a consequence one proposes
refinements of Betti numbers and Novikov–Betti numbers provided by a
continuous real or angle valued map defined on a compact ANR. These
refinements can be interpreted as monic polynomials of degree the Betti
numbers or Novikov–Betti numbers. One shows that these polynomials depend
continuously on the real or the angle valued map and satisfy a Poincaré
duality property in case the underlying space is a closed manifold. Our work
offers an alternative perspective on Morse–Novikov theory which can be applied
to a considerably larger class of spaces and maps and provides features
inexistent in classical Morse–Novikov theory.

Part of this work was done while the second author enjoyed the hospitality of
the Ohio State University. The present version of this paper was finalized
while first author was visiting the Bernoulli Center at EPFL Lausanne and MPIM
Bonn. The first author acknowledges partial support from NSF grant MCS
0915996. The second author acknowledges the support of the Austrian Science
Fund, grant P19392-N13.

###### Contents

  1. 1 The results
  2. 2 Graph representations
  3. 3 Bar codes and Jordan blocks via graph representations
  4. 4 Proof of Theorem 1.1.
  5. 5 Stability for configurations $C_{r}(f).$ Proof of Theorem 1.2
  6. 6 Poincaré duality for configurations $C_{r}(f).$ Proof of Theorem 1.3
  7. 7 The mixed bar codes. Proof of Theorem 1.5
  8. 8 Linear relations and monodromy. Proof of Theorem 1.4
  9. 9 Appendix (an example)

##  1\. The results

In this paper a nice space is a friendlier name for a locally compact ANR. In
particular a metrizable, locally compact, finite dimensional locally
contractible space is nice. Finite dimensional simplicial complexes and finite
dimensional topological manifolds are nice spaces but the class is
considerably larger. A tame map is a proper continuous map $f:X\to\mathbb{R}$
or $f:X\to\mathbb{S}^{1},$ defined on a nice space $X$ which satisfies:

  1. (i)

each fiber of $f$ is a neighborhood deformation retract,

  2. (ii)

away from a discrete set $\Sigma\subset\mathbb{R}$ or
$\Sigma\subset\mathbb{S}^{1}$ the restriction of $f$ to $X\setminus
f^{-1}(\Sigma)$ is an Hurewicz fibration, cf. [1].

All proper simplicial maps, and proper smooth generic maps defined on a smooth
manifold 111 here ”generic” means that for any $x\in M$ the quotient algebra
of germs of smooth functions at $x$ by the ideal of partial derivatives is a
finite dimensional vector space, in particular proper real or angle valued
Morse maps, are tame.

The subspace of tame maps is residual in the space of continuous maps when
equipped with the compact open topology and weakly homotopy equivalent to the
space of all continuous maps (equipped with compact open topology)222we are
unable to locate a reference in literature for this statement, however in case
that the space $X$ is homeomorphic to a finite simplicial complex, it is a
straightforward consequence of the approximability of continuous maps by pl-
maps. Since our invariants are based on homology we fix once for all a field
$\kappa$ and write $H_{r}(X)$ for the singular homology of $X$ with
coefficients in $\kappa$. A vector space without additional specifications
will be over the field $\kappa$.

We consider a tame map, $f\colon X\to\mathbb{S}^{1}$, and as in [1] associate
to it:

  1. (i)

the critical angles $0<\theta_{1}<\theta_{2}<\cdots<\theta_{m}\leq 2\pi$,

and for any $r=0,1,\dotsc,\dim X$,

  2. (ii)

four type of intervals of real numbers, subsequently called _$r$ -bar codes_,
$r=0,1,\cdots$ whose ends mod $2\pi$ are the critical angles

    1. (1)

closed $[a,b]$,

    2. (2)

open $(a,b)$,

    3. (3)

closed–open $[a,b)$,

    4. (4)

open–closed $(a,b]$,

and

  3. (iii)

a collection of _Jordan blocks_ , i.e. isomorphism classes of indecomposable
pairs $J=(V,T)$, $V$ a finite dimensional $\kappa$-vector space, $T$ a linear
isomorphism.

We will denote by $\mathcal{B}_{r}^{c}(f)$, $\mathcal{B}_{r}^{o}(f)$,
$\mathcal{B}_{r}^{co}(f)$, $\mathcal{B}_{r}^{oc}(f)$ the collections (multi-
sets) of closed, open, closed-open and open-closed $r$-bar codes and by
$\mathcal{J}_{r}(f)$ the collection of $r$-Jordan blocks. Each bar code or
Jordan block appears in its collection with a multiplicity possibly larger
than one. For $u\in\kappa\setminus 0$ we denote by $\mathcal{J}_{r,u}(f)$ the
sub collection $\\{(V,T)\in\mathcal{J}_{r}(f)\mid
u\in\operatorname{spect}(T)\\}$.

In the Appendix the reader can see an example. As shown in[1] these invariants
are effectively computable.

In this paper the bar codes will be recorded as the finite configurations of
points in $\mathbb{C}\setminus 0,$ denoted by $C_{r}(f)$ and
$C_{r}^{m}(f)$333actually $C_{r}^{m}(f)$ is a configuration of points in
$\mathbb{C}\setminus\\{S^{1}\sqcup 0\\}$respectively, see below.

A pair $(V,T)$ as in (iii) above is _indecomposable_ if not isomorphic to the
sum of two nontrivial pairs. Note that if $T$ has $\lambda\in\kappa$ as an
eigenvalue all other eigenvalues are equal to $\lambda$ and $(V,T)$ is
isomorphic to $(\kappa^{k},T(\lambda,k))$ where

$T(\lambda,k)=\begin{pmatrix}\lambda&1&0&\cdots&0\\\
0&\lambda&1&\ddots&\vdots\\\ 0&0&\ddots&\ddots&0\\\
\vdots&\ddots&\ddots&\lambda&1\\\ 0&\cdots&0&0&\lambda\end{pmatrix}_{.}$ (1)

In [1] the indecomposable pairs $(\kappa^{k},T(\lambda,k))$ were called
_Jordan cells_. When $\kappa$ is algebraically closed all Jordan blocks are
Jordan cells.

Each tame map with $X$ compact has finitely many bar codes and Jordan blocks.

These type of invariants, are based on changes in the homology of the fibers
and have been introduced in [4] and [1] using graph representations (in [4]
only for real valued maps).

Let $\xi_{f}\in H^{1}(X;\mathbb{Z})$ be the integral cohomology class
represented by $f$. The first result we prove in this paper is:

######  Theorem 1.1 (Homotopy invariance).

If $f\colon X\to\mathbb{S}^{1}$ is a tame map then:

  1. (1)

$\sharp\mathcal{B}^{c}_{r}(f)+\sharp\mathcal{B}^{o}_{r-1}(f)$ is a homotopy
invariant of the pair $(X,\xi_{f})$, more precisely equal to the Novikov–Betti
number $\beta^{N}_{r}(X,\xi_{f})$ (see the definition in Section 4).

  2. (2)

The collection $\mathcal{J}_{r}(f)$ is a homotopy invariant of the pair
$(X,\xi_{f})$. More precisely, $\bigoplus_{J\in\mathcal{J}_{r}}(V(J),T(J))$ is
the monodromy of $(X;\xi_{f})$ (see the definition in Section 4).

  3. (3)

$\sharp\mathcal{B}^{c}_{r}(f)+\sharp\mathcal{B}^{o}_{r-1}(f)+\sharp\mathcal{J}_{r,1}(f)+\sharp\mathcal{J}_{r-1,1}(f)$
is a homotopy invariant of $X$, more precisely the Betti number
$\beta_{r}(X)$.

Here $\sharp$ denotes cardinality of multi set. Item (3) has been already
established in [1] and is included in Theorem 1.1 only for the completeness of
the topological information derived from bar codes and Jordan blocks.

In view of Theorem 1.1 it is natural to put together $\mathcal{B}_{r}^{c}(f)$
and $\mathcal{B}^{o}_{r-1}(f).$ For this purpose consider
$\mathbb{T}=\mathbb{C}/\mathbb{Z}$ and $\Delta_{\mathbb{T}}=\Delta/\mathbb{Z}$
where the $\mathbb{Z}$-action on $\mathbb{C}$ is given by $(n,z)=z+(2\pi
n+i2\pi n)$ and $\Delta=\\{z=a+ib\mid a=b\\}$. We will record the collections
$\mathcal{B}_{r}^{c}(f)\sqcup\mathcal{B}_{r-1}^{o}(f)$ as a finite
configuration of points in $\mathbb{T}$, denoted by $C_{r}(f)$, and the
collection $\mathcal{B}_{r}^{co}(f)\sqcup\mathcal{B}_{r}^{oc}(f)$ as a finite
configuration of points in $\mathbb{T}\setminus\Delta_{\mathbb{T}}$, denoted
by $C_{r}^{m}(f)$. Precisely in the first case a closed $r$-bar code $[a,b]$
will be written as the complex number $z=a+ib$ mod the action of $\mathbb{Z}$
and an open $(r-1)$-bar code $(\alpha,\beta)$ as the complex number
$z=\beta+i\alpha$ mod the action of $\mathbb{Z}$. Similarly, in the second
case, a closed-open $r$-bar code $[a,b)$ will be written as the complex number
$z=a+ib$ mod the action of $\mathbb{Z}$ and an open-closed $r$-bar code
$(\alpha,\beta]$ as the complex number $z=\beta+i\alpha$ mod the action of
$\mathbb{Z}.$

In Section 4 we will provide a direct definition of the configuration
$C_{r}(f)$ of which we derive the $r-$closed and $(r-1)-$open bar codes of $f$
and in Section 7 we will do the same for the configuration $C^{m}_{r}(f).$ The
direct definition of $C^{m}_{r}(f)$ is essentially a reformulation of the
definition of persistence diagrams used in [5] but the one for $C_{r}(f)$ is
not closed to anything considered so far. It should be noticed that the
configuration $C_{r}(f)$ makes sense for any continuous map and implicitly the
close and open bar codes can be defined for any such map.

In view of Theorem 1.1 if $f$ is in the homotopy class defined by $\xi\in
H^{1}(X;\mathbb{Z})$ then the configuration $C_{r}(f)$ has the support of
cardinality444the cardinality of the support of a configuration is the sum of
the multiplicities of its points exactly $\beta^{N}_{r}(X;\xi)$, see below,
and can be regarded as a point in the $n$-fold symmetric product
$S^{n}(\mathbb{T}),n=\beta^{N}(X,\xi)$ of $\mathbb{T}$. Note also that
$\mathbb{T}$ can be identified to $\mathbb{C}\setminus 0$ via the map $z\to
e^{i\overline{z}-\frac{(z+\overline{z})}{2}}$. Therefore each $C_{r}(f)$, and
in fact any element of $S^{n}(\mathbb{T})$, can be regarded as a monic
polynomial $P_{r}^{f}(z)$ of degree $n$ with non-vanishing free coefficient,
hence $S^{n}(\mathbb{T})$ identifies to
$\mathbb{C}^{n-1}\times(\mathbb{C}\setminus 0)$. We equip $S^{n}(\mathbb{T})$
with the topology of the symmetric product or equivalently with the topology
of $\mathbb{C}^{n-1}\times(\mathbb{C}\setminus 0)$.

Let $C(X,\mathbb{S}^{1})$ denote the space of all continuous maps equipped
with the compact open topology and let $C_{\xi}(X,\mathbb{S}^{1})$ be the
connected component corresponding to $\xi$. Let $C_{\xi,t}(X,\mathbb{S}^{1})$
be the subspace of tame maps in $C_{\xi}(X,\mathbb{S}^{1})$. Our next result
and in some sense the least expected is the following theorem.

######  Theorem 1.2 (Stability).

The assignment $C_{\xi,t}(X,\mathbb{S}^{1})\ni f\leadsto C_{r}(f)\in
S^{n}(\mathbb{T}),$ $n=\beta^{N}_{r}(X,\xi),$ is continuous. Moreover, if $X$
is homeomorphic to a simplicial complex, it extends to a continuous assignment
$C_{\xi}(X,\mathbb{S}^{1})\ni f\leadsto C_{r}(f)\in S^{n}(\mathbb{T}).$

The configuration $C_{r}(f)$, equivalently the polynomial $P_{r}^{f}(z)$, can
be viewed as a refinement of the $r$-Novikov–Betti number. The Poincaré
duality for closed manifolds extends from Novikov–Betti numbers to these
refinements and we have:

######  Theorem 1.3 (Poincaré duality).

If $M^{n}$ is a closed $\kappa$-orientable555If $\kappa$ has characteristic
$2$ any manifold is $\kappa$-orientable if not the manifold should be
orientable. topological manifold with $f\colon M\to\mathbb{S}^{1}$ a tame map
then $C_{r}(f)(z)=C_{n-r}(\overline{f})(z^{-1})$ where $\mathbb{S}^{1}$ is
viewed as the set of complex numbers of absolute value equal to $1,$
$\overline{f}\colon X\to\mathbb{S}^{1}\subset\mathbb{C}$ denotes the
composition of $f$ with the complex conjugation and $C_{r}(f)$ and $C_{n-r}$
are viewed as configurations of points in $\mathbb{C}\setminus 0.$

The proofs of Theorems 1.2 and 1.3 we provide use an alternative definition of
the configuration $C_{r}(f)$. More precisely, one defines the function
$\delta^{f}_{r}$ on $\mathbb{T}$ with values in $\mathbb{Z}_{\geq 0},$ one
checks that it is equal to the configuration $C_{r}(f)$ and one verifies
Theorems 1.2 and 1.3 for $\delta^{f}_{r}$ instead of $C_{r}(f).$

Similarly, the Jordan blocks introduced in [1] via graph representations, can
be recovered in a different manner, more precisely, as the regular part of a
linear relation, as stated in Theorem 1.4 below.

Recall that a linear relation $R\colon V\leadsto V$, concept generalizing
linear map, discussed in more details in Section 8, has a canonical linear
isomorphism $R_{\mathrm{reg}}\colon V_{\mathrm{reg}}\to V_{\mathrm{reg}}$
associated with it, cf. Section 8. We continue to write $R_{\mathrm{reg}}$ for
the pair $(V_{\mathrm{reg}},R_{\mathrm{reg}})$.

Given a tame map $f\colon X\to\mathbb{S}^{1}$ the infinite cyclic covering
$\tilde{f}\colon\tilde{X}\to\mathbb{R}$ is defined by the pullback diagram

$\textstyle{\tilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{f}}$$\textstyle{\mathbb{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{S^{1}.}$

For any $\theta\in\mathbb{S}^{1}$ regular angle, one obtains a linear relation
$R^{\theta}_{r}$ by passing to homology in the diagram

$f^{-1}(\theta)=\tilde{f}^{-1}(\tilde{\theta})\hookrightarrow\tilde{f}^{-1}\bigl{(}[\tilde{\theta},\tilde{\theta}+2\pi]\bigr{)}\hookleftarrow\tilde{f}^{-1}(\tilde{\theta}+2\pi)=f^{-1}(\theta).$

Here the real number $\tilde{\theta}\in\mathbb{R}$ corresponds to the angle
$\theta$. We have the following theorem.

######  Theorem 1.4.

If $f$ is a tame map then for any angle $\theta,$ and any $r,$ nonnegative
integer, the pair $(R^{\theta}_{r})_{\mathrm{reg}}$ is isomorphic to
$\bigoplus_{J=(V,T)\in\mathcal{J}_{r}(f)}(V,T).$

Finally we note that the collection $\mathcal{B}_{r}^{co}(f)$ can be
identified to the collection of persistence intervals considered in [12] or
[5] for the map $\tilde{f}:\tilde{X}\to\mathbb{R},$ made equivalent modulo
$2\pi-$translation. Similarly the collection $\mathcal{B}_{r}^{oc}(f),$ after
changing $(a,b]$ into $[-b,-a)$ can be identified to the collection of
persistence intervals of $-\tilde{f}.$ The stability result of [5] can be
reformulated as a stability result for the configuration $C^{m}_{r}(f).$ The
configurations $C^{m}_{r}(f)$s do not have the supports of constant
cardinality when $f$ varies in a fixed homotopy class. To give meaning to
”stability” the set of finite configurations of points in
$\mathbb{T}\setminus\Delta_{\mathbb{T}}$ has to be equipped with the topology
induced from the bottle neck metric introduced by the authors of [5]. This
metric can make arbitrary ”close” configurations with supports of different
cardinality, provided the difference is caused by points close to
$\Delta_{\mathbb{T}}.$ A statement of the result in [5] (in a slightly weaker
form), in terms of the configuration $C^{m}_{r}(f)$ is provided in Section 7,
see Theorem 7.1. In this case one can not extend the assignment
$f\rightsquigarrow C^{m}_{r}(f)$ continuously to the entire space
$C_{\xi}(X;\mathbb{S}^{1}).$

Poincaré duality holds for the configuration $C_{r}^{m}(f)$ but in analogy
with the Poincaré duality for the torsion of the integral homology for closed
orientable manifolds. Precisely we have the following result.

######  Theorem 1.5.

$($Poincaré Duality$)$ If $M^{n}$ is a closed $\kappa$-orientable topological
manifold and $f\colon M\to\mathbb{S}^{1}$ a tame map then
$C^{m}_{r}(f)([a,b])=C^{m}_{n-1-r}(-f)([-a,-b])$ with $[a,b]$ denotes the
image of $(a,b)$ in $\mathbb{T}.$

When $f$ is real valued $C_{r}(f)$ and $C_{r}^{m}(f)$ can be considered as a
finite configuration of points in $\mathbb{R}^{2}$ without passing to
$\mathbb{T}.$ The cardinality of the support of $C_{r}(f)$ is the standard
Betti number $\beta_{r}(X),$ the Poincaré dualities become
$C_{r}(f)(a,b)=C_{n-r}(-f)(-a,-b)$ and
$C^{m}_{r}(f)(a,b)=C_{n-1-r}(-f)(-a,-b)$ and there are no Jordan blocks. These
configurations can be recovered from the information derived via zigzag
persistence proposed in [4].

We like to regard the elements (i), (ii), (iii) associated to a tame angle
valued map $f\colon X\to\mathbb{S}^{1}$ in analogy to the rest points, the
isolated trajectories between rest points and the closed trajectories
(actually Poincaré return maps for closed trajectories) of
$\operatorname{grad}_{g}f$ when $(M,g)$ is a closed Riemannian manifold and
$f\colon M\to\mathbb{S}^{1}$ a Morse map. These are the elements which enter
the classical Morse–Novikov theory.

The generality of the class of spaces and maps which our theory can handle,
the finiteness of the number of the elements (i), (ii) and (iii), the
computability (by implementable algorithms) at least for $X$ simplicial
complex and $f$ simplicial map), cf. [1], end especially the robustness of
$C_{r}(f)$ to small perturbations of $f,$ make this theory “computer friendly”
and hopefully of some relevance outside mathematics.

The paper contains in addition to the present section, which summarizes the
results, seven more sections and one appendix. In Section 2 we review and
prove simple results about graph representations of the two relevant graphs
for this paper, $G_{2m}$ and $\mathcal{Z}.$ In Sections 3 and 4 we provide the
background and intermediate results for the proof of Theorem 1.1 and the
verification that $\delta^{f}_{r}$ and $C_{r}(f)$ are equal. We also prove
Theorem 1.1. In Section 5 we define the function $\delta^{f}_{r}$ and prove
Theorem 1.2. In Sections 6 and 7 we discuss the Poincaré duality for the
configurations $C_{r}(f)$ and $C^{m}_{r}(f)$ and establish Theorems 1.3 and
1.5. In Section 8 we discuss some linear algebra of linear relations and prove
Theorem 1.4. The appendix provides an example of tame map and describes its
bar codes and Jordan cells. The example is taken from [1].

The algebraic topology-minded reader can easily realize that the collection of
bar codes described in this paper can be derived from the Leray spectral
sequence of the map $f:X\to S^{1}$ whose $E_{2}-$ term is the homology of
$S^{1}$ with coefficients in the constructible sheaf defined by the homology
of $f^{-1}(U),$ $U\subset\mathbb{S}^{1}.$ The interpretation of the stability
results (Theorems 1.2 and 7.1) in terms of such spectral sequence is an
interesting problem.

Prior work: The approach of relating the topology of a space to the
homological behavior of the levels of a real or angle valued map expands the
ideas of “persistence theory” introduced in [12]. It also owes to the
apparently forgotten efforts and ideas of R. Deheuvels to extend Morse theory
to all continuous functions (fonctionelles) cf. [8], ideas which preceded
persistence theory. The stability phenomena for bar codes in classical
persistence theory was first established in [5]. The first use of graph
representations in connection with persistence appears first in [4] under the
name of zigzag persistence. The definition of bar codes and of Jordan cells
for $\mathbb{S}^{1}$-valued tame maps was first provided in [1] based on graph
representations.

##  2\. Graph representations

Let $\kappa$ be a fixed field and $\Gamma$ an oriented graph, possibly with
infinitely many vertices. A $\Gamma$-representation $\rho$ is an assignment
which to each vertex $x$ of $\Gamma$ assigns a finite dimensional vector space
$V_{x}$ and to each oriented arrow from the vertex $x$ to the vertex $y$ a
linear map $V_{x}\to V_{y}$. The concepts of morphism, isomorphism=
equivalence, sum, direct summand, zero and nontrivial representations are
obvious.

If $\rho_{\alpha},\ \alpha\in\mathcal{A},$ is a family of $\Gamma-$
representations with the property that for any $x$ all but finitely vector
spaces $V^{\alpha}_{x}$ are zero dimensional, then one considers
$\sum_{\alpha\in\mathcal{A}}\ \rho_{\alpha}$ the $\Gamma-$representation whose
vector space for the vertex $x$ is the direct sum
$\oplus_{\alpha}V^{\alpha}_{x}$ and for each oriented arrow the linear map is
the direct sum $\oplus_{\alpha}V^{\alpha}_{x}\to\oplus
V^{\alpha}_{y}=\bigoplus_{\alpha}(V^{\alpha}_{x}\to V^{\alpha}_{y}).$

The $\Gamma-$representation $\rho$ is called:

_regular_ , if all the linear maps are isomorphisms,

with _finite support_ , if $V_{x}=0$ for all but finitely many vertices and

_indecomposable_ , if not the sum of two nontrivial representations.

In this paper the oriented graph $\Gamma$ of primary concern will be $G_{2m}$
and for technical reasons we will need the infinite oriented graph
$\mathcal{Z}.$ The graph $\Gamma=G_{2m}$ has vertices
$x_{1},x_{2},\dotsc,x_{2m}$ and arrows $a_{i}\colon x_{2i-1}\to x_{2i},$
$1\leq i\leq m$, and $b_{i}\colon x_{2i+1}\to x_{2i}$, $1\leq i\leq m-1$ and
$b_{m}\colon x_{1}\to x_{2m}$. The graph $\Gamma=\mathcal{Z}$ has vertices
$x_{i}$, $i\in\mathbb{Z}$, and arrows $a_{i}\colon x_{2i-1}\to x_{2i}$ and
$b_{i}\colon x_{2i+1}\to x_{2i}.$

Both $G_{2m}$ and $\mathcal{Z}-$representations $\rho$ will be recorded as

$\rho:=\\{V_{r},\ \alpha_{i}:V_{2i-1}\to V_{2i},\ \beta_{i}:V_{2i+1}\to
V_{2i}\\}$

in the first case with $1\leq r\leq 2m,1\leq i\leq m,$ with the convention
that $V_{m+1}=V_{1},$ in the second case with $r,i\in\mathbb{Z}.$

Any regular $G_{2m}$-representation $\rho=\\{V_{r},\alpha_{i},\beta_{i}\\},$
not necessary indecomposable, is equivalent = isomorphic to the representation

$\rho(V,T)=\\{V^{\prime}_{r}=V,\alpha^{\prime}_{1}=T,\alpha^{\prime}_{i}=Id\
i\neq 1,\ \beta^{\prime}_{i}=Id\\}$

with $T=\beta^{-1}_{m}\cdot\alpha_{m}^{-1}\cdots\beta_{1}^{-1}\cdot\alpha_{1}$

The $\mathcal{Z}-$representations we consider are either with finite support
or periodic. The representation is periodic if for some integer $N,$
$V_{r}=V_{r+2N},$ $\alpha_{i}=\alpha_{i+N},\beta_{i}=\beta_{i+N}.$ Both type
of $\mathcal{Z}-$representations, periodic and with finite support, as well as
a finite direct sum of of such representations will be referred to as _good_
$\mathcal{Z}-$representations.

###  2.1. The indecomposable $G_{2m}$ and good $\mathcal{Z}-$representations.

The indecomposable $G_{2m}-$ representations are of two types, (cf. [1]).

Type I (bar codes): They are indexed by the four types of intervals $I$ with
integer valued ends $r$ and $s,$ $r\leq s,$ $1\leq r\leq m,$ namely $[r,s]$
with $r\leq s,$ and $(r,s),[r,s),(r,s]$ with $r<s,$

They are denoted by $\rho^{I}(\\{r,s\\})$ with ”$\\{$” notation for either
”$[$” or ”$($” and ”$\\}$” for either ”$]$” or”$)$” and described as follows.

Suppose the vertices $x_{1},x_{2},\cdots x_{2m-1},x_{2m}$ are located counter-
clockwise on the unit circle say at the the angles
$0<t_{1}<\theta_{1}<t_{2}<\theta_{2}<\cdots<t_{m}<\theta_{m}\leq 2\pi$ with
the $t_{i}$ angle corresponding to an odd vertices and the $\theta_{i}$ to an
even vertices.

To describe the representation $\rho^{I}(\\{i,j+mk\\}),\ 1\leq i,j\leq m,$
draw the counterclockwise spiral curve from $a=\theta_{i}$ to
$b=\theta_{j}+2\pi k$ with the ends a black or an empty circle if the end is
closed or open. Black circle indicates that the end is on our spiral empty
circle that the end is not.

Let $V_{i}$ be the vector space generated by the intersection points of the
spiral with the radius corresponding to the vertex $x_{i}$ and let
$\alpha_{i}$ and $\beta_{i}$ be defined on bases as follows: a generator $e$
of $V_{2i\pm 1}$ is sent to the generator $e^{\prime}$ of $V_{2i}$ if
connected by a piece of spiral and to $0$ otherwise. The spiral in Figure 1
below corresponds to $k=2.$

Figure 1. The spiral for $[i,j+2m)$.

Type II (Jordan blocks/cells): They are indexed by Jordan blocks $J=(V,T)$ and
denoted by $\rho^{II}(J).$ Recall that a Jordan block is an isomorphism class
of indecomposable pairs $(V,T),$ $V$ a vector space $T:V\to V$ an isomorphism.
The representation $\rho^{II}(J)$ has all vector spaces equal to $V,$
$\alpha_{1}=T$ and $\beta_{1}=\alpha_{i}=\beta_{i}=Id$ for $2\leq i\leq m.$

One refers to both the interval $\\{r,s\\}$ and the representation
$\rho^{I}(\\{r,s\\})$ as bar code and to the indecomposable pair $J$ and the
representation $\rho^{II}(J)$ as Jordan block. One denotes by
$\mathcal{B}(\rho)$ the collection of all bar codes (with proper multiplicity
when appear multiple times as independent summands) and by
${\mathcal{B}}^{c}(\rho),\mathcal{B}^{o}(\rho),\mathcal{B}^{c,o}(\rho)$ and
$\mathcal{B}^{o,c}(\rho)$ the sub collections of barcodes with both ends
closed, open, the left closed right open and left open right closed. One
denotes by $\mathcal{J}(\rho)$ the collection of all Jordan blocks (with
proper multiplicity when appear multiple times as independent summands).

For $\lambda\in\kappa\setminus 0$ one denotes by $\mathcal{J}_{\lambda}(\rho)$
the collection of Jordan blocks $J=(V,T)$ with T having $\lambda$ as an
eigenvalue, hence of the form $(\kappa^{k},T(\lambda,k)).$

By Remak-Schmidt theorem any $G_{2m}-$representation $\rho$ can be decomposed
as

$\displaystyle\rho=\bigoplus_{I\in\mathcal{B}(\rho)}\rho^{I}(I)\oplus\bigoplus_{J\in\mathcal{J}(\rho)}\rho^{II}(J).$
(2)

The indecomposable factors and their multiplicity are unique. The above
description is implicit in [13] and [10].

The indecomposable $\mathcal{Z}-$representations with finite support are all
bar codes indexed by four type of intervals $I$ with ends $i$ and $j,$ $[i,j]$
with $i\leq j$, or $[i,j),(i,j],(i,j)$ with $i<j$ and denoted by $\rho(I).$
The only periodic indecomposable representation is denoted by $\rho_{\infty}.$
The representation denoted by $\rho(I)$ has all vector spaces either $=\kappa\
\rm{or}\ 0$ and the linear maps $\alpha_{i},\beta_{j}$ equal to the identity
if both the source and the target are nontrivial and zero otherwise.
Precisely,

  1. (i)

$\rho([i,j]),i\leq j$ has $V_{r}=\kappa$ for $r=\\{2i,2i+1,\cdots 2j\\}$ and
$V_{r}=0$ otherwise,

  2. (ii)

$\rho([i,j)),i<j$ has $V_{r}=\kappa$ for $r=\\{2i,2i+2,\cdots 2j-1\\}$ and
$V_{r}=0$ otherwise,

  3. (iii)

$\rho((i,j]),i<j$ has $V_{r}=\kappa$ for $r=\\{2i+1,2i+2,\cdots 2j\\}$ and
$V_{r}=0$ otherwise,

  4. (iv)

$\rho((i,j)),i<j$ has $V_{r}=\kappa$ for $r=\\{2i+1,2i+2,\cdots 2j-1\\}$ and
$V_{r}=0$ otherwise.

Both the labeling interval $I$ and the representation $\rho(I)$ will be
referred to as bar codes.

The indecomposable representation $\rho_{\infty},$ has all vector spaces
$V_{r}=\kappa$ and all linear maps $\alpha_{i}=\beta_{i}=Id.$

One denotes by $\mathcal{B}(\rho)$ the collection of all bar codes (with
multiplicity) with $\mathcal{B}^{c}(\rho),$ $\mathcal{B}^{o}(\rho),$
$\mathcal{B}^{co}(\rho)$ and $\mathcal{B}^{oc}(\rho)$ the sub collections of
closed, open, closed-open and open-closed bar codes and by $\mathcal{J}(\rho)$
the collection of all copies of $\rho_{\infty}$ which can appear as
independent direct summands in $\rho.$

The Remak-Schmidt decomposition for representations with finite support
extends to all good $\mathcal{Z}-$representations. Precisely, any such
representation $\rho$ is a sum (in the sense described above) of possibly
infinitely many indecomposables either with finite support or isomorphic to
$\rho_{\infty},$

$\rho=\bigoplus_{I\in\mathcal{B}(\rho)}\rho(I)\oplus\bigoplus_{n}\rho_{\infty},$
(3)

with indecomposable factors and their multiplicity unique up to isomorphism.
Here $\bigoplus_{n}\rho_{\infty}$ denotes the sum of $n$ copies of
$\rho_{\infty}.$ Each indecomposable $\rho(I)$ or $\rho_{\infty}$ appears with
finite multiplicity. The statements about $G_{2m}-$representations or good
$\mathcal{Z}-$representations formulated in this paper will be verified first
for the indecomposable representations described above and if hold true, in
view of the Remak–Schmidt decomposition theorem, concluded for arbitrary
representations.

###  2.2.  Two basic constructions.

The infinite cyclic covering of a $G_{2m}-$representation
$\rho=\\{V_{r},a_{i},b_{i},1\leq r\leq 2m,1\leq i\leq m\\}$ is the periodic
$\mathcal{Z}-$representation
$\tilde{\rho}:=\\{\tilde{V}_{r},\tilde{a}_{i},\tilde{b}_{i},r,i\in\mathbb{Z}\\}$
defined by
$\tilde{V}_{r+2mk}=V_{r},\tilde{a}_{i+km}=a_{i},\tilde{b}_{i+km}=b_{i}.$ When
applied to indecomposable $\rho^{I}(I)$ or $\rho^{II}(J)$ one obtains :

$\displaystyle\widetilde{\rho^{I}(I)}=$
$\displaystyle\bigoplus_{k\in\mathbb{Z}}\rho(I+mk)$ (4)
$\displaystyle\widetilde{\rho^{II}(J)}=$
$\displaystyle\bigoplus_{n}\rho_{\infty},n=\sum_{J\in\mathcal{J}(\rho)}\dim
V,J=(V,T).$

where $I+a,$ $a\in\mathbb{Z}$ denotes translate of the interval $I,$ with $a$
units.

The truncation $T_{k,l}$ of a $\mathcal{Z}-$representation is defined for any
pair of integers $k,l,\ k\leq l$ and of a $G_{2m}-$representation for a any
pair of integers $k,l,1\leq k\leq l\leq m.$

If $\rho=\\{V_{r},\alpha_{i},\beta_{i}\\}$ and
$T_{k,l}(\rho)=\\{V^{\prime}_{r},\alpha^{\prime}_{i},\beta^{\prime}_{i}\\}$
then

$\displaystyle V^{\prime}_{r}=$ $\displaystyle\begin{cases}V_{r}\ \ 2k\leq
r\leq 2l\\\ 0\ \ \rm{otherwise}\end{cases}$ (5)
$\displaystyle\alpha^{\prime}_{r}=$ $\displaystyle\begin{cases}\alpha_{r}\ \
k+1\leq r\leq l\\\ 0\ \ \rm{otherwise}\end{cases}$
$\displaystyle\beta^{\prime}_{r}=$ $\displaystyle\begin{cases}\beta_{r}\ \
k\leq r\leq l-1\\\ 0\ \ \rm{otherwise}.\end{cases}$

More precisely for the indecomposable $\mathcal{Z}-$representations one
obtains

$\displaystyle T_{k,l}(\rho_{\infty})=$ $\displaystyle\rho([k,l])$ (6)
$\displaystyle T_{k,l}(\rho(I))=$ $\displaystyle\rho(I\cap[k,l])$

and for the indecomposable $G_{2m}-$representations one obtains

$\displaystyle T_{k,l}(\rho^{I}(\\{i,l\\})=$
$\displaystyle\rho^{I}(\\{i,l\\}\cap[k,l])$ (7) $\displaystyle
T_{k,l}(\rho^{II}(J))=$ $\displaystyle\bigoplus_{n}{\rho^{I}([k,l])},\ n=\dim
V.$

Given a $G_{2m}-$representation $\rho$ we write:

$\tilde{\mathcal{J}}(\rho)$ for the collection which contains with any Jordan
block $J\in\mathcal{J}(\rho),$ a number of $n(J)=\dim(V)$ copies of $J=(V,T)$
and

$\tilde{\mathcal{B}}^{\cdots}(\rho):=\\{I+2\pi k\mid
I\in\mathcal{B}^{\cdots}(\rho),k\in\mathbb{Z}\\}$ with
$\tilde{\mathcal{B}}^{\cdots}$ any of
$\tilde{\mathcal{B}},\tilde{\mathcal{B}}^{c},\tilde{\mathcal{B}}^{o},\tilde{\mathcal{B}}^{co},\tilde{\mathcal{B}}^{oc}.$

With the above notation one has :

######  Observation 2.1.

1.If $\rho$ is a $G_{2m}-$representation then

$\displaystyle\mathcal{B}^{\cdots}(\tilde{\rho})=$
$\displaystyle\tilde{\mathcal{B}}^{\cdots}(\rho),$
$\displaystyle\mathcal{J}(\tilde{\rho})=$
$\displaystyle\tilde{\mathcal{J}}(\rho).$

2\. If $\rho$ is a good $\mathcal{Z}$ or a $G_{2m}-$representation then

$\displaystyle\mathcal{B}^{c}(T_{l,k}(\rho))=$
$\displaystyle\\{I\in\mathcal{B}(\rho),I\cap[k,l]\neq\emptyset\ \rm{and\
closed}\\}\sqcup\tilde{\mathcal{J}}(\rho),$
$\displaystyle\mathcal{B}^{o}(T_{l,k}(\rho))=$
$\displaystyle\\{I\in\mathcal{B}^{o}(\rho),I\subset[k,l]\\}.$

###  2.3.  The matrix $M(\rho)$ and the representation $\rho_{u}$

For a $G_{2m}-$representation $\rho=\\{V_{r},\alpha_{i},\beta_{i}\\}$ $1\leq
r\leq 2m,1\leq i\leq m,$ the linear map $M(\rho):\bigoplus_{1\leq i\leq
m}V_{2i-1}\to\bigoplus_{1\leq i\leq m}V_{2i}$ is defined by the block matrix

$\begin{pmatrix}\alpha_{1}&-\beta_{1}&0&\dots&\dots&0\\\
0&\alpha_{2}&-\beta_{2}&\dots&\dots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\\\
0&\dots&\dots&\dots&\dots\alpha_{m-1}&-\beta_{m-1}\\\
-\beta_{m}&\dots&\dots&\dots&\dots&\alpha_{m}\end{pmatrix}_{.}$

and the $G_{2m}-$representation
$\rho_{u}=\\{V^{\prime}_{r},\alpha^{\prime}_{i},\beta^{\prime}_{i}\\}$ by

$V^{\prime}_{r}=V_{r},$ $\alpha^{\prime}_{1}=u\alpha_{1},$
$\alpha^{\prime}_{i}=\alpha_{i}$ for $i\neq 1$ and
$\beta^{\prime}_{i}=\beta_{i}.$ For a $\mathcal{Z}-$representation
$\rho=\\{V_{r},\alpha_{i},\beta_{i}\\}$ the linear map
$M(\rho):\bigoplus_{i\in\mathbb{Z}}V_{2i-1}\to\bigoplus_{i\in\mathbb{Z}}V_{2i},$
is defined by the infinite block matrix with entries

$M(\rho)_{2r-1,2s}=\begin{cases}&\alpha_{r},\ \ \ \rm{if}\ \ s=r\\\
&\beta_{r-1},\rm{if}\ s=r-1\\\ &0\ \ \ \ \ \ \rm{otherwise}\ .\end{cases}$

Denote by:

  1. (i)

$\dim(\rho):\Gamma\to\mathbb{Z}_{\geq 0}$ with $\dim(\rho)(x_{r})=\dim V_{r},$

  2. (ii)

$d\ker(\rho):=\dim\ker M(\rho)$ and

  3. (iii)

$d\operatorname{coker}(\rho):=\dim\operatorname{coker}M(\rho).$

As noticed in [1]

######  Observation 2.2.

  1. (i)

$\dim(\rho_{u})=\dim(\rho),$

  2. (ii)

$(\rho_{1}\oplus\rho_{2})_{u}=(\rho_{1})_{u}\oplus(\rho_{2})_{u},$

  3. (iii)

$\rho^{II}(\lambda,k)_{u}=\rho^{II}(u\lambda,k),$

  4. (iv)

$\rho^{I}(\\{i,j\\};k))_{u}\equiv\rho^{I}(\\{i,j\\};k),$

  5. (v)

$\dim(\rho_{1}\oplus\rho_{2})=\dim(\rho_{1})+\dim(\rho_{2})$,

  6. (vi)

$d\ker(\rho_{1}\oplus\rho_{2})=d\ker(\rho_{1})+d\ker(\rho_{2})$,

  7. (vii)

$d\operatorname{coker}(\rho_{1}\oplus\rho_{2})=d\operatorname{coker}(\rho_{1})+d\operatorname{coker}(\rho_{2}).$

and one has

######  Proposition 2.3.

1\. For indecomposable $G_{2m}-$representations of type I

  1. (i)

$d\ker\rho^{I}([i,j])=0$, $d\operatorname{coker}\rho^{I}([i,j])=1$,

  2. (ii)

$d\ker\rho^{I}([i,j))=0$, $d\operatorname{coker}\rho^{I}([i,j))=0$,

  3. (iii)

$d\ker\rho^{I}((i,j])=0$, $d\operatorname{coker}\rho^{I}((i,j])=0$,

  4. (iv)

$d\ker\rho^{I}((i,j))=1$, $d\operatorname{coker}\rho^{I}((i,j))=0$

and for indecomposable $\mathcal{Z}-$representations with finite support

  1. (i)

$d\ker\rho([i,j])=0$, $d\operatorname{coker}\rho([i,j])=1$,

  2. (ii)

$d\ker\rho([i,j))=0$, $d\operatorname{coker}\rho([i,j))=0$,

  3. (iii)

$d\ker\rho((i,j])=0$, $d\operatorname{coker}\rho((i,j])=0$,

  4. (iv)

$d\ker\rho((i,j))=1$, $d\operatorname{coker}\rho((i,j))=0$.

2\. For indecomposable $G_{2m}-$representations of type II

  1. (i)

$d\ker\rho^{II}(J)=0$ if $J\neq(\kappa^{k},T(1,k));$
$d\ker\rho^{II}(\kappa^{k},T(1,k))=1$

  2. (ii)

$d\operatorname{coker}\rho^{II}(J)=0$ if $J\neq(\kappa^{k},T(1,k));$
$d\operatorname{coker}\rho^{II}(\kappa^{k},T(1,k))=1$

and for the representation $\rho_{\infty}$

  1. (i)

$d\ker(\rho_{\infty})=0$ $d\operatorname{coker}(\rho_{\infty})=1.$

To check Proposition 2.3 one notices that the calculation of the kernel of
$M(\rho)$ boils down to the description of the space of solutions of the
linear system

$\displaystyle\alpha_{1}(v_{1})=$ $\displaystyle\beta_{1}(v_{3})$
$\displaystyle\alpha_{2}(v_{3})=$ $\displaystyle\beta_{2}(v_{5})$
$\displaystyle\cdots$ $\displaystyle\alpha_{m}(v_{2m-1})=$
$\displaystyle\beta_{m}(v_{1})$

which in the case of indecomposable are easy to do.

Proposition 2.3 can be refined. For each indecomposable consider the concrete
description presented above and specify a nonzero vector in $\ker M(\rho)$ or
$\operatorname{coker}(M(\rho)$ when the case. For example for Jordan blocks
such choice is needed only for the Jordan cells of form $(1,k)$ since the
kernels and cokernels are otherwise zero dimensional. This additional
specification will be regarded as part of the concrete realization of the
indecomposable representation and referred to as the _model for the
indecomposable_.

Recall that for a set $S$ one denotes by $\kappa[S]$ the vector space
generated by $S,$ equivalently the vector space of $\kappa-$valued maps on $S$
with finite support.

######  Proposition 2.4.

1\. Let $\rho$ be a $G_{2m}-$representation equipped with a decomposition
$\rho=\bigoplus_{I\in\mathcal{B}(\rho)}\rho^{I}(I)\oplus\bigoplus_{J\in\mathcal{J}(\rho)}\rho^{II}(J).$
The decomposition induces the canonical isomorphisms

$\displaystyle\Psi^{c}:\kappa$
$\displaystyle[\mathcal{B}^{c}(\rho)\sqcup\mathcal{J}(\rho)(1)]\to\operatorname{coker}M(\rho)$
$\displaystyle\Psi^{o}:\kappa$
$\displaystyle[\mathcal{B}^{o}(\rho)\sqcup\mathcal{J}(\rho)(1)]\to\ker
M(\rho).$

compatible with truncations.

2\. Let $\rho$ be a good $\mathcal{Z}-$representation equipped with a
decomposition
$\rho=\bigoplus_{I\in\mathcal{B}(\rho)}\rho(I)\oplus\bigoplus_{n}\rho_{\infty},\
$ $n=\sharp J(\rho).$ The decomposition induces the canonical isomorphisms

$\displaystyle\Psi^{c}:\kappa$
$\displaystyle[\mathcal{B}^{c}(\rho)\sqcup\mathcal{J}(\rho)]\to\operatorname{coker}M(\rho)$
$\displaystyle\Psi^{o}:\kappa$ $\displaystyle[\mathcal{B}^{o}(\rho)]\to\ker
M(\rho).$

compatible with truncations.

The construction of $\Psi^{c}$ and $\Psi^{o}$ is tautological for the model of
indecomposables as presented above. For an arbitrary representation the
decomposition permits to assemble the tautological $\Psi^{c}$’s and
$\Psi^{o}$’s into isomorphisms as stated. Note that a specified decomposition
of $\rho$ provides, in view of Observation 2.1, a decomposition of
$\tilde{\rho}$ and of the truncations $T_{k,l}(\tilde{\rho})$ and
$T_{k,l}(\rho).$

Let us explain in more details what ”compatible with the truncations” means.

The inclusions of sets $\\{i\mid k\leq i\leq l\\}\subseteq\\{i\mid
k^{\prime}\leq i\leq l^{\prime}\\}\subset\mathbb{Z}$ for $i^{\prime}\leq
i,l^{\prime}\geq l,$ induce the commutative diagram

$\textstyle{\oplus_{k\leq i\leq
l}V_{2i-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{M(T_{k,l}(\rho))}$$\textstyle{\oplus_{k^{\prime}\leq
i\leq
l^{\prime}}V_{2i-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{M(T_{k^{\prime},l^{\prime}}(\rho))}$$\textstyle{\oplus_{i}V_{2i-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{M((\rho)}$$\textstyle{\oplus_{k\leq
i\leq
l}V_{2i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\oplus_{k^{\prime}\leq
i\leq
l^{\prime}}V_{2i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\oplus_{i}V_{2i}}$
(8)

and then the linear maps

$\textstyle{\ker
M(T_{k,l}(\rho))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\ker
M(T_{k^{\prime},l^{\prime}}(\rho))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\prime}}$$\textstyle{\ker
M(\rho)}$ (9)

and

$\textstyle{\operatorname{coker}M(T_{k,l}(\rho))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{\operatorname{coker}M(T_{k^{\prime},l^{\prime}}(\rho))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{\prime}}$$\textstyle{\operatorname{coker}M(\rho).}$
(10)

The linear maps $i$ and $i^{\prime}$ are injective since by Observation 2.1
(2.) we have the inclusions
$\mathcal{B}(T_{k,l}(\rho))^{o}\subseteq\mathcal{B}(T_{k^{\prime},l^{\prime}}(\rho))^{o}\subseteq\mathcal{B}(\rho)^{o}\subseteq\mathcal{B}(\rho)^{o}\sqcup\mathcal{J}(1),$
which make the linear maps

$\textstyle{\kappa[\mathcal{B}^{o}(T_{k,l}(\rho))]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\kappa[\mathcal{B}^{o}(T_{k^{\prime},l^{\prime}}(\rho))]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\kappa[\mathcal{B}^{o}(\rho)\sqcup\mathcal{J}(1)]}$
(11)

injective.

We also have the linear maps

$\textstyle{\kappa[\mathcal{B}^{c}(T_{k,l}(\rho))\sqcup\mathcal{J}(1)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\kappa[\mathcal{B}^{c}(T_{k^{\prime},l^{\prime}}(\rho))\sqcup\mathcal{J}(1)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\kappa[\mathcal{B}^{c}(\rho)\sqcup\mathcal{J}(1)]}$
(12)

which are not necessary injective, defined as follows. As the elements of
$\mathcal{B}^{c}(T_{k,l}(\rho))$ are elements of $\mathcal{B}(\rho),$ the
linear maps in the sequence (12) send a bar code
$I\in\mathcal{B}^{c}(T_{k,l}(\rho))$ to itself if it belongs to the next set
and to zero otherwise and any element of $\mathcal{J}(1)$ to itself. The
compatibility with truncation means the commutativity of the diagrams.

---
$\textstyle{\ker
M(T_{k,l}(\rho))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\ker
M(T_{k^{\prime},l^{\prime}}(\rho))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\prime}}$$\textstyle{\ker
M(\rho)}$$\textstyle{\kappa[\mathcal{B}^{o}(T_{k,l}(\rho))]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\kappa[\mathcal{B}^{o}(T_{k^{\prime},l^{\prime}}(\rho))]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\kappa[\mathcal{B}^{o}(\rho)\sqcup\mathcal{J}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
(13)

with the vertical arrows $\Psi^{o}$s and

---
$\textstyle{\operatorname{coker}M(T_{k,l}(\rho))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{\operatorname{coker}M(T_{k^{\prime},l^{\prime}}(\rho))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{\prime}}$$\textstyle{\operatorname{coker}M(\rho)}$$\textstyle{\kappa[\mathcal{B}^{c}(T_{k,l}(\rho))\sqcup\mathcal{J}(1)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\kappa[\mathcal{B}^{c}(T_{k^{\prime},l^{\prime}}(\rho))\sqcup\mathcal{J}(1)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\kappa[\mathcal{B}^{c}(\rho)\sqcup\mathcal{J}(1).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
(14)

with vertical arrows $\Psi^{c}$s.

We finish this section with an observation about the
$\mathcal{Z}-$representations $\tilde{\rho}$ when $\rho$ is a
$G_{2k}-$representation. The shift of indices $r\to r+2k$ for vector spaces
and $i\to i+k$ for linear maps induces the linear endomorphism $\tau_{k}$ on
the kernel and cokernel of the associated matrices $M(\tilde{\rho}).$ We will
need the compositions

$(\Psi^{o})^{-1}\cdot\tau_{k}\cdot\Psi^{o}:\kappa[\mathcal{B}^{o}(\tilde{\rho})]\to\kappa[\mathcal{B}^{o}(\tilde{\rho})]$

and

$(\Psi^{c})^{-1}\cdot\tau_{k}\cdot\Psi^{c}:\kappa[\mathcal{B}^{c}(\tilde{\rho})\sqcup\mathcal{J}(\tilde{\rho})]\to\kappa[\mathcal{B}^{c}(\tilde{\rho})\sqcup\mathcal{J}(\tilde{\rho})]$

to provide a $\kappa[T^{-1},T]-$module structure (multiplication by $T$) on
$\ker M(\tilde{\rho})$ and $\operatorname{coker}M(\tilde{\rho}).$

It suffices to describe these compositions separately, for
$G_{2k}-$representations $\rho$ with $\mathcal{J}(\rho)=\emptyset$ and with
$\mathcal{B}(\rho)=\emptyset.$ In the second case $\rho$ is regular, hence
isomorphic with the representation
$\\{V_{r}=V,\alpha_{1}=T,\beta_{1}=\beta_{i}=\beta_{i}=Id,i\geq 2\\}$ with
$T:V\to V$ isomorphism.

######  Observation 2.5.

1\. If $\rho$ is a $G_{2k}-$representation with $\mathcal{J}(\rho)=\emptyset$
then the compositions above are induced by the map on bar codes which sends
the interval $\\{r,s\\}$ into the interval $\\{r+k,l+k\\}.$

2\. If $\rho$ is a $G_{2k}-$representation with $\mathcal{B}(\rho)=\emptyset$
then $\mathcal{B}^{o}(\tilde{\rho})=\mathcal{B}^{c}(\tilde{\rho})=\emptyset$
and the pair $(V,T)$ is isomorphic to
$(\kappa[\mathcal{J}(\tilde{\rho})],(\Psi^{c})^{-1}\cdot\tau_{k}\cdot\Psi^{c}\
).$

Recall that
$\sharp\mathcal{J}(\tilde{\rho})=\sum_{(V,T)\in\mathcal{J}(\rho)}\dim V.$

##  3\. Bar codes and Jordan blocks via graph representations

Let $f:X\to S^{1}$ be a tame map and
$0<\theta_{1}<\theta_{2}<\cdots\theta_{m}\leq 2\pi$ be the critical angles
(the angles of the set $\Sigma$ in the definition of tameness). Choose the
regular values $t_{1}<t_{2},\cdots<t_{m}$ with $\theta_{i-1}<t_{i}<\theta_{i}$
and $0<t_{1}<\theta_{1}.$ In order to differentiate between regular and
singular fibers we write $R_{i}:=f^{-1}(t_{i})$ and
$X_{i}:=f^{-1}(\theta_{i}).$

The tameness of $f$ induces the maps $a_{i}:R_{i}\to X_{i}$ for $1\leq i\leq
m,$ $b_{i}:R_{i+1}\to X_{1}$ for $i\leq m-1$ and $b_{m}:R_{1}\to X_{m}$ which
are unique up to homotopy; this means that different choices of the regular
values, say $t^{\prime}_{i}$ instead of $t_{i},$ lead to homotopy equivalences
$\omega_{i}:R_{i}\to R^{\prime}_{i}$ s.t. $a^{\prime}_{i}\cdot\omega_{i}$ is
homotopic to $a_{i}$ and $b^{\prime}_{i}\cdot\omega_{i}$ is homotopic to
$b_{i}.$ Indeed the fiber $R_{i}$ identifies up to homotopy to regular fiber
$f^{-1}(t)$ and $f^{-1}(t^{\prime})$ with $t$ a regular value closed enough to
$\theta_{i}$ and $t^{\prime}$ a regular value closed enough to
$\theta^{\prime}_{i-1}$ to insure that $f^{-1}(t)$ resp. $f^{-1}(t^{\prime})$
is contained in an open set which retracts to $X_{i}$ resp. $X_{i-1}.$ The
maps $a_{i}$ or $b_{i-1}$ are the composition of such identifications with
these retractions to $X_{i}$ resp. $X_{i-1}.$ We leave the reader to do the
tedious verification that the homotopy classes of $a_{i}$ and $b_{i-1}$ are
independent of the choices made. Passing to $r-$homology one obtains the
$G_{2m}-$representation $\rho_{r}=\rho_{r}(f)$ whose vector spaces are
$V_{2s}=H_{r}(X_{s})$ and $V_{2s-1}=H_{r}(R_{s})$ and linear maps $\alpha_{i}$
and $\beta_{i}$ are induced by the continuous maps $a_{i}$ and $b_{i}.$

The representation $\rho_{r}(f)$ has bar codes whose ends are $i,j+km,$ $1\leq
i,j\leq m.$ Denote by $\mathcal{B}_{r}(f),$ the collections of intervals
defined by the bar codes of $\rho_{r}(f)$ with ends $i$ and $j+km$ replaced by
$\theta_{i}$ and $\theta_{j}+2\pi k.$ Denote by $\mathcal{J}_{r}(f)$ the
collection of Jordan blocks of the representation $\rho_{r}(f)$.

One can think to these bar codes in a way more consistent with points in the
space $\mathbb{T}$. If $\tilde{f}:\tilde{X}\to\mathbb{R}$ is the infinite
cyclic covering of $f$ then the real numbers $\theta_{i}+2\pi k$ are the
critical values and $t_{i}+2\pi k$ are regular values (between consecutive
critical values) and the tameness of $\tilde{f}$ gives the maps
$a_{i+km}:\tilde{X}_{t_{i+1}+2\pi k}\to\tilde{X}_{\theta_{i}+2\pi k}$ and
$b_{i+km}:\tilde{X}_{t_{i}+2\pi k}\to\tilde{X}_{\theta_{i}+2\pi k}.$ By
passing to homology in dimension $r$ one obtains a good
$\mathcal{Z}-$representation $\rho_{r}(\tilde{f})$ which is exactly the
infinite cyclic covering $\widetilde{\rho_{r}(f)}.$ The collections
$\mathcal{B}_{r}(\tilde{f}),\mathcal{B}^{c}_{r}(\tilde{f}),$
$\mathcal{B}^{o}_{r}(\tilde{f}),$ $\mathcal{B}^{co}_{r}(\tilde{f}),$
$\mathcal{B}^{oc}_{r}(\tilde{f})$ are invariants w.r to the $2\pi$ translation
and the collections $\mathcal{B}_{r}(f),\mathcal{B}^{c}_{r}(f)$
$\mathcal{B}^{o}_{r}(f)$ $\mathcal{B}^{co}_{r}(f)$ $\mathcal{B}^{oc}_{r}(f)$
can be viewed as equivalence (= modulo the $2\pi$ translation) classes of
elements of $\mathcal{B}^{c}_{r}(\tilde{f}),$
$\mathcal{B}^{o}_{r}(\tilde{f}),$ $\mathcal{B}^{co}_{r}(\tilde{f}),$
$\mathcal{B}^{oc}_{r}(\tilde{f}).$

Given $\xi\in H^{1}(X;\mathbb{Z})$ and $u\in\kappa\setminus 0,$ the pair
$(\xi,u)$ denotes the rank one representation
$H_{1}(X;\mathbb{Z})\to\mathbb{Z}\to\kappa\setminus 0,$ where the first arrow
is given by $\xi$ and the second by the homomorphism
$<u>:\mathbb{Z}\to\kappa\setminus 0$ defined by $<u>(n)=u^{n}.$ One denotes by
$H_{r}(X;(\xi,u))$ the homology of $X$ with coefficients in the local system
defined by the representation $(\xi,u),$ which for $u=1$ satisfies
$H_{r}(X;(\xi,1))=H_{r}(X).$ When restricted to $R_{i}$ and $X_{i}$ the local
system is the constant one with fiber $\kappa$ so by passing to homology the
$G_{2m}-$representation obtained will have the same vector spaces for all
$u^{\prime}$s but not necessary the same $\alpha_{i}^{\prime}$s and
$\beta_{i}^{\prime}$s. The $G_{2m}-$representation obtained will be isomorphic
to $(\rho_{r}(f))_{u}.$ More general for
$X_{[\theta_{1},\theta_{2}]}=f^{-1}([\theta_{1},\theta_{2}])$ with
$\theta_{2}-\theta_{1}<2\pi,$ the restriction of the local system considered
above is isomorphic to the constant local system with fiber $\kappa$ and the
inclusion $X_{[\theta_{1},\theta_{2}]}\subset X$ induces the homomorphism

$H_{r}(X_{[\theta_{1},\theta_{2}]})\to H_{r}(X;(\xi,u)).$

###  3.1. The relevant exact sequences. (cf. [1])

The tool which permits the calculation of the homology of $X$, $\tilde{X}$ and
various pieces of these spaces is provided by Proposition 3.1 below.

Observe that for $\theta_{i}\leq\theta_{j}$ critical angles of $f$ and
$f_{[\theta_{i},\theta_{j}]}$ denoting the restriction of $f$ to
$X_{[\theta_{i},\theta_{j}]}=f^{-1}[\theta_{i},\theta_{j}]$ one has

$\rho_{r}(f_{[\theta_{i},\theta_{j}]})=T_{i,j}(\rho_{r}(f)).\ $

Similarly, for $c_{i}\leq c_{j}$ critical values of $\tilde{f}$ and
$\tilde{f}_{[c_{i},c_{j}]}$ denoting the restriction $\tilde{f}$ to
$\tilde{X}_{[c_{i},c_{j}]}=\tilde{f}^{-1}[c_{i},c_{j}]$ one has

$\rho_{r}(\tilde{f}_{[c_{i},c_{j}]})=T_{i,j}(\tilde{\rho}_{r}(f)).$

######  Proposition 3.1.

Let $f:X\to\mathbb{S}^{1}$ a tame map, $\tilde{f}:\tilde{X}\to\mathbb{R}$ its
infinite cyclic covering. Let $\rho_{r}=\rho_{r}(f)$ and
$\tilde{\rho}_{r}=\rho_{r}(\tilde{f})=\widetilde{\rho}_{r}(f)$ be the
representations associated with $f$ and $\tilde{f}.$ One has the following
short exact sequences:

$0\to\operatorname{coker}M((\rho_{r})_{u})\to H_{r}(X;(\xi_{f},u))\to\ker
M((\rho_{r-1})_{u})\to 0,$ (15)

which for $u=1$ becomes

$0\to\operatorname{coker}M(\rho_{r})\to H_{r}(X)\to\ker M(\rho_{r-1})\to 0,$
(16)

and

$0\to\operatorname{coker}M(\tilde{\rho}_{r})\to H_{r}(\tilde{X})\to\ker
M(\tilde{\rho}_{r-1})\to 0.$ (17)

The sequences are compatible with the truncations with respect to the pairs of
critical angles $(\theta_{i},\theta_{j})$ and
$(\theta_{i^{\prime}},\theta_{j^{\prime}}),$
$0<\theta_{i}\leq\theta_{i^{\prime}}\leq\theta_{j^{\prime}}\leq\theta_{j}\leq
2\pi$ resp. the pairs of critical values $(c_{i},c_{j})$ and
$(c_{i^{\prime}},c_{j^{\prime}})$ with $c_{i}\leq c_{i^{\prime}}\leq
c_{j^{\prime}}\leq c_{j}.$

In the case of $G_{2m}-$ representation $\rho_{r}(f)$ ”compatibility with
truncation” means the commutativity of the diagram (18)

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{coker}M(T_{i^{\prime},j^{\prime}}(\rho_{r}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v_{l}}$$\textstyle{H_{r}(X_{[\theta_{i^{\prime}},\theta_{j^{\prime}}]})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{v}$$\textstyle{\ker
M(T_{i^{\prime},j^{\prime}}(\rho_{r-1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v_{r}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{coker}M(T_{i,j}(\rho_{r}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v^{\prime}_{l}}$$\textstyle{H_{r}(X_{[\theta_{i},\theta_{j}]})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi"}$$\scriptstyle{v^{\prime}}$$\textstyle{\ker
M(T_{i,j}(\rho_{r-1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v^{\prime}_{r}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{coker}M((\rho_{r})_{u})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{r}(X;(\xi_{f},u))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\ker
M((\rho_{r-1})_{u})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
(18)

and in the case of the $\mathcal{Z}-$representation $\tilde{\rho}_{r}$ the
commutativity of the diagram (19)

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{coker}M(T_{{i^{\prime}},{j^{\prime}}}(\tilde{\rho}_{r}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v_{l}}$$\textstyle{H_{r}(\tilde{X}_{[c_{i^{\prime}},c_{j^{\prime}}]})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{v}$$\textstyle{\ker
M(T_{i^{\prime},j^{\prime}}(\tilde{\rho}_{r-1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v_{r}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{coker}M(T_{i,j}(\tilde{\rho}_{r}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v^{\prime}_{l}}$$\textstyle{H_{r}(\tilde{X}_{[c_{i},c_{j}]})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi"}$$\scriptstyle{v^{\prime}}$$\textstyle{\ker
M(T_{i,j}(\tilde{\rho}_{r-1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v^{\prime}_{r}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{coker}M(\tilde{\rho}_{r})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{r}(\tilde{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\ker
M(\tilde{\rho}_{r-1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$
(19)

To establish these diagrams denote by $\mathcal{R}:=\sqcup_{1\leq i\leq
m}R_{i},$ $\tilde{\mathcal{R}}:=\sqcup_{i\in\mathbb{Z}}R_{i},$
$\mathcal{X}:=\sqcup_{1\leq i\leq m}X_{i}$ and
$\tilde{\mathcal{X}}:=\sqcup_{i\in\mathbb{Z}}X_{i}.$

The short exact sequences (15) and (16) follow from the long exact sequence

$\cdots\to
H_{r}(\mathcal{R})\xrightarrow{M((\rho_{r})_{u})}H_{r}(\mathcal{X})\to
H_{r}(X;(\xi,u))\to
H_{r-1}(\mathcal{R})\xrightarrow{M(\rho_{r-1})}H_{r-1}(\mathcal{X})\to\cdots.$
(20)

with $H_{r}(\mathcal{R})=\bigoplus_{1\leq i\leq m}H_{r}(R_{i})$ and
$H_{r}(\mathcal{X})=\bigoplus_{1\leq i\leq m}H_{r}(X_{i})$ (16 for $u=1$) and
the short exact sequence (17) from the long exact sequence

$\cdots\to
H_{r}(\tilde{\mathcal{R}})\xrightarrow{M(\rho_{r})}H_{r}(\tilde{\mathcal{X}})\to
H_{r}(\tilde{X})\to
H_{r-1}(\tilde{\mathcal{R}})\xrightarrow{M(\rho_{r-1})}H_{r-1}(\tilde{\mathcal{X}})\to\cdots.$
(21)

Since both long exact sequences (20) and (21) are derived in the same way we
will work only on (20) and for simplicity only for $u=1$.

First choose an $\epsilon>0$ small enough so that $2\epsilon<t_{1}$ and
$\theta_{i-1}+2\epsilon<t_{i}<\theta_{i}-2\epsilon.$ To simplify the writing,
since $i\leq m,$ introduce $\theta_{m+1}=\theta_{1}+2\pi$ and define
$f^{-1}([\theta_{m}\pm\epsilon,\theta_{m+1}\pm\epsilon):=\tilde{f}^{-1}([\theta_{m}\pm\epsilon,\theta_{1}+2\pi\pm\epsilon]).$

Define

  1. (i)

$\mathcal{P}^{\prime}=\sqcup_{1\leq i\leq
m}f^{-1}([\theta_{i},\theta_{i+1}-\epsilon))$

  2. (ii)

$\mathcal{P}^{\prime\prime}=\sqcup_{1\leq i\leq
m}f^{-1}((\theta_{i}+\epsilon,\theta_{i+1}])$

and observe that in view of the choice of $\epsilon$ and the tameness of $f$
the inclusions $\mathcal{X}\subset\mathcal{P}^{\prime},$
$\mathcal{X}\subset\mathcal{P}"$ and
$\mathcal{X}\sqcup\mathcal{R}\subset\mathcal{P}^{\prime}\cap\mathcal{P}^{\prime\prime}$
are homotopy equivalences.

The Mayer Vietoris long exact sequence for
$X=\mathcal{P}^{\prime}\cup\mathcal{P}^{\prime\prime}$ gives the diagram

$\textstyle{H_{r}(\mathcal{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{M(\rho_{r}(f))}$$\textstyle{H_{r}(\mathcal{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{r+1}(\mathcal{T})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{r+1}}$$\textstyle{H_{r}(\mathcal{R})\oplus
H_{r}(\mathcal{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pr_{1}}$$\scriptstyle{N}$$\textstyle{H_{r}(\mathcal{X})\oplus
H_{r}(\mathcal{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}$$\scriptstyle{(Id,-Id)}$$\scriptstyle{(i^{r},-i^{r})}$$\textstyle{H_{r}(\mathcal{T})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{r}(\mathcal{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{in_{2}}$$\scriptstyle{Id}$$\textstyle{H_{r}(\mathcal{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$
(22)

where $\Delta$ denotes the diagonal, $in_{2}$ the inclusion on the second
component, $pr_{1}$ the projection on the first component, $i^{r}$ the linear
map induced in homology by the inclusion $\mathcal{X}\subset\mathcal{T}.$

The matrix $M(\rho_{r}(f))$ is defined by

$\begin{pmatrix}\alpha^{r}_{1}&-\beta^{r}_{1}&0&\cdots&0\\\
0&\alpha^{r}_{2}&-\beta^{r}_{2}&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\ddots&0\\\
0&\cdots&0&\alpha^{r}_{m-1}&-\beta^{r}_{m-1}\\\
-\beta^{r}_{m}&0&\cdots&0&\alpha^{r}_{m}\end{pmatrix}$

with $\alpha^{r}_{i}\colon H_{r}(R_{i})\to H_{r}(X_{i})$ and
$\beta^{r}_{i}\colon H_{r}(R_{i+1})\to H_{r}(X_{i})$ induced by the maps
$a_{i}$ and $b_{i}$ and the block matrix $N$ defined by

$\begin{pmatrix}\alpha^{r}&\operatorname{Id}\\\
-\beta^{r}&\operatorname{Id}\end{pmatrix}$

where $\alpha^{r}$ and $\beta^{r}$ are the matrices

$\begin{pmatrix}\alpha^{r}_{1}&0&\cdots&0\\\ 0&\alpha^{r}_{2}&\ddots&\vdots\\\
\vdots&\ddots&\ddots&0\\\
0&\cdots&0&\alpha^{r}_{m-1}\end{pmatrix}\quad\text{and}\quad\begin{pmatrix}0&\beta^{r}_{1}&0&\dots&0\\\
0&0&\beta^{r}_{2}&\ddots&\vdots\\\ \vdots&\vdots&\ddots&\ddots&0\\\
0&0&\dots&0&\beta^{r}_{m-1}\\\ \beta^{r}_{m}&0&\dots&0&0\end{pmatrix}_{.}$

The long exact sequence (20) is the top sequence in the diagram (22).

By carefully following the above construction one verifies the commutativity
of the diagrams. q.e.d.

##  4\. Proof of Theorem 1.1.

Consider the pair $(X,\xi\in H^{1}(X;\mathbb{Z}))$ with $X$ a compact ANR and
denote by $\tilde{X}\to X$ the infinite cyclic covering associated to $\xi$.
Recall from Section 1 that for $\xi=\xi_{f}$ the covering $\tilde{X}\to X$ is
the pull back by $f$ of the universal covering $\mathbb{R}\to\mathbb{S}^{1}$

$\textstyle{\tilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{f}}$$\textstyle{\mathbb{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{S^{1}.}$

The vector space $H_{r}(\tilde{X})$ is actually a $\kappa[T^{-1},T]$-module
666$\kappa[T^{-1},T]$ denotes the ring of Laurent polynomials with
coefficients in $\kappa$ where the multiplication by $T$ is the linear
isomorphism induced by the deck transformation
$\tau\colon\tilde{X}\to\tilde{X}.$

Let $\kappa[T^{-1},T]]$ be the field of Laurent power series and define

$H^{N}_{r}(X;\xi):=H_{r}(\tilde{X})\otimes_{\kappa[T^{-1},T]}\kappa[T^{-1},T]].$

The $\kappa[T^{-1},T]]$-vector spaces $H^{N}_{r}(X;\xi)$ is called the $r-$th
Novikov homology777instead of $\kappa[T^{-1},T]]$ one can consider the field
$\kappa[[T^{-1},T]$ of Laurent power series in $T^{-1}$, which is isomorphic
to $\kappa[T^{-1},T]]$ by an isomorphism induced by $T\to T^{-1}$. The
(Novikov) homology defined using this field has the same Novikov–Betti numbers
as the the one defined using $\kappa[T^{-1},T]]$. and its dimension over the
field $\kappa[T^{-1},T]]$, the _Novikov–Betti number_ $\beta^{N}_{r}(X;\xi).$
Consider $H_{r}(\tilde{X})\to H^{N}_{r}(X;\xi)$ the $\kappa[T^{-1},T]$-linear
map induced by taking the tensor product with $\kappa[T^{-1},T]]$ over
$\kappa[T^{-1},T].$ The $\kappa[T^{-1},T]-$ module $V(\xi),$

$V(\xi):=\ker(H_{r}(\tilde{X})\to H^{N}_{r}(X;\xi)),$

when regarded as a $\kappa-$vector space equipped with the linear isomorphism
$T(\xi)$ provided by the multiplication by $T$ is referred to as the
$r-$monodromy of $(X,\xi)$. As a $\kappa[T^{-1},T]-$module $V_{r}(\xi)$ is
exactly the torsion of the $\kappa[T^{-1},T]-$module $H_{r}(\tilde{X}).$

Since $X$ is a compact ANR all numbers $\dim H_{r}(X),\ \beta^{N}_{r},\dim
V(\xi)$ are finite.

A nonempty subset $K$ of $\mathbb{S}^{1}$ or $\mathbb{R},$ will be called a
_closed multi-interval_ if it is a finite union of disjoint closed intervals
$[\theta_{1},\theta_{2}]$ with $0\leq\theta_{1}\leq\theta_{2}<2\pi$ in the
case of $\mathbb{S}^{1},$ and $[a,b]$ with $a\leq b$ or $(-\infty,a]$ or
$[b,\infty)$ in the case of $\mathbb{R}$. One denotes by $X_{K}:=f^{-1}(K)$ if
$K\subset\mathbb{S}^{1}$ and by $\tilde{X}_{K}=f^{-1}(K)$ if
$K\subset\mathbb{R}.$

In case $K\subset\mathbb{S}^{1}$ one considers

  1. (i)

$\mathcal{B}^{c}_{r,K}(f)=\\{I\in\mathcal{B}^{c}_{r}(f)\mid I\cap
K\neq\emptyset\\}$

  2. (ii)

$\mathcal{B}^{o}_{r-1,K}(f)=\\{I\in\mathcal{B}^{o}_{r-1}(f)\mid I\subset K\\}$

and for $u\in\kappa\setminus 0$ the sets:

  3. (iii)

$S_{r,K,u}(f)=\mathcal{B}^{c}_{r,K}(f)\sqcup\mathcal{B}^{o}_{r-1,K}(f)\sqcup\mathcal{J}_{r,u}(f)\quad\rm$

  4. (iv)

$S_{r,u}(f)=\mathcal{B}^{c}_{r}(f)\sqcup\mathcal{B}^{o}_{r-1}(f)\sqcup\mathcal{J}_{r,u}(f)\sqcup\mathcal{J}_{r-1,u}(f).$

In case $K\subset\mathbb{R}$ one considers

  1. (i)

$\tilde{\mathcal{B}}^{c}_{r,K}(f)=\\{I\in\tilde{\mathcal{B}}^{c}_{r}(f)\mid
I\cap K\neq\emptyset\\}$

  2. (ii)

$\tilde{\mathcal{B}}^{o}_{r-1,K}(f)=\\{I\in\tilde{\mathcal{B}}^{o}_{r-1}(f)\mid
I\subset K\\}$ and the sets:

  3. (iii)

$\tilde{S}_{r,K}(f)=\tilde{\mathcal{B}}^{c}_{r,K}(f)\sqcup\tilde{\mathcal{B}}^{o}_{r-1,K}(f)\sqcup\tilde{\mathcal{J}}_{r}(f)\quad\rm$

  4. (iv)

$\tilde{S}_{r}(f)=\tilde{\mathcal{B}}^{c}_{r}(f)\sqcup\tilde{\mathcal{B}}^{o}_{r-1}(f)\sqcup\tilde{\mathcal{J}}_{r}(f).$

These sets have the following properties:

  1. (i)

If $K_{1},K_{2},K$ are closed multi-intervals in $\mathbb{S}^{1}$ or
$\mathbb{R}$ with $K_{1}\cap K_{2}=\emptyset$ and $K=K_{1}\cup K_{2}$ then
$S_{r,K,u}=S_{r,K_{1},u}\cup S_{r,K_{2},u}$ and
$\tilde{S}_{r,K}=\tilde{S}_{r,K_{1}}\cup\tilde{S}_{r,K_{2}}$

  2. (ii)

If $K_{1},K_{2},K$ are closed multi-intervals in $\mathbb{S}^{1}$ or
$\mathbb{R}$ with $K_{1}\cap K_{2}=K$ then $S_{r,K,u}=S_{r,K_{1},u}\cap
S_{r,K_{2},u}$ and
$\tilde{S}_{r,K}=\tilde{S}_{r,K_{1}}\cap\tilde{S}_{r,K_{2}},$

  3. (iii)

If $K_{1},K_{2}$ closed multi-intervals with $K_{1}\subset K_{2}$ then
$S_{r,K_{1},u}\subseteq S_{r,K_{2},u}$ and
$\tilde{S}_{r,K_{1}}\subseteq\tilde{S}_{r,K_{2}}.$

For $K$ a multi-interval in $\mathbb{S}^{1}$ or $\mathbb{R}$ denote by:

$\mathbb{I}_{r}(f;K,u):=\operatorname{img}(H_{r}(X_{K})\to H_{r}(X;(\xi,u))),$
and

$\mathbb{I}_{r}(\tilde{f};K):=\operatorname{img}(H_{r}(\tilde{X}_{K})\to
H_{r}(\tilde{X})).$

With the notations and definitions above we have the following result which
calculates the homologies of $X$ and $\tilde{X}.$

######  Proposition 4.1.

Let $f:X\to\mathbb{S}^{1}$ be a tame map and suppose that for each $r$ a
decomposition of the representation $\rho_{r}(f)$ as a sum of bar code
representations and Jordan block representations is given. Let
$u\in\kappa\setminus 0.$

1\. One can provide the isomorphism

$\omega_{r,u}:\kappa[S_{r,u}(f)]\to H_{r}(X;(\xi_{f},u))$

and for any closed multi interval $K\subset\mathbb{S}^{1}$ the isomorphism

$\omega_{r,K,u}:\kappa[S_{r,K,u}(f)]\to\mathbb{I}_{r}(f;K,u)$

such that for $K^{\prime},K$ closed multi-intervals in $\mathbb{S}^{1}$ with
$K^{\prime}\subset K,$ the diagram

$\textstyle{\mathbb{I}_{r}(f;K^{\prime},u)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$$\textstyle{\mathbb{I}_{r}(f;K,u)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$$\textstyle{H_{r}(X;(\xi_{f},u))}$$\textstyle{\kappa[S_{r,K^{\prime},u}(f)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega_{r,K^{\prime},u}}$$\textstyle{\kappa[S_{r,K,u}(f)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega_{r,K,u}}$$\textstyle{\kappa[S_{r,u}(f)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega_{r,u}}$
(23)

is commutative. The horizontal arrows of the bottom line in the diagram are
induced by the inclusions of the sets in brackets.

2\. One can provide the isomorphism

$\tilde{\omega}_{r}:\kappa[\tilde{S}_{r}(f)]\to H_{r}(\tilde{X})$

and for any closed multi interval $K\subset\mathbb{R}$ the isomorphism

$\tilde{\omega}_{r,K}:\kappa[\tilde{S}_{r,K}(f)]\to\mathbb{I}_{r}(\tilde{f};K)$

such that for $K^{\prime},K$ closed multi-intervals in $\mathbb{R}$ with
$K^{\prime}\subset K,$ the diagram

$\textstyle{\mathbb{I}_{r}(\tilde{f};K^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$$\textstyle{\mathbb{I}_{r}(\tilde{f};K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$$\textstyle{H_{r}(\tilde{X})}$$\textstyle{\kappa[\tilde{S}_{r,K^{\prime}}(f)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\omega}_{r,K^{\prime}}}$$\textstyle{\kappa[\tilde{S}_{r,K}(f)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\omega}_{r;K}}$$\textstyle{\kappa[\tilde{S}_{r}(f)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\omega}_{r}}$
(24)

is commutative. The horizontal arrows in the bottom line are induced by the
inclusions of the sets in brackets.

3\. One can provide an isomorphism $\omega^{N}_{r}:\kappa[T^{-1},T]][S_{r}]\to
H_{r}^{N}(X;\xi_{f}).$

It is also possible to calculate $H_{r}(X_{K})$ for $K\subset\mathbb{S}^{1}$
and $H_{r}(\tilde{X}_{K})$ for $K\subset\mathbb{R}.$ In this case, in addition
to closed and open bar codes and to Jordan blocks, mixed bar codes will
appear. In this case it suffices to state the result for $K$ consisting of
only one interval say $[\theta_{1},\theta_{2}],\
0\leq\theta_{1}\leq\theta_{2}<2\pi$ in case of $\mathbb{S}^{1}$ and
$[a,b],-\infty<a\leq b<\infty$ in case of $\mathbb{R}.$

To formulate the result one extends the sets $S_{r,K}(f),\tilde{S}_{r,K}(f)$
to $S^{\prime}_{r,K}(f),\tilde{S}^{\prime}_{r,K}(f),$ $K$ a closed interval in
$\mathcal{S}^{1}$ or $\mathbb{R}$ as follows.

For $K\subset\mathbb{S}^{1}$ define

$S^{\prime}_{r,K}(f)={{\mathcal{B}^{\prime}}_{r,K}}(f)\sqcup\mathcal{B}^{o}_{r-1,K}(f)\sqcup\mathcal{J}_{r}(f)$

where ${\mathcal{B}}^{\prime}_{r,K}(f)=\\{I\in{\mathcal{B}}_{r}\mid I\cap
K\neq\emptyset,\rm{and\ closed}\\}$ and for $K\subset\mathbb{R}$ define

$\tilde{S}^{\prime}_{r,K}(f)=\tilde{\mathcal{B}}^{\prime}_{r,K}(f)\sqcup\tilde{\mathcal{B}}^{o}_{r-1,K}(f)\sqcup\mathcal{J}_{r}(f)$

where
$\tilde{\mathcal{B}}^{\prime}_{r,K}(f)=\\{I\in\tilde{\mathcal{B}}_{r}\mid
I\cap K\rm{\ closed\ and}\neq\emptyset,\\}.$

######  Proposition 4.2.

Under the same hypothesis as in Proposition 4.1 one has:

1\. For any pair of angles $\theta^{\prime},\theta^{\prime\prime},$
$0<\theta^{\prime}\leq\theta^{\prime\prime}<2\pi$ one can provide the
isomorphisms
$\omega^{\prime}_{r,[\theta^{\prime},\theta^{\prime\prime}]}:\kappa[S^{\prime}_{r,[\theta^{\prime},\theta^{\prime\prime}]}(f)]\to
H_{r}(X_{\theta^{\prime},\theta^{\prime\prime}})$ so that for
$0<\theta_{1}\leq\theta_{2}\leq\theta_{3}\leq\theta_{4}<2\pi$ the following
diagram

$\textstyle{H_{r}(X_{[\theta_{2},\theta_{3}]})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v_{r}}$$\textstyle{H_{r}(X_{[\theta_{1},\theta_{4}]})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v^{\prime}_{r}}$$\textstyle{H_{r}(X;(\xi_{f},u))}$$\textstyle{\kappa[S^{\prime}_{r,[\theta_{2},\theta_{3}]}(f)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega^{\prime}_{r,[\theta_{2},\theta_{3}]}}$$\textstyle{\kappa[S^{\prime}_{r,[\theta_{1},\theta_{4}]}(f)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega^{\prime}_{r,[\theta_{1},\theta_{4}]}}$$\textstyle{\kappa[S_{r,u}(f)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{\omega_{r,u}}$
(25)

is commutative.

2\. For any pair of numbers $a^{\prime},b^{\prime},$ $a^{\prime}\leq
b^{\prime}$ or $a^{\prime}=-\infty$ or $b^{\prime}=\infty$ one can provide the
isomorphisms
$\tilde{\omega}^{\prime}_{r,[a,b]}:\kappa[\tilde{S}^{\prime}_{r,[a,b]}(f)]\to
H_{r}(\tilde{X}_{[a,b]})$ so that for $a\leq b\leq c\leq d$ the following
diagram

$\textstyle{H_{r}(\tilde{X}_{[b,c]})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v_{r}}$$\textstyle{H_{r}(\tilde{X}_{[a,d]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v^{\prime}_{r}}$$\textstyle{H_{r}(\tilde{X})}$$\textstyle{\kappa[\tilde{S}^{\prime}_{r,[b,c]}(f)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\omega}_{c,d}}$$\textstyle{\kappa[\tilde{S}^{\prime}_{r,[a,d]}(f)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\omega}_{a,d}}$$\textstyle{\kappa[\tilde{S}_{r}(f)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\omega}_{r}}$
(26)

is commutative.

In both cases the horizontal arrows in the top line are inclusion induced
linear maps in homology, while in the bottom line are defined as follows: a
bar code in the set $S^{\prime}_{r,\cdots}$ or in
${\tilde{S}^{\prime}}_{r,\cdots}$ is sent to itself if continues to belong to
the next set or if not to the zero vector in the next vector space.

The isomorphisms claimed in these propositions are uniquely determined by the
decomposition of $\rho_{r}^{\prime}$s and by the choice of a splitting in the
short exact sequences (16), (17), (15).

Let $\alpha\leq a\leq b\leq\beta,$
$i(a,b;\alpha,\beta):\tilde{X}_{[a,b]}\subseteq\tilde{X}_{[\alpha,\beta]}$ be
the inclusion and $i_{r}(a,b:\alpha,\beta):H_{r}(X_{[a,b]})\to
H_{r}(X_{[\alpha,\beta]})$ be the inclusion induced linear maps. The following
corollary of Proposition 4.2 will be of use later.

######  Proposition 4.3.

Under the same hypothesis as in Proposition 4.1 one has:

$\displaystyle\dim H_{r}(\tilde{X}_{[a,b]})=$
$\displaystyle\sharp\\{I\in\tilde{\mathcal{B}}_{r}(f)\mid
I\cap[a,b]\neq\emptyset\ \rm{and\ closed}\\}$ $\displaystyle\
+\sharp\\{I\in\tilde{\mathcal{B}}^{o}_{r-1}(f)\mid
I\subset[a,b]\\}+\sharp\tilde{\mathcal{J}}_{r}(f)$
$\displaystyle\dim\operatorname{img}i_{r}(a,b;\alpha,\beta)=$
$\displaystyle\sharp\\{I\in\tilde{\mathcal{B}}_{r}(f)\mid
I\cap[\alpha,\beta]\neq\emptyset\ \rm{and\
closed},I\cap[a,b]\neq\emptyset\\}+$ $\displaystyle\
+\sharp\\{I\in\tilde{\mathcal{B}}^{o}_{r-1}(f)\mid
I\subset[a,b]\\}+\sharp\tilde{\mathcal{J}}_{r}(f)$
$\displaystyle\dim\operatorname{coker}i_{r}(a,b;\alpha,\beta)=$
$\displaystyle\sharp\\{I\in\tilde{\mathcal{B}}_{r}\mid
I\cap[\alpha,\beta]\neq\emptyset\ \rm{and\ closed},I\cap[a,b]=\emptyset\\}$
$\displaystyle\ +\sharp\\{I\in\tilde{\mathcal{B}}^{o}_{r-1}\mid
I\subset[\alpha,\beta],I\nsubseteq[a,b]\\}$ $\displaystyle\dim\ker
i_{r}(a,b;\alpha,\beta)=$
$\displaystyle\sharp\\{I\in\tilde{\mathcal{B}}_{r}\mid
I\cap[a,b]\neq\emptyset\ \rm{and\ closed},I\cap[\alpha,\beta]\ \rm{not\
closed}\\}$ $\displaystyle\dim
H_{r}(\tilde{X}_{[\alpha,\beta]},\tilde{X}_{[a,b]})=$
$\displaystyle\dim\operatorname{coker}i_{r}(a,b;\alpha,\beta)+\dim\ker
i_{r-1}(a,b;\alpha,\beta)$

Proof of Propositions 4.1 and 4.2.

###### Proof.

In view the properties of the sets $S_{K,\cdots}$ and $\tilde{S}_{K,\cdots}$
it suffices to prove the statements for $K$ consisting of one single interval
and in view the tameness of $f$ one can suppose that $\theta_{1},\theta_{2}$
are critical angles and $a,b$ critical values.

For each $r$ choose a decomposition of $\rho_{r}$ in bar codes and Jordan
blocks, which implies decompositions of $T_{k,l}(\rho_{r})$s and choose a
linear splitting $s:\ker(M((\rho_{r-1})_{u})\to H_{r}(X;(\xi_{f},u))$ of $\pi$
in diagram (18).

We treat first the item (1.) in both propositions.

In view of the injectivity of $v_{r}$ and $v^{\prime}_{r},$ in diagrams (18)
and (19) in Proposition 3.1, the splitting $s$ provides by restriction the
compatible splittings

$s_{[\theta_{1},\theta_{4}]}:\ker(M((T_{\theta_{1},\theta_{4}}(\rho_{r-1}))\to
H_{r}(X;_{[\theta_{1},\theta_{4}]})$

and

$s_{[\theta_{2},\theta_{3}]}:\ker(M((T_{\theta_{2},\theta_{3}}(\rho_{r-1}))\to
H_{r}(X;_{[\theta_{2},\theta_{3}]}).$

This leads to the commutative diagram (27) with horizontal arrows isomorphisms

$\textstyle{{\operatorname{coker}M(T_{\theta_{2},\theta_{3}}(\rho_{r}))\oplus\ker
M(T_{\theta_{2},\theta_{3}}(\rho_{r-1}))}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v_{l}\oplus
v_{r}}$$\textstyle{H_{r}(X_{[\theta_{2},\theta_{3}]})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v}$$\textstyle{{\operatorname{coker}M(T_{\theta_{1},\theta_{4}}(\rho_{r}))\oplus\ker
M(T_{\theta_{1},\theta_{4}}(\rho_{r-1}))}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v^{\prime}_{l}\oplus
v^{\prime}_{r}}$$\textstyle{H_{r}(X_{[\theta_{1},\theta_{4}]})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v^{\prime}}$$\textstyle{\quad{\operatorname{coker}M((\rho_{r})_{u})\oplus\ker
M((\rho_{r-1})_{u})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{r}(X;(\xi_{f},u)).}$
(27)

Proposition 2.4 combined with Observation 2.1 gives the commutative diagram

$\textstyle{\kappa[S^{\prime}_{r,\theta_{1},\theta_{4}}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{coker}M(T_{\theta_{2},\theta_{3}}(\rho_{r}))\oplus\ker
M(T_{\theta_{2},\theta_{3}}(\rho_{r-1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v_{l}\oplus
v_{r}}$$\textstyle{\kappa[S^{\prime}_{r,\theta_{2},\theta_{3}}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{coker}M(T_{\theta_{1},\theta_{4}}(\rho_{r}))\oplus\ker
M(T_{\theta_{1},\theta_{4}}(\rho_{r-1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v^{\prime}_{l}\oplus
v^{\prime}_{r}}$$\textstyle{\kappa[S_{r,u}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{coker}M((\rho_{r})_{u})\oplus\ker
M((\rho_{r-1})_{u}).}$ (28)

The isomorphism $\omega_{u}$ (in Proposition 4.1) is the composition of
horizontal arrows in the last line of diagrams (27) (28) while the isomorphism
$\omega^{\prime}_{r,[\theta_{2},\theta_{3}]}$ and
$\omega^{\prime}_{r,[\theta_{1},\theta_{4}]}$ (in Proposition 4.2) are the
compositions of the horizontal arrows in the first and second lines of the
same diagrams. The isomorphisms $\omega_{r,[\theta_{2},\theta_{3}],u}$and
$\omega_{r,[\theta_{1},\theta_{4}],u}$ are restrictions of $\omega_{r,u}.$ The
commutativity of the diagrams claimed in Proposition 4.1 and 4.2 is the
consequence of the commutativity of the diagrams (27), (28). This establishes
item (1.) in both Propositions 4.1 and 4.2.

Item (2.) is verified essentially in the same way. More precisely:

The decompositions of $\rho_{r}^{\prime}s$ imply decompositions of
$\tilde{\rho}_{r}^{\prime}s$ and $T_{k,l}(\tilde{\rho}_{r})^{\prime}s.$
Observe that the commutative diagrams (27), (28) remain valid when we replace
$X$ by $\tilde{X}$, the representation $\rho_{r}$ by $\tilde{\rho}_{r},$ and
$\theta_{1},\theta_{2},\theta_{3},\theta_{4}$ by $a,b,c,d.$ In this case
$\tilde{\omega}$ is defined as $\omega_{u}$ was, namely as the composition of
the horizontal arrows of the last lines in the replaced diagrams (27), (28).

To check item (3.) in Proposition 4.1 one observes that
$\omega^{N}=\omega\otimes\kappa[T^{-1},T]].$

∎

Proof of Theorem 1.1.

###### Proof.

Item (1.) and item (3.) follow from Proposition 4.1 (3.) and (1.) To check
item (2.) we first observe that the sequence (17)

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{coker}M((\tilde{\rho}_{r}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{r}(\tilde{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\ker
M((\tilde{\rho}_{r-1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

is actually a sequence of $\kappa[T^{-1},T]-$modules where the multiplication
by $T$ on the first and third term in given by the $m-$shift described in the
end of Section 2.

Next we consider the diagram (29), whose horizontal arrows on the second line
are induced by inclusion and projection (cf. the definitions of the sets
$\tilde{S}_{r}(f)$ and $\widetilde{\mathcal{J}}_{r}(f)$). Observe that the
diagram is actually a commutative diagram of $\kappa[T^{-1},T]-$ modules, with
the module structure on the vector spaces located on the last two horizontal
lines of the diagram (29) as described in Observation 2.5.

$\textstyle{H_{r}(\tilde{X})\otimes_{\kappa[T^{-1},T]}\kappa[T^{-1},T]]}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{coker}M((\tilde{\rho}_{r}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{r}(\tilde{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ker
M(\tilde{\rho}_{r-1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\kappa[\mathcal{B}^{c}(\tilde{\rho}_{r})\sqcup\widetilde{\mathcal{J}(\rho_{r})}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi^{c}}$$\textstyle{\kappa[\tilde{S}_{r}(f)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\omega}_{r}}$$\textstyle{\kappa[\mathcal{B}^{o}(\tilde{\rho}_{r-1})]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi^{o}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\kappa[\widetilde{\mathcal{J}(\rho_{r})}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{\kappa[\widetilde{\mathcal{J}}_{r}(f)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
(29)

In view of Observation 2.5 the $\kappa[T^{-1},T]-$module
$\kappa[\widetilde{\mathcal{J}(\rho_{r})}]=\kappa[\widetilde{\mathcal{J}}_{r}(f)]$
is the $\kappa-$vector space $\bigoplus_{J\in\mathcal{J}_{r}}V(J)$ with the
multiplication by $T$ given by the linear isomorphism
$\bigoplus_{J\in\mathcal{J}_{r}}T(J).$ This is exactly the torsion of the
$\kappa[T^{-1},T]-$module $\kappa[\tilde{S}_{r}(f)]$ isomorphic to
$H_{r}(\tilde{X})$ hence $V(\xi_{f}).$ This verifies item (2.) and then
finishes the proof. ∎

##  5\. Stability for configurations $C_{r}(f).$ Proof of Theorem 1.2

The proof of Theorems 1.2 and 1.3 will require an alternative definition of
the configurations $C_{r}(f).$ This will be provided by the integer valued
functions $\delta_{r}^{f}$ which will be defined for an arbitrary real valued
tame map and then, via the infinite cyclic covering for an angle valued tame
map.

###  5.1. Real valued maps.

For $f:X\to\mathbb{R}$ a map and $a,b\in\mathbb{R}$ denote by:

  1. (i)

$X(a)=f^{-1}(a),$ $X^{f}_{a}=f^{-1}(-\infty,a]),\
X^{b}_{f}=f^{-1}([b,\infty)),$ $X^{b}_{a}=X^{a}\cap X_{b}$ and $i_{a}:X_{a}\to
X,\ i^{b}:X^{b}\to X$ the obvious inclusions,

  2. (ii)

$\mathbb{I}^{f}_{a}(r):=\operatorname{img}(i_{a}(r):H_{r}(X_{a})\to
H_{r}(X)),\ $
$\mathbb{I}^{b}_{f}(r):=\operatorname{img}(i^{b}(r):H_{r}(X^{b})\to
H_{r}(X)),$ and then

  3. (iii)

$F^{f}_{r}(a,b):=\dim(\mathbb{I}^{f}_{a}(r)\cap\mathbb{I}^{b}_{f}(r))$ and
$G^{f}_{r}(a,b):=\dim H_{r}(X)/(\mathbb{I}^{f}_{a}(r)+\mathbb{I}^{b}_{f}(r)).$

and observe that:

######  Observation 5.1.

1\. For $a\leq a^{\prime}$ $b^{\prime}\leq b$ $F^{f}_{r}(a,b)\leq
F^{f}_{r}(a^{\prime},b^{\prime})$ and $G^{f}_{r}(a,b)\geq
G^{f}(a^{\prime},b^{\prime})$

2\. If $|f-g|<\epsilon$ and $a\leq b$ then
$F^{f}_{r}(a-\epsilon,b+\epsilon)\leq F^{g}(a,b)$ and $G^{f}_{r}(a,b)\leq
G^{g}(a-\epsilon,b+\epsilon)$

3\. $F^{f}_{r}(a,b)=F^{-f}_{r}(-b,-a)$ and $G^{f}_{r}(a,b)=G^{-f}_{r}(-b,-a)$

To check (1.) notice that $X^{f}_{a}\subseteq X^{f}_{a^{\prime}}$ and
$X^{b^{\prime}}_{f}\supseteq X^{b}_{f}$ which imply
$\mathbb{I}^{f}_{a}\subseteq\mathbb{I}^{f}_{a^{\prime}}$ and
$\mathbb{I}^{b^{\prime}}_{f}\subseteq\mathbb{I}^{b}_{f}$ hence
$\mathbb{I}^{f}_{a}\cap\mathbb{I}^{b}_{f}\subseteq\mathbb{I}^{f}_{a^{\prime}}\cap\mathbb{I}_{f}^{b^{\prime}}$
and then the statement.

To check (2.) notice that $|f-g|<\epsilon$ implies $f-\epsilon<g<f+\epsilon$
which implies to $X^{f}_{a-\epsilon}\subseteq X^{g}_{a}$ and
$X^{f}_{b+\epsilon}\subseteq X^{g}_{b}.$ These inclusions imply
$\mathbb{I}^{f}_{a-\epsilon}\subseteq\mathbb{I}^{g}_{a}$ and
$\mathbb{I}^{b+\epsilon}_{f}\subseteq\mathbb{I}_{g}^{b}$ hence
$F^{f}(a-\epsilon,b+\epsilon)\leq F^{g}(a,b).$ The arguments for $G$ are
similar.

To check (3.) one uses the fact that
$f^{-1}((-\infty,a])=(-f)^{-1}([-a,\infty))$ q.e.d

If $X$ is a compact ANR it is immediate that both $F^{f}_{r}(a,b)$ and
$G^{f}_{r}(a,b)$ are finite since $\dim H_{r}(X)$ is finite. The same remains
true for $f:X\to\mathbb{R}$ a tame map with $X$ not compact but this statement
requires arguments since $\dim H_{r}(X)$ is not necessary finite. We have the
following:

######  Proposition 5.2.

For $f:X\to\mathbb{R}$ a tame map then:

1\. $F_{r}^{f}(a,b)<\infty,$

2\. $G^{f}_{r}(a,b)<\infty,$

3\. If $a\geq b$ then $F_{r}^{f}(a,b)=\operatorname{img}(H_{r}(X^{b}_{a})\to
H_{r}(X))$

###### Proof.

(1.) : In view of Observation 5.1 it suffices to check the statements for
$a>b.$ Consider

$i_{a}(r)-i^{b}(r):H_{r}(X_{a})\oplus H_{r}(X^{b})\to H_{r}(X)$

and

$i_{a}(r)+i^{b}(r):H_{r}(X_{a})\oplus H_{r}(X^{b})\to H_{r}(X)$

and observe that
$\mathbb{I}^{f}_{a}(r)\cap\mathbb{I}^{b}_{f}(r))=(i_{a}(r)+i^{b}(r))(\ker((i_{a}(r)-i^{b}(r))).$
Then

$\dim(\mathbb{I}^{f}_{a}(r)\cap\mathbb{I}^{b}_{f}(r))\leq\dim(\ker((i_{a}(r)-i^{b}(r))).$

Since $a\geq b$ we have $X=X_{a}\cup X^{b}.$ In view of the Mayer-Vietoris
long exact sequence associated with $X=X_{a}\cup X^{b}$

$\ker(i_{a}(r)-i^{b}(r))=\operatorname{img}(:H_{r}(X^{b}_{a})\to
H_{r}(X_{a})\oplus H_{r}(X^{b}))$

has finite dimension since $\dim H_{r}(X^{b}_{a})$ is finite.

( 2.): If $a<b$ one uses the exact sequence of the pair $(X,X_{a}\sqcup
X^{b})$ to conclude that
$H_{r}(X)/(\mathbb{I}^{f}_{a}(r)+\mathbb{I}^{b}_{f}(r))$ is isomorphic to a
subspace of $H_{r}(X,X_{a}\sqcup X^{b})=H_{r}(X^{b}_{a},X(a)\sqcup X(b))$
which is of finite dimension. Indeed $f$ tame implies $X(a),X(b)$ and
$X_{a}^{b},$ compact ANRs, hence with finite dimensional homology.

If $a\geq b$ one use the Mayer-Vietoris exact sequence associated with
$X_{a},X^{b}$ to conclude that
$H_{r}(X)/(\mathbb{I}^{f}_{a}(r)+\mathbb{I}^{b}_{f}(r))$ is isomorphic to a
subspace of $H_{r}(X^{b}_{a})$ which is of finite dimension. This long exact
sequence implies item (3.) as well.

∎

Let $a<b,c<d.$ We refer to the set

$B(a,b:c,d)=(a,b]\times[c,d)\subset\mathbb{R}^{2},\ a<b,\ c<d$

as a ”box ” and define

$\displaystyle\mu_{r}^{F,f}(B)=$ $\displaystyle
F_{r}^{f}(a,d)+F_{r}^{f}(b,c)-F_{r}^{f}(a,c)-F_{r}^{f}(b,d)$ (30)
$\displaystyle\mu_{r}^{G,f}(B)=$ $\displaystyle-
G_{r}^{f}(a,d)-G_{r}^{f}(b,c)+G_{r}^{f}(a,c)+G_{r}^{f}(b,d).$

One has

######  Proposition 5.3.

If $X$ is compact or $f$ is a tame map then:

1\. $\mu_{r}^{F,f}(B)=\mu_{r}^{G,f}(B).$

Let $\mu^{f}_{r}(B):=\mu_{r}^{F,f}(B)=\mu_{r}^{G,f}(B).$

2\. $\mu^{f}_{r}(B)$ is a nonnegative integer number.

3\. If $B=B_{1}\cup B_{2},B_{1}\cap B_{2}=\emptyset$ with $B_{1},B_{2},B_{3}$
boxes then $\mu^{f}(B)=\mu^{f}(B_{1})+\mu^{f}(B_{2}),$ in particular if the
$B^{\prime}$ and $B^{\prime\prime}$ are two boxes with $B^{\prime}\subset
B^{\prime\prime}$ one has $\mu^{f}(B^{\prime})\leq\mu^{f}(B^{\prime\prime}).$

###### Proof.

To ease the writing, we drop $f$ and $r$ from the notations involving
$\mathbb{I}$ and $f$ and introduce:

  1. (i)

$I_{1}:=\dim(\mathbb{I}_{a}\cap\mathbb{I}^{d})$

  2. (ii)

$I_{2}:=\dim(\mathbb{I}_{a}\cap\mathbb{I}^{c}/\mathbb{I}_{a}\cap\mathbb{I}^{d})$

  3. (iii)

$I_{3}:=\dim(\mathbb{I}_{b}\cap\mathbb{I}^{d}/\mathbb{I}_{a}\cap\mathbb{I}^{d})$

  4. (iv)

$I_{4}:=\dim(\mathbb{I}_{b}\cap\mathbb{I}^{c}/\mathbb{I}_{a}\cap\mathbb{I}^{c}+\mathbb{I}_{b}\cap\mathbb{I}^{d})$

  5. (v)

$I_{5}:=\dim\mathbb{I}_{b}/\mathbb{I}_{a}+\mathbb{I}_{b}\cap\mathbb{I}^{c}$

  6. (vi)

$I_{6}:=\dim\mathbb{I}^{c}/\mathbb{I}_{a}\cup\mathbb{I}^{d}+\mathbb{I}^{d}$

  7. (vii)

$I_{7}:=\dim H/\mathbb{I}_{b}+\mathbb{I}^{c}$ with $H=H_{r}(X).$

$H$$\mathbb{I}_{b}$$\mathbb{I}_{a}$$\mathbb{I}^{c}$$\mathbb{I}^{d}$

Using the picture above is not hard to notice that:

$F(a,d)=I_{1}$

$F(b,c)=(I_{1}+I_{2}+I_{3}+I_{4})$

$F(a,c)=(I_{1}+I_{2})$

$F(b,d)=(I_{1}+I_{3})$

and

$G(a,d)=(I_{7}+I_{6}+I_{5}+I_{4})$

$G(b,c)=I_{7}$

$G(a,c)=(I_{7}+I_{5})$

$G(b,d)=(I_{7}+I_{6})$

Then we have:

$F(a,d)+F(b,c)-F(a,c)-F(b,d)=I_{1}+(I_{1}+I_{2}+I_{3}+I_{4})-(I_{1}+I_{2})-(I_{1}+I_{3})=I_{4}$
and

$G(a,d)+G(b,c)-G(a,c)-G(b,d)=(I_{7}+I_{6}+I_{5}+I_{4})+I_{7}-(I_{7}+I_{5})-(I_{7}+I_{6})=I_{4}.$
These equalities establish items (1.) and (2.). Item (3.) follows from
definition by inspecting the relative positions of $B_{1}$ and $B_{2}.$

∎

Define the jump function

$\delta^{f}_{r}(a,b):=\lim_{\epsilon\to
0}\mu^{f}((a-\epsilon,a+\epsilon]\times[b-\epsilon,b+\epsilon)),$ (31)

The limit exists since by Proposition 5.3 the right side decreases when
$\epsilon$ decreases. This function has values in $\mathbb{Z}_{\geq 0},$ since
the critical values of a tame map are discrete, has discrete support and
satisfies the following proposition.

######  Proposition 5.4.

If $X$ compact or $f$ is a tame map then:

1\. For $a<b,c<d$ one has $\mu^{f}_{r}((a,b]\times[c,d))=\sum_{a<x\leq b,c\leq
y<d}\delta^{f}_{r}(x,y),$

2\. $F^{f}_{r}(b,c)=\sum_{-\infty,x\leq b;c\leq y,\infty}\delta^{f}_{r}(x,y),$

3\. $G^{f}_{r}(a,d)=\sum_{a\leq x<\infty;-\infty<y\leq c}\delta^{f}_{r}(x,y).$

###### Proof.

Item (1.) follows from Proposition 5.3 ( 3.)

Item (2.) follows from item (1.) by making $a$ goes to $-\infty$ and $d$ to
$\infty$ and item (3.) follows from item (1.) by making $b$ goes to $\infty$
and $c$ to $-\infty.$ ∎

For a tame map $f$ the set of critical values is discrete so they can be
written as $\cdots c_{i}<c_{i+1}<\cdots.$ Define

$\epsilon(f)=\inf_{i\in\mathbb{Z}}(c_{i+1}-c_{i}).$

Clearly if $f:X\to\mathbb{R}$ is tame with $X$ compact then $\epsilon(f)>0$
and if $f:X\to\mathbb{S}^{1}$ is tame then the infinite cyclic covering
$\tilde{f}:\tilde{X}\to\mathbb{R}$ is tame and $\epsilon(\tilde{f})>0.$

######  Proposition 5.5.

Let $f:X\to\mathbb{R}$ be a tame map with $\epsilon(f)>0.$

1\. For any $\epsilon,\epsilon^{\prime}<\epsilon(f)$ one has:

$F^{f}_{r}(c_{i},c_{j})=F^{f}_{r}(c_{i}+\epsilon,c_{j}-\epsilon^{\prime})=F^{f}_{r}(c_{i+1}-\epsilon,c_{j-1}+\epsilon^{\prime}),$

2\.
$\delta^{f}_{r}(c_{i},c_{j})=F^{f}_{r}(c_{i-1},c_{j+1})+F^{f}_{r}(c_{i},c_{j})-F^{f}_{r}(c_{i-},c_{j})-F^{f}_{r}(c_{i},c_{j+1}).$

###### Proof.

The tameness of $f$ and of the hypothesis the inclusions
$X^{f}_{c_{i}}\subseteq X^{f}_{c_{i}+\epsilon},$ $X^{f}_{c_{i}}\subseteq
X^{f}_{c_{i+1}-\epsilon^{\prime}}$ and $X_{f}^{c_{j}-\epsilon}\supseteq
X_{f}^{c_{j}},$ $X_{f}^{c_{j-1}+\epsilon^{\prime}}\supseteq X_{f}^{c_{j}}$
induce isomorphisms in homology. These facts imply that
$\mathbb{I}^{f}_{c_{i}}=\mathbb{I}^{f}_{c_{i}+\epsilon}=\mathbb{I}^{f}_{c_{i+1}-\epsilon^{\prime}}$
and
$\mathbb{I}_{f}^{c_{j-1}+\epsilon}=\mathbb{I}_{f}^{c_{j}-\epsilon^{\prime}}=\mathbb{I}_{f}^{c_{j}}$
which imply item (1.). To check item (2.) recall that in view of the
definition, for $\epsilon$ very small, one has
$\delta^{f}(c_{i},c_{j})=F(c_{i}-\epsilon,c_{j}+\epsilon)+F(c_{i}+\epsilon,c_{j}-\epsilon)-F(c_{i}-\epsilon,c_{j}-\epsilon)-F(c_{i}+\epsilon,c_{j}+\epsilon).$
Item (2.) follows then from item (1.) by taking $\epsilon<\epsilon(f).$

∎

For a pair $(a,b)\in\mathbb{R}^{2}$ and $\epsilon>0$ consider the box
$B(a,b;2\epsilon)=(a-2\epsilon,a+2\epsilon]\times[b-2\epsilon,b+2\epsilon).$

######  Proposition 5.6.

Let $f:X\to\mathbb{R}$ be a tame map. For any $\epsilon<\epsilon(f)/6,$ $g$ an
tame map with $|f-g|<\epsilon$ and $(a,b)\in\rm{supp}\ \delta^{f}_{r}$ one
has:

1\. $\rm{supp}\ \delta^{f}_{r}\cap B(a,b;2\epsilon)\equiv(a,b)$

2\. $\sharp(\rm{supp}\ \delta^{g}\cap(\sqcup_{(a,b)\in\rm{supp}\
\delta^{f}}B(a,b;2\epsilon)))=\sharp(\rm{supp}\ \delta^{f}_{r}).$

In particular if the cardinality of the supports888recall that the cardinality
of the support is the sum of multiplicity of the elements in the support of
$\delta^{f}_{r}$ and $\delta^{g}_{r}$ are equal and $|g-f]<\epsilon,$ then the
support of $\delta^{g}_{r}$ lies in an $\epsilon-$neighborhood 999 here
$\epsilon-$neighborhood of $(a,b)$ means the domain
$(a-\epsilon,a+\epsilon)\times(b-\epsilon,b+\epsilon)$ of the support of
$\delta^{f}_{r}.$

###### Proof.

To simplify the writing the index $r$ will be omitted from the notation.

Item (1.) follows from definition of $\delta^{f}.$

To prove item (2.) observe that if $(a,b)\in\rm{supp}\ \delta^{f}$ both
numbers have to be critical values, hence the $a=c_{i},b=c_{j}.$ In view of
Proposition 5.5, for any
$\epsilon^{\prime},\epsilon^{\prime\prime}<\epsilon(f)/2$ one has

$\displaystyle F^{f}(c_{i-1},c_{j+1})=$ $\displaystyle
F^{f}(a-\epsilon^{\prime},b+\epsilon^{\prime\prime})$ (32) $\displaystyle
F^{f}(c_{i},c_{j})=$ $\displaystyle
F^{f}(a+\epsilon^{\prime},b-\epsilon^{\prime\prime})$ $\displaystyle
F^{f}(c_{i},c_{j+1})=$ $\displaystyle
F^{f}(a+\epsilon^{\prime},b+\epsilon^{\prime\prime})$ $\displaystyle
F^{f}(c_{i-1},c_{j})=$ $\displaystyle
F^{f}(a-\epsilon^{\prime},b-\epsilon^{\prime\prime}).$

Since $|f-g|<\epsilon,$ in view of Observation 5.1 one has

$\displaystyle F^{f}(a-3\epsilon,b+3\epsilon)\leq$ $\displaystyle\
F^{g}(a-2\epsilon,b+2\epsilon)\leq F^{f}(a-\epsilon,b+\epsilon),$ (33)
$\displaystyle F^{f}(a+\epsilon,b-\epsilon)\leq$ $\displaystyle
F^{g}(a+2\epsilon,b-2\epsilon)\leq F^{f}(a+3\epsilon,b-3\epsilon),$
$\displaystyle F^{f}(a+\epsilon,b+3\epsilon)\leq$ $\displaystyle
F^{g}(a+2\epsilon,b+2\epsilon)\leq F^{f}(a+3\epsilon,b+\epsilon),$
$\displaystyle F^{f}(a-3\epsilon,b-\epsilon)\leq$ $\displaystyle
F^{g}(a-2\epsilon,b-2\epsilon)\leq F^{f}(a-\epsilon,b-3\epsilon).$

Since $\epsilon<\epsilon(f)/6,$ (32) and (33) imply that

$\displaystyle F^{g}(a-2\epsilon,b+2\epsilon)=F^{f}(c_{i-1},c_{j+1})$ (34)
$\displaystyle F^{g}(a+2\epsilon,b-2\epsilon)=F^{f}(c_{i},c_{j})$
$\displaystyle F^{g}(a+2\epsilon,b+2\epsilon)=F^{f}(c_{i},c_{j+1})$
$\displaystyle F^{g}(a-2\epsilon,b-2\epsilon)=F^{f}(c_{i-1},c_{j}).$

In view of Proposition 5.4

$\displaystyle\sharp(\rm{supp}\ \delta^{g}\cap
B(a,b:2\epsilon))=\mu^{g}(B(a,b:2\epsilon))=$ $\displaystyle
F^{g}(a-2\epsilon,b+2\epsilon)+F^{g}(a+2\epsilon,b-2\epsilon)$
$\displaystyle-F^{g}(a-2\epsilon,b-2\epsilon)-F^{g}(a+2\epsilon,b+2\epsilon)$

which in view of (33) and (34) and Proposition 5.5 (2.) leads to

$\sharp(\rm{supp}\ \delta^{g}\cap B(a,b:2\epsilon))=\sharp(\rm{supp}\
\delta^{f}\cap B(a,b:2\epsilon))=\delta^{f}(a,b).$

∎

###  5.2. Angle valued maps.

Let $f:X\to\mathbb{S}^{1}$ be a tame map and
$\tilde{f}:\tilde{X}\to\mathbb{R}$ its infinite cyclic covering. Recall that
$\epsilon(\tilde{f})>0$ and observe that

$\delta^{\tilde{f}}_{r}(a,b)=\delta^{\tilde{f}}_{r}(a+2\pi,b+2\pi).$ (35)

Consider the projection Let
$p:\mathbb{R}^{2}\to\mathbb{T}=\mathbb{R}^{2}/\mathbb{Z},$ with $\mathbb{T}$
the quotient space of $\mathbb{R}^{2}$ by the action
$\mu:\mathbb{Z}\times\mathbb{R}^{2}\to\mathbb{R}^{2}$ given by
$\mu(n,(a,b))=(a+2\pi n,b+2\pi n).$

Define

$\epsilon(f):=\epsilon(\tilde{f})$

and

$\delta^{f}_{r}(p(a,b)):=\delta^{\tilde{f}}_{r}(a,b).$ (36)

In view of (35) $\delta^{f}_{r}$ is a well defined integer valued function
with finite support. and Proposition 5.6 holds for $f:X\to\mathbb{S}^{1}$ with
exactly the same conclusion.

Proposition 5.6 equally implies that the cardinality of the support of
$\delta^{g}_{r}$ with $g$ closed enough to $f$ in $C^{0}$ topology is larger
or equal to the cardinality of the support of $\delta^{f}_{r}$ and therefore
the cardinality of the supports of tame maps in the same connected components
is constant, a fact we already knew by Theorem 1.1.

For the proof of Theorem 1.2 we also need to show that $\delta^{f}_{r}$ and
$C_{r}(f)$ when viewed as functions on $\mathbb{T}$ are equal.

######  Proposition 5.7.

If $f$ is a tame real or angle valued map defined on $X,$ a compact ANR, then
$\delta^{f}_{r}$ and $C_{r}(f)$ are equal as functions.

###### Proof.

We check the case of an angle valued map $f:X\to\mathbb{S}^{1}$ only. The real
valued case can be regarded as a particular case of this one.First note that
$\epsilon(f)>0.$ In view of the definition of $\delta^{\tilde{f}}_{r}$ it
suffices to check that:

  1. (i)

If at least one, $a$ or $b,$ is not a critical value then we have
$\delta^{\tilde{f}}_{r}(a,b)=0.$

  2. (ii)

If $a=c_{i}$ $b=c_{j}$ are critical value with $c_{i}\geq c_{j}$

$\delta^{\tilde{f}}_{r}(c_{i},c_{j})=\sharp\\{I\in\tilde{\mathcal{B}}^{c}_{r}(f)\mid
I=[c_{j},c_{i}]\\}.$

  3. (iii)

If $a=c_{i}$ $b=c_{j}$ are critical value with $c_{i}<c_{j}$

$\delta^{\tilde{f}}_{r}(c_{i},c_{j})=\sharp\\{I\in\tilde{\mathcal{B}}^{o}_{r-1}(f)\mid
I=(c_{j},c_{j})\\}.$

Recall that $\delta_{r}(a,b):=\lim_{\epsilon\to
0}(-F_{r}(a-\epsilon,b-\epsilon)-F_{r}(a+\epsilon,b+\epsilon)+F_{r}(a-\epsilon,b+\epsilon)+F_{r}(a+\epsilon,b-\epsilon).$

In view of Proposition 5.5 if $a$ is not critical value, for $\epsilon$
sufficiently small
$F^{\tilde{f}}_{r}(a-\epsilon,\cdots)=F^{\tilde{f}}_{r}(a+\epsilon,\cdots)$
which implies $\delta^{\tilde{f}}_{r}(a,\cdots)=0,$ and if $b$ is not critical
value for $\epsilon$ sufficiently small
$F^{\tilde{f}}_{r}(\cdots,b-\epsilon)=F^{\tilde{f}}_{r}\cdots,b+\epsilon)$
which implies $\delta^{\tilde{f}}_{r}(\cdots,b)=0.$ This establishes statement
(i)

Suppose that $a=c_{i}$ and $b=c_{j}$ critical values. In view of Proposition
5.5 and of the definition of $\delta^{\tilde{f}}$ one obtains

$\delta^{\tilde{f}}_{r}(c_{i},c_{j})=-F^{\tilde{f}}_{r}(c_{i-1},c_{j})-F^{\tilde{f}}_{r}(c_{i},c_{j+1})+F^{\tilde{f}}_{r}(c_{i-1},c_{j+1})+F^{\tilde{f}}_{r}(c_{i},c_{j})$
(37)

By Propositions 5.2 and 4.3, when $c_{i}\geq c_{j},$ one has

$F^{\tilde{f}}_{r}(c_{i},c_{j})=\sharp\begin{cases}\begin{aligned}
\\{I\in&\tilde{\mathcal{B}}_{r}^{c}(f)\mid I\cap[c_{j},c_{i}]\neq\emptyset\\}\
\sqcup\\\ \\{I\in&\tilde{\mathcal{B}}^{o}_{r-1}(f)\mid
I\subset(c_{j},c_{i})\sqcup\\\
&\tilde{\mathcal{J}}_{r}(f)\end{aligned}\end{cases}$ (38)

and when $c_{i}>c_{j},$ in view of Proposition 4.1 one has

$F^{\tilde{f}}_{r}(c_{i},c_{j})=\sharp\begin{cases}\begin{aligned}
\\{I\in&\tilde{\mathcal{B}}_{r}^{c}(f)\mid I\supset[c_{i},c_{j}]\ \sqcup\\\
&\tilde{\mathcal{J}}_{r}(f)\end{aligned}\end{cases}$ (39)

Comparing the collections of bar codes whose cardinality are given by
$F^{\tilde{f}}_{r}(c_{i-1},c_{j}),$ $F^{\tilde{f}}_{r}(c_{i},c_{j+1}),$
$F^{\tilde{f}}_{r}(c_{i-1},c_{j+1})$ and $F^{\tilde{f}}_{r}(c_{i},c_{j})$ and
using (37) and (38) one derives the statement ii), and using (37) and 39) one
derives the statement iii).

∎

###  5.3. Proof of Theorem 1.2.

We begin with a few observations.

  1. (i)

Consider the space of continuous maps $C(X,\mathbb{S}^{1}),$ $X$ a compact
ANR, with the compact open topology. This topology is induced from the metric
$D(f,g):=\sup_{x\in\mathbb{X}}d(f(x),g(x)),$ with
${}^{\prime\prime}d^{\prime\prime}$ the geodesic distance on $\mathbb{S}^{1}$
given by
$d(\theta_{1},\theta_{2})=\inf(|\theta_{1}-\theta_{2}|,2\pi-|\theta_{1}-\theta_{2}|),$
$0\leq\theta_{1},\theta_{2}<2\pi.$ With this metric $(C(X,\mathbb{S}^{1}),D)$
is complete.

  2. (ii)

Consider
$S^{N}\mathbb{T}=(\mathbb{T}\times\mathbb{T}\cdots\mathbb{T})/\Sigma_{N},$
with $\Sigma_{N}$ is the $N-$symmetric group acting on the $N-$fold cartesian
product of $\mathbb{T}$ by permutations equipped equipped with the induced
metric $\underline{D}$ induced from the complete metric on
$\mathbb{T}/\mathbb{Z}.$ With this metric $(S^{N}(\mathbb{T}),\underline{D})$
is complete.

  3. (iii)

Observe that if $f,g$ are in a connected component $C_{\xi}(X,\mathbb{S}^{1})$
of $C(X,\mathbb{S}^{1})$ and $D(f,g)<\pi$ then for any $t\in[0,1]$ the map
$h_{t}:=h_{t}(f,g)\in C(X;\mathbb{S}^{1})$ defined by the formulae

$h_{t}(x)=\begin{cases}tf(x)+(1-t)g(x)\ \rm{if}\ 0\leq g(x),f(x)<2\pi,f(x)\leq
g(x)\\\ (1-t)f(x)+tg(x)\ \rm{if}\ 0\leq g(x),f(x)<2\pi,f(x)\geq
g(x)\end{cases}$

is continuous and lies in the connected component of
$C_{\xi}(X,\mathbb{S}^{1})$ and for any $0=t_{0}<t_{1}\cdots t_{N-1}<t_{N}=1$
one has

$D(f,g)=\sum_{0\leq i<N}D(h_{t_{i+1}},h_{t_{i}}).$ (40)

  4. (iv)

If $X$ is a simplicial complex and $\mathcal{U}\subset
C_{\xi}(X,\mathbb{S}^{1})$ denotes the subset of p.l-maps then:

1\. $\mathcal{U}$ is a dense subset

2\. $f,g\in\mathcal{U}$ implies $h_{t}\in\mathcal{U}$ hence
$\epsilon(h_{t})>0$ hence for any $t\in[0,1]$ there exists $\delta(t)>0$ so
that $|t^{\prime}-t|<\delta(t)$ implies
$D(h_{t^{\prime}},h_{t})<\epsilon(h_{t})/6.$

Recall that $f$ is p.l on $X$ if with respect to some subdivision is
simplicial (i.e. the liftings to $\mathbb{R}$ of the restriction of $f$ to
simplexes are linear) and for any two p.l maps $f,g$ there exists a common
subdivision of $X$ which makes $f$ and $g$ simultaneously simplicial, hence
any $h_{t}$ is a simplicial map. Item (1.) follows from approximability of
continuous maps by p.l maps and item (2.) from the continuity in $t$ of the
family $h_{t}$ and of the compacity of $X.$

  5. (v)

Proposition 5.6 states that $f,g\in C(X,\mathbb{S}^{1})_{t,\xi}$ and
$D(f,g)<\epsilon(f)/6$ implies

$\underline{D}(\delta^{f}_{r},\delta^{g}_{r})<2D(f,g).$ (41)

The above observations combined imply Theorem 1.2. Indeed, Item (v.) makes
$\delta:C(X;\mathbb{S}^{1})_{t,\xi}\to S^{N}(\mathbb{T})$ a continuous map and
establishes the continuity of the assignment $C(X,\mathbb{S}^{1})_{t,\xi}\ni
f\to\delta^{f}_{r}\in S^{N}(\mathbb{T})$ $N=\beta^{N}_{r}(X,\xi).$ To conclude
the existence of a continuous extension of $\delta_{r}$ to the entire
$C(X,\mathbb{S}^{1}),$ in view of item (i) and (ii) and (iv), it suffices to
show that for a Cauchy sequence $\\{f_{n}\\},$ $f_{n}\in\mathcal{U},$
$\delta_{r}^{f_{n}}$ is a Cauchy sequence in $S^{N}(\mathbb{T}).$ This will
follow once we can show that for any two $f,g\in\mathcal{U}$ with $d(f,g)<\pi$
we have $\underline{D}(\delta_{r}^{f},\delta^{g})\leq 2D(f,g).$ To establish
this last fact we proceed as in [5].

Start with $f,g\in\mathcal{U}$ with $D(f,g)<\pi$ and consider $h_{t},$
$t\in[o,1]$ defined above.

Choose a sequence $0=t_{0}<t_{2}<t_{4},\cdots t_{2N-2}<t_{2N}=1$ so that the
open intervals $I_{2i}=(t_{2i}-\delta(t_{2i}),t_{2i}+\delta(t_{2i}))$ cover
$[0,1].$ The compacity of $[0,1]$ makes this possible.

By possibly removing some of the points $t_{2i}$ and decreasing
$\delta(t_{2i})$ one can make $I_{2i}\cap I_{2i+2}\neq\emptyset$ and
$t_{2t_{-}2},t_{2i+2}\notin I_{2i}.$ Choose $t_{1}<t_{3}<\cdots t_{2N-1}$ with
$t_{2i}<t_{2i+1}<t_{2i}$ and $t_{2i+1}\in I_{2i}\cap I_{2i+2}.$ We have then
$|t_{2i+1}-t_{2i}|<\delta(t_{2i})$ and $|t_{2i+2}-t_{2i+1}|<\delta(t_{2i+2}).$

In view of item (iv) $|t_{2i+1}-t_{2i}|<\delta(t_{2i})$ implies
$D(h_{t_{2i}},h_{t_{2i+1}})<\epsilon(h_{t_{2i}})/6$ and
$|t_{2i+2}-t_{2i+1}|<\delta(t_{2i+2})$ implies
$D(h_{t_{2i+2}},h_{t_{2i+1}})<\epsilon(h_{t_{2i+2}})/6.$ In view of item (v)
the last inequalities imply
$\underline{D}(\delta^{h_{t_{2i+1}}}_{r},\delta^{h_{t_{2i}}}_{r})<2D(h_{t_{2i}},h_{t_{2i+1}})$
and
$\underline{D}(\delta^{h_{t_{2i+2}}}_{r},\delta^{h_{t_{2i+1}}}_{r})<2D(h_{t_{2i+2}},h_{t_{2i+1}}).$
Therefore, for any $0\leq k\leq{2N-1}$ one has
$\underline{D}(\delta^{h_{t_{k+1}}}_{r},\delta^{h_{t_{k}}}_{r})<2D(h_{t_{k+1}},h_{t_{k}}).$
Then

$\underline{D}(\delta^{f},\delta^{g})\leq\sum_{0\leq
i<2N-1}D(\delta^{h}(t_{i+1},\delta^{h}(t_{i}))\leq 2\sum_{0\leq
i<2N-1}D(h_{t_{i+1}},h_{t_{i}}).$

which by item (iii) is exactly $D(d,g).$

This finishes the proof of Theorem 1.2.

##  6\. Poincaré duality for configurations $C_{r}(f).$ Proof of Theorem 1.3

For an $n-$dimensional manifold $Y,$ not necessary compact, Poincaré Duality
can be better formulated using Borel–Moore homology, cf. [3], especially
tailored for locally compact spaces $Y$ and pairs $(Y,K)$, $K$ closed subset
of $Y.$ Borel Moore homology coincides with the standard homology when $Y$ is
compact. In general, for a locally compact space $Y$ can be described as the
inverse limit of the homology $H_{r}(Y,Y\setminus U)$ for all $U$ open sets
with compact closure. One denotes the Borel–Moore homology in dimension $r$ by
$H^{BM}_{r}.$ For $Y$ a $n-$dimensional topological $\kappa-$orientable
manifold, $g:Y\to\mathbb{R}$ a tame map and $a$ a regular value of
$g,$101010i.e. $f:f^{-1}(a-\epsilon,a+\epsilon)\to(a-\epsilon,a+\epsilon)$ is
a fibration Poincaré Duality provides the commutative diagrams

---
$\textstyle{H^{BM}_{r}(Y_{a})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\quad}$$\textstyle{H^{BM}_{r}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{BM}_{r}(Y,Y_{a})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{n-r}(Y,Y^{a})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\quad}$$\textstyle{H^{n-r}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{n-r}(Y^{a})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(H_{n-r}(Y,Y^{a}))^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\quad}$$\textstyle{(H_{n-r}(Y))^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(H_{n-r}(Y^{a}))^{\ast}}$.
(42)

---
$\textstyle{H^{BM}_{r}(Y^{a})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\quad}$$\textstyle{H^{BM}_{r}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{BM}_{r}(Y,Y^{a})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{n-r}(Y,Y_{a})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\quad}$$\textstyle{H^{n-r}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{n-r}(Y_{a})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(H_{n-r}(Y,Y_{a}))^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\quad}$$\textstyle{(H_{n-r}(Y))^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(H_{n-r}(Y_{a}))^{\ast}}$.
(43)

The first vertical arrow in each column of both diagrams is the Poincaré
Duality isomorphism, the second is the the isomorphism between cohomology and
the dual of homology with coefficients in a field. The horizontal arrows are
induced by the inclusions of $Y_{a}$ or $Y^{a}$ in $Y$ and the inclusion of
pairs $(Y,\emptyset)$ in $(Y,Y_{a})$ or $(Y,Y^{a}).$

We apply diagrams (42) and (43) to $Y=\tilde{M}^{n}$ and $g=\tilde{f},$ with
$\tilde{f}:\tilde{M}\to\mathbb{R}$ the infinite cyclic covering of
$f:M^{n}\to\mathbb{S}^{1},$ a tame map defined on a closed $\kappa-$
orientable topological manifold and obtain

$\textstyle{H^{BM}_{r}(\tilde{M}_{a})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\quad}$$\scriptstyle{i_{a}(r)}$$\textstyle{H^{BM}_{r}(\tilde{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{a}(r)}$$\textstyle{H^{BM}_{r}(\tilde{M},\tilde{M}_{a})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{n-r}(\tilde{M},\tilde{M}^{a})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\quad}$$\scriptstyle{s^{a}(n-r)}$$\textstyle{H^{n-r}(\tilde{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r^{a}(n-r))}$$\textstyle{H^{n-r}(\tilde{M}^{a})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(H_{n-r}(\tilde{M},\tilde{M}^{a}))^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\quad}$$\scriptstyle{(j^{a}(n-r))^{\ast}}$$\textstyle{(H_{n-r}(\tilde{M}))^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(i^{a}(n-r))^{\ast}}$$\textstyle{(H_{n-r}(\tilde{M}^{a}))^{\ast}}$
(44)

---
$\textstyle{H^{BM}_{r}(\tilde{M}^{b})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\quad}$$\scriptstyle{i^{b}(r)}$$\textstyle{H^{BM}_{r}(\tilde{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{b}(r)}$$\textstyle{H^{BM}_{r}(\tilde{M},\tilde{M}^{b})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{n-r}(\tilde{M},\tilde{M}_{b})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\quad}$$\scriptstyle{(s_{b}(n-r))^{\ast}}$$\textstyle{H^{n-r}(\tilde{M}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{b}(n-r)}$$\textstyle{H^{n-r}(\tilde{M}_{b})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(H_{n-r}(\tilde{M},\tilde{M}_{b}))^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\quad}$$\scriptstyle{(j_{b}(n-r))^{\ast}}$$\textstyle{(H_{n-r}(\tilde{M}))^{\ast}\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(i_{b}(n-r))^{\ast}}$$\textstyle{(H_{n-r}(\tilde{M}_{b}))^{\ast}}$.
(45)

For $\tilde{M},\tilde{M}_{a},\tilde{M}^{a}$ the Borel–Moore homology can be
described as the following inverse limits :

$\displaystyle H^{BM}_{r}(\tilde{M})=$ $\displaystyle\varprojlim_{\tiny
0<l\to\infty}H_{r}(\tilde{M},\tilde{M}_{-l}\sqcup\tilde{M}^{l}),$ (46)
$\displaystyle H^{BM}_{r}(\tilde{M}_{a})=$ $\displaystyle\varprojlim_{\tiny
0<l\to\infty}H_{r}(\tilde{M}_{a},\tilde{M}_{a-l}),$ $\displaystyle
H^{BM}_{r}(\tilde{M}^{a})=$
$\displaystyle\varprojlim_{0<l\to\infty}H_{r}(\tilde{M},\tilde{M}^{a+l}),$
$\displaystyle H^{BM}_{r}(\tilde{M},\tilde{M}_{a})=$
$\displaystyle\varprojlim_{0<l\to\infty}H_{r}(\tilde{M},\tilde{M}_{a}\sqcup\tilde{M}^{a+l}),$
$\displaystyle H^{BM}_{r}(\tilde{M},\tilde{M}^{a})=$
$\displaystyle\varprojlim_{0<l\to\infty}H_{r}(\tilde{M},\tilde{M}^{a}\sqcup\tilde{M}_{a-l}).$

The inclusion of pairs
$(\tilde{M},\tilde{M}_{-l^{\prime}}\sqcup\tilde{M}^{l^{\prime}})\subseteq(\tilde{M},\tilde{M}_{-l}\sqcup\tilde{M}^{l})$
for $l^{\prime}>l$ induces in homology an invers system whose limit is
$H^{BM}_{r}(\tilde{M}).$ Similar inclusions of pairs associated with
$l^{\prime}>l$ induce inverse systems whose limits are the remaining
Borel–Moore homology vector spaces considered above.

The horizontal arrows in both diagrams are inclusion (possibly of pairs)
induced linear maps in homology when denoted by $i(\cdots)$s and $j(\cdots)$s
or cohomology when denoted by $r(\cdots)$s and $s(\cdots)$s .

In view of the above involvement of Borel–Moore homology, in addition to
$\mathbb{I}^{\tilde{f}}_{a}(r)$ and $\mathbb{I}_{\tilde{f}}^{a}(r),$ consider

$\mathbb{I}^{BM,\tilde{f}}_{a}(r)=\operatorname{img}(H^{BM}_{r}(\tilde{X}_{a})\to
H^{BM}_{r}(\tilde{X})),$
$\mathbb{I}^{BM,a}_{\tilde{f}}(r)=\operatorname{img}(H^{BM}_{r}(\tilde{X}^{a})\to
H^{BM}_{r}(\tilde{X})),$

and
$F^{BM,f}_{r}(a,b)=\dim(\mathbb{I}^{BM,\tilde{f}}_{a}(r)\cap\mathbb{I}^{BM,b}_{\tilde{f}}(r)).$

Note that the exact sequences in Borel–Moore homology of the pairs
$(\tilde{M},\tilde{M}_{a})$ or $(\tilde{M},\tilde{M}^{b}),$ the top lines of
the two diagrams, give

$F^{BM,\tilde{f}}(a,b)=\mathbb{I}^{BM,\tilde{f}}_{a}(r)\cap\mathbb{I}^{BM,b}_{\tilde{f}}(r)=\ker(j^{BM}_{a}(r),j^{BM,b}(r)).$
(47)

Looking to the right side corners of the diagrams (44) and (45) one concludes
that

$\ker(j^{BM}_{a}(r),j^{BM,b}(r))\equiv\ker(r^{a}(n-r),r_{b}(n-r)).$ (48)

In view of the canonical isomorphism between cohomology the dual of homology
one obtains

$\ker((r^{a}(n-r),r_{b}(n-r))\equiv(\operatorname{coker}(i^{a}(n-r)+i_{b}(n-r)))^{\ast}.$
(49)

In view of the definition and of the finite dimensionality of
$G^{\tilde{f}}(a,b)$ one obtains

$G^{\tilde{f}}_{n-r}(b,a):=\dim(\operatorname{coker}(i_{b}(n-r)+i^{a}(n-r))=\dim(\operatorname{coker}(i_{b}(n-r)+i^{a}(n-r)))^{\ast}.$
(50)

Note also that

$G^{\tilde{f}}(a,b)=G^{-\tilde{f}}(-b,-a).$ (51)

Consequently $F^{BM,\tilde{f}}_{r}(a,b)=G_{n-r}^{-\tilde{f}}(-a,-b).$

In order to conclude that

$\delta^{\tilde{f}}_{r}(a,b)=\delta^{-\tilde{f}}_{n-r}(-a,-b).$ (52)

it suffices to show that the function $\delta^{BM,\tilde{f}}_{r}$ calculated
from $F^{BM,\tilde{f}}_{r}$ using (31) is the same as the function
$\delta^{\tilde{f}}_{r}.$ If so we obtain

$\delta^{f}_{r}(z)=\delta^{\overline{f}}_{n-r}(z^{-1})$ (53)

for $z=e^{ia+(b-a)},$which establishes Theorem 1.3.

For this purpose we need the following proposition.

######  Proposition 6.1.

$F^{BM,\tilde{f}}_{r}(a,b)+\sharp\tilde{\mathcal{J}}_{r}(f)=F_{r}^{\tilde{f}}(a,b)$
with $\sharp$ meaning ” cardinality”.

Proposition 6.1 is proved by comparing $F^{BM,\tilde{f}}_{r}(a,b)$ and
$F_{r}^{\tilde{f}}(a,b)$ calculated in terms of number of bar codes with the
help of Propositions 4.1 and 4.2.

The final outcome of the calculation can be summarized as follows:
$F^{BM,\tilde{f}}_{r}(a,b)=\sharp S^{\prime}$ and
$F_{r}^{\tilde{f}}(a,b)=\sharp\\{S^{\prime}\sqcup S^{\prime\prime}\\}$ where

when $a\leq b$ $S^{\prime}=\\{I\in\tilde{\mathcal{B}}^{c}_{r}\mid
I\supseteq[a,b]\\}$ and

when $a>b$ $S^{\prime}=\\{I\in\tilde{\mathcal{B}}^{c}_{r}\mid
I\cap[b,a]\neq\emptyset\\}\sqcup\\{I\in\mathcal{B}^{o}_{r-1}\mid
I\subset(b,a]\\}$

and for any $a,b\in\mathbb{R},$ $S^{\prime\prime}=\tilde{\mathcal{J}}_{r}.$

$F^{\tilde{f}}_{r}(a,b)$ can be read off from Proposition 4.1 directly. To
calculate $F^{BM,\tilde{f}}_{r}(a,b)$ one has to describe
$H_{r}^{BM}(\tilde{X}_{a})\to H^{BM}_{r}(\tilde{X})$ and
$H_{r}^{BM}(\tilde{X}^{b})\to H^{BM}_{r}(\tilde{X}).$

Recall that for an interval $I$ we denote by
$\tilde{X}_{I}:=\tilde{f}^{-1}(I).$

Notice that the long exact sequence of the pair
$(\tilde{X},\tilde{X}\setminus\tilde{X}_{(-a,a)})$ and the inclusion of pairs
$(\tilde{X},\tilde{X}\setminus\tilde{X}_{(-a^{\prime},a^{\prime})})\subset(\tilde{X},\tilde{X}\setminus\tilde{X}_{(-a,a)})$
for $a^{\prime}>a,$ gives rise to the commutative diagram whose lines are
short exact sequences

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{coker}_{r}(-a,a)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{r}(\tilde{X},\tilde{X}\setminus\tilde{X}_{(-a,a)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ker_{r-1}(-a,a)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{coker}_{r}(-a^{\prime},a^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{r}(\tilde{X},\tilde{X}\setminus\tilde{X}_{(-a^{\prime},a^{\prime})})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ker_{r-1}(-a^{\prime},a^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where

$\displaystyle\operatorname{coker}_{r}(-a,a)=$
$\displaystyle\operatorname{coker}(H_{r}(\tilde{X}\setminus{\tilde{X}}_{(-a,a)})\to
H_{r}(\tilde{X}))$ $\displaystyle\ker_{r-1}(-a,a)=$
$\displaystyle\ker((H_{r-1}(\tilde{X}\setminus\tilde{X}_{(-a,a)})\to
H_{r-1}(\tilde{X}))$

In view of Proposition 4.1 one has

$\varprojlim_{\tiny
a\to\infty}\ker(H_{r-1}(\tilde{X}\setminus\tilde{X}_{(-a,a)})\to
H_{r-1}(\tilde{X}))=0$

and then

$H^{BM}_{r}(\tilde{X})=\varprojlim_{\tiny
a\to\infty}\operatorname{coker}(H_{r}(\tilde{X}\setminus{\tilde{X}}_{(-a,a)})\to
H_{r}(\tilde{X})).$ (54)

By similar arguments one derives

$\displaystyle H^{BM}_{r}(\tilde{X}_{a})=$ $\displaystyle\varprojlim_{\tiny
a^{\prime}\to-\infty}\operatorname{coker}(H_{r}(\tilde{X}_{a}\setminus{\tilde{X}}_{(a^{\prime},a]})\to
H_{r}(\tilde{X}_{a})),$ (55) $\displaystyle H^{BM}_{r}(\tilde{X}^{b})=$
$\displaystyle\varprojlim_{\tiny
b^{\prime}\to\infty}\operatorname{coker}(H_{r}(\tilde{X}^{b}\setminus{\tilde{X}}_{[b,b^{\prime}})\to
H_{r}(\tilde{X})).$

From Proposition (4.1) for $a<b$ one derives that

$\operatorname{coker}(H_{r}(\tilde{X}\setminus{\tilde{X}}_{(a,b)})\to
H_{r}(\tilde{X}))=H_{r}(\tilde{X})/\mathbb{I}^{f}_{a}(r)+\mathbb{I}^{b}_{f}(r))=\kappa[\overline{S_{r,[a,b]}]}$

where

$\overline{S_{r,[a,b]}}=\\{I\in\tilde{\mathcal{B}}^{c}_{r}(f)\mid
I\subset(a,b)\\}\sqcup\\{I\in\tilde{\mathcal{B}}_{r-1}^{o}(f)\mid
I\cap(a,b)\neq\emptyset\\}.$

which implies

$H_{r}^{BM}(\tilde{X})=Maps(\tilde{\mathcal{B}}^{c}_{r}(f)\sqcup\tilde{\mathcal{B}}^{o}_{r-1}(f),\kappa)$
(56)

and identifies the canonical homomorphism $H_{r}(\tilde{X})\to
H_{r}^{BM}(\tilde{X})$ to

$\kappa[\tilde{\mathcal{B}}^{c}_{r}(f)\sqcup\tilde{\mathcal{B}}^{o}_{r-1}(f)\sqcup\tilde{J}_{r}(f)]\to
Maps(\tilde{\mathcal{B}}^{c}_{r}(f)\sqcup\tilde{\mathcal{B}}^{o}_{r-1}(f),\kappa)$
(57)

induced by sending the elements of $\tilde{\mathcal{J}}_{r}(f)$ to zero and
the other to their characteristic map.Similarly one obtains

$\displaystyle H_{r}^{BM}(\tilde{X}_{a})=$ $\displaystyle
Maps(\overline{S}_{r,(-\infty,a]},\kappa)$ (58) $\displaystyle
H_{r}^{BM}(\tilde{X}^{b})=$ $\displaystyle
Maps(\overline{S}_{r,[b,\infty)},\kappa)$

where

$\displaystyle\overline{S}_{r,(-\infty,a]}=$
$\displaystyle\\{I\in\tilde{\mathcal{B}}_{r}(f)\mid I\cap(-\infty,a]\
\rm{closed\ end}\
\neq\emptyset\\}\sqcup\\{I\in\tilde{\mathcal{B}}_{r-1}^{o}(f)\mid
I\subset(-\infty,a)\\}$ $\displaystyle\overline{S}_{r,[b,\infty)}=$
$\displaystyle\\{I\in\tilde{\mathcal{B}}_{r}(f)\mid I\cap[b,\infty)\
\rm{closed\ end}\
\neq\emptyset\\}\sqcup\\{I\in\tilde{\mathcal{B}}_{r-1}^{o}(f)\mid
I\subset(b,\infty)\\}.$

with $H_{r}^{BM}(\tilde{X}_{a})\to H^{BM}_{r}(\tilde{X}))$ and
$H_{r}^{BM}(\tilde{X}^{b})\to H^{BM}_{r}(\tilde{X}))$ identified to

$\displaystyle Maps(\overline{S}_{r,(-\infty,a]},\kappa)$ $\displaystyle\to
Maps(\tilde{\mathcal{B}}^{c}_{r}(f)\sqcup\tilde{\mathcal{B}}^{o}_{r-1}(f),\kappa)$
$\displaystyle Maps(\overline{S}_{r,[b,\infty)},\kappa)$ $\displaystyle\to
Maps(\tilde{\mathcal{B}}^{c}_{r}(f)\sqcup\tilde{\mathcal{B}}^{o}_{r-1}(f),\kappa)$

defined as follows: If $l\in Maps(\overline{S}_{r,\cdots},\kappa)$ its image
$\hat{l}\in
Maps((\tilde{\mathcal{B}}^{c}_{r}(f)\sqcup\tilde{\mathcal{B}}^{o}_{r-1}(f),\kappa))$
takes the same value as $l$ on any barcode in $\overline{S}_{r,\cdots}$ which
belongs to
$\tilde{\mathcal{B}}^{c}_{r}(f)\sqcup\tilde{\mathcal{B}}^{o}_{r-1}(f)$ and
zero on all others. Using the definition of $F^{BM,\tilde{f}}_{r}(a,b)$ one
obtains $F^{BM}_{r}(a,b)=\sharp S^{\prime}.$ q.e.d

##  7\. The mixed bar codes. Proof of Theorem 1.5

As pointed out in Section 1 for a tame map $f:X\to\mathbb{S}^{1}$ the set
$\tilde{\mathcal{B}}_{r}^{co}(f)$ and the collection
$\tilde{\mathcal{B}}_{r}^{oc}(f)$ coincides with the collection of finite
persistence bar codes associated to the filtration by the sub-levels and sup-
levels of $\tilde{f}$ respectively, as defined in [12]. Precisely the
multiplicity of the $r-$persistence barcode $(a,b)$ of the map $\tilde{f}$ is
the multiplicity of the closed-open bar code $[a,b)$ in the collection
$\tilde{\mathcal{B}}_{r}^{co}(f)$ and the multiplicity of the $r-$persistence
bar code $(-b,-a)$ for $-\tilde{f}$ is the multiplicity of the open-closed bar
code $(a,b]$ in the collection $\tilde{\mathcal{B}}_{r}^{oc}(f).$ This can be
easily derived from Proposition4.3 and the relationship between persistence
bar codes and persistent homology.

As indicated in Section 1 one can record the closed open $r-$bar code $[a,b)$
as the point $(a,b)\in\mathbb{R}^{2}\setminus\Delta$ (above the diagonal) and
to open closed $r-$bar code$(c,d]$ as the point
$(d,c)\in\mathbb{R}^{2}\setminus\Delta$ (below diagonal), equivalently we put
together the $r-$persistence diagrams of $\tilde{f}$ and of $-\tilde{f}$. We
obtain in this way a configuration $C_{r}^{m}(\tilde{f})$ of points in
$\mathbb{R}^{2}\setminus\Delta,$ which defines the configuration
$C_{r}^{m}(f)$ of points in $\mathbb{T}\setminus\Delta_{\mathbb{T}}.$ There is
no interaction between points above diagonal and below diagonal when the map
$f$ varies, so associating closed-open $r-$bar codes with open-closed
$r-$barcodes is only an issue of economy rather than meaning.

One can derive the configuration $C^{m}_{r}(f)$ as the ”jump function” of the
two variable function
$T^{\tilde{f}}_{r}:\mathbb{R}^{2}\setminus\Delta\to\mathbb{Z}_{\geq 0}$ in the
manner described in section 5 for the configuration $C_{r}(f).$ The function
$T^{\tilde{f}}_{r}$ is defined by:

$T^{\tilde{f}}(a,b)=\begin{cases}\dim\ker(H_{r}(\tilde{X}_{a})\to
H_{r}(\tilde{X}_{b}))\ \mbox{if}\ a<b\\\ \dim\ker(H_{r}(\tilde{X}^{b})\to
H_{r}(\tilde{X}^{a}))\ \mbox{if}\ a>b\end{cases}$

If $f$ is tame then so is $\tilde{f}$ and the limit

$\delta^{m,\tilde{f}}_{r}(a,b)=\lim_{\epsilon\to
0}(-T^{\tilde{f}}_{r}(a-\epsilon,b+\epsilon)-T^{\tilde{f}}_{r}(a+\epsilon,b-\epsilon)+T^{\tilde{f}}_{r}(a-\epsilon,b-\epsilon)+T^{\tilde{f}}_{r}(a+\epsilon,b+\epsilon)$

exists and defines a function which satisfies
$\delta^{m,\tilde{f}}_{r}(a+2\pi,b+2\pi)=\delta^{m,\tilde{f}}_{r}(a+2\pi,b+2\pi)$
and then, as in section 5, the function
$\delta^{m,f}_{r}:\mathbb{T}\setminus\Delta_{T}\to\mathbb{Z}_{\geq 0}.$ Using
Proposition 4.3 on can show that $\delta^{m,f}_{r}$ and $C^{m}_{r}(f)$ are
equal. The definition above is essentially the description of the persistence
diagrams of $\tilde{f}$ and $-\tilde{f},$ cf [11], and will not be pursued
further in this paper.

The stability phenomena discovered in [5] can be formulated in terms of
configuration $C_{r}^{m}(f)$ when one equips the set of finite configurations
of points in $\mathbb{T}\setminus\Delta_{\mathbb{T}}$ with the topology
induced by the bottle neck distance defined [5].Note that in this case the
configurations do not have the same cardinality and, in this topology, the
definition of ” proximity” largely ignores the points points near the diagonal
$\Delta_{\mathbb{T}}.$

Here is the definition for such topology on the space
$\mathcal{C}onfg(X\setminus K)$ of finite configurations of points in
$X\setminus K,$ $X$ locally compact space and $K$ a closed subset of $X.$
Recall that a configuration is a map with finite support, $\delta:X\setminus
K\to\mathbb{Z}_{\geq 0}.$ Define a base for the topology by specifying a
collection of open sets indexed by systems
$S=\\{(U_{1},k_{1}),\cdots(U_{r},k_{r}),V\\}$ with:

(1) $U_{i},i=1\cdots r$ open subsets of $X\setminus K,$ $V$ open neighborhood
of $K,$

(2) $k_{1},k_{2},\cdots k_{r}$ positive integers.

The ”open set” of configurations corresponding to $S$ is
$\mathcal{U}(S):=\\{\delta\in\mathcal{C}onfg(X\setminus K)\mid\rm{support}\
(\delta)\subset U_{1}\cup U_{2}\cdots\cup U_{r}\cup V,\ \sum_{x\in
U_{i}}\delta(x)=k_{i}\\}$ .

The MAIN THEOREM in [5] implies

######  Theorem 7.1.

The assignment $f\rightsquigarrow C^{m}_{r}(f)$ is a continuous map from the
space $C_{t}(X,\mathbb{S}^{1})$ of tame maps to
$\mathcal{C}onfg(\mathbb{T}\setminus\Delta)$ when the first space is equipped
with the compact open topology and the second with the topology described
above in case $(X,K)=(\mathbb{T},\Delta).$

Poincaré duality also holds for the configuration $C_{r}^{m}(f).$ Theorem 1.5
formulates this duality. We understand that for $f$ a real valued function it
is implicit in the work of Edelsbrunner and others. We treat however the angle
valued maps rather than real valued maps and derive its proof as a corollary
to Proposition 4.2. We provide below the arguments.

###  7.1.  Proof of Theorem 1.5.

In consistency with the notation in previous sections for
$\tilde{f}:\tilde{X}\to\mathbb{R}$ the infinite cyclic covering of the tame
map $f:X\to\mathbb{S}^{1}$ we denote by

  1. (i)

$i_{a}(r):H_{r}(\tilde{X}_{a})\to H_{r}(\tilde{X})\ \mbox{and}\
i^{BM}_{a}(r):H^{BM}_{r}(\tilde{X}_{a})\to H_{r}(\tilde{X}),$

  2. (ii)

$i^{a}(r):H_{r}(\tilde{X}^{a})\to H_{r}(\tilde{X})\ \mbox{and}\
i^{BM,a}(r):H^{BM}_{r}(\tilde{X}_{a})\to H_{r}(\tilde{X}),$

and for $a\leq b$

  3. (iii)

$i_{a,b}(r):H_{r}(\tilde{X}_{a})\to H_{r}(\tilde{X}_{b})\ \mbox{and}\
i^{BM}_{a,b}(r):H^{BM}_{r}(\tilde{X}_{a})\to H_{r}(\tilde{X}_{b}),$

  4. (iv)

$i^{b,a}(r):H_{r}(\tilde{X}^{b})\to H_{r}(\tilde{X}^{a})\ \mbox{and}\
i^{BM,b,a}(r):H^{BM}_{r}(\tilde{X}^{b})\to H_{r}(\tilde{X}^{a})$

the inclusion induced linear maps in homology and Borel-Moore homology.

We introduce

  1. (i)

$\mathbb{K}_{a}(r):=\ker i_{a}(r)\ \mbox{and}\ \mathbb{K}^{BM}_{a}(r):=\ker
i^{BM}_{a}(r),$

  2. (ii)

$\mathbb{K}^{a}(r):=\ker i^{a}(r)\ \mbox{and}\ \mathbb{K}^{BM,a}(r):=\ker
i^{BM,a}(r)$

and denote by

${}^{\prime}i_{a,b}(r):\mathbb{K}_{a}(r)\to\mathbb{K}_{b}(r)$ and
${}^{\prime}i^{BM}_{a,b}(r):\mathbb{K}^{BM}_{a}(r)\to\mathbb{K}^{BM}_{b}(r),$
${}^{\prime}i^{b,a}(r):\mathbb{K}^{b}(r)\to\mathbb{K}^{a}(r)$ and
${}^{\prime}i^{BM,b,a}(r):\mathbb{K}^{BM,b}(r)\to\mathbb{K}^{BM,a}(r)$

the restrictions of of $i_{a,b}(r),$ $i^{BM}_{a,b}(r)$ and of $i^{b,a}(r),$
$i^{BM,b,a}(r)$ to the respective

kernels $\mathbb{K}^{\cdots}_{\cdots}(r).$

Note that in view of the calculations of Borel–Moore homology of
$\tilde{X}^{a},\tilde{X}_{a},\tilde{X}$ and of the canonical homomorphism
$H_{r}(\tilde{M}\cdots)\to H^{BM}_{r}(\tilde{M}\cdots)$ one concludes that

$\mathbb{K}(r)=\mathbb{K}^{BM}(r)\ \rm{and}\
^{\prime}i(r)=^{\prime}i^{BM}(r).$

Proposition 4.2 permits to describe the vector spaces
$\mathbb{K}_{a}(r),\mathbb{K}^{a}(r),$ $\ker\ ^{\prime}i_{a,b}(r),$
$\operatorname{coker}\ ^{\prime}i_{a,b}(r),$ $\ker\
^{\prime}i^{b,a}(r),\operatorname{coker}\ ^{\prime}i^{b,a}(r)$ in terms of
mixed bar codes as summarized in the next proposition.

######  Proposition 7.2.

Suppose $f:X\to\mathbb{S}^{1}$ is a tame map with
$\tilde{f}:\tilde{X}\to\mathbb{R}$ its infinite cyclic covering, and $a,b$
real numbers with $a\leq b.$ Then

1\.
$\mathbb{K}^{\tilde{f}}_{a}(r)=\kappa[\\{I\in\tilde{\mathcal{B}}^{co}_{r}(f)\mid
I\ni a\\}]$

2\.
$\mathbb{K}_{\tilde{f}}^{a}(r)=\kappa[\\{I\in\tilde{\mathcal{B}}^{oc}_{r}(f)\mid
I\ni a\\}]$

3\. $\ker\
^{\prime}i_{a,b}(r)=\kappa[\\{I\in\tilde{\mathcal{B}}^{co}_{r}(f)\mid I\ni a,\
b\notin I\\}]$

$\operatorname{coker}\
^{\prime}i_{a,b}(r)=\kappa[\\{I\in\tilde{\mathcal{B}}^{co}_{r}(f)\mid I\ni b,\
a\notin I\\}]$

4\. $\ker\
^{\prime}i^{b,a}(r)=\kappa[\\{I\in\tilde{\mathcal{B}}^{oc}_{r}(f)\mid I\ni b,\
a\notin I\\}]$

$\operatorname{coker}\
^{\prime}i^{b,a}(r)=\kappa[\\{I\in\tilde{\mathcal{B}}^{oc}_{r}(f)\mid I\ni a,\
b\notin I\\}]$

The long exact sequence for the pair $(\tilde{X},\tilde{X}_{a})$

$\textstyle{H_{n-r}(\tilde{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{a}(n-r)}$$\textstyle{H_{n-r}(\tilde{X},\tilde{X}^{a})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta^{a}(n-r)}$$\textstyle{H_{n-r-1}(\tilde{X}^{a})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{a}(n-r-1)}$$\textstyle{H_{n-1-r}(\tilde{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
(59)

gives rise to the canonical isomorphism

$\delta^{a}(n-r):\operatorname{coker}j^{a}(n-r)\to\ker
i^{a}(n-r)=\mathbb{K}^{a}(n-r-1)$ (60)

which being ”natural” w.r. to the inclusion of pairs
$(\tilde{X},\tilde{X}^{b})\subseteq(\tilde{X},\tilde{X}^{a})$ for $a\leq b$
implies the commutativity of the diagram

$\textstyle{\operatorname{coker}j^{b}(n-r)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\delta^{b}_{n-r}}$$\textstyle{\mathbb{K}^{b}(n-r-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{b,a}(n-r-1)}$$\textstyle{\operatorname{coker}j^{a}(n-r)\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\delta^{a}(n-r)}$$\textstyle{\mathbb{K}^{a}(n-r-1)}$ (61)

Suppose that $X=M^{n}$ is a closed $\kappa-$orientable manifold and $a$ is a
regular value of $\tilde{f}.$ Poincaré Duality for the manifold
$\tilde{M}^{n}$ and for the pairs $(\tilde{M},\tilde{M}_{a})$ and
$(\tilde{M},\tilde{M}^{a})$ provides the commutative diagram

$\textstyle{\mathbb{K}_{a}(r)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{H_{r}(\tilde{M}_{a})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{a}(r)}$$\textstyle{H_{r}(\tilde{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{K}^{BM}_{a}(r)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{PD}$$\textstyle{H^{BM}_{r}(\tilde{M}_{a})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{PD}$$\scriptstyle{i^{BM}_{a}(r)}$$\textstyle{H^{BM}_{r}(\tilde{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{PD}$$\textstyle{\ker(j^{a}(n-r))^{\ast})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(H_{n-r}(\tilde{M},\tilde{M}^{a}))^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(H_{n-r}(\tilde{M}))^{\ast}}$
(62)

with the bottom vertical arrows the Poincaré Duality isomorphisms considered
in Section 6. The diagram is natural w.r. to the inclusion of pairs
$(X,X_{a})\subseteq(X,X_{b}),$ provided $a$ and $b$ are regular values, and
leads to the commutative diagram (63) whose vertical arrows are all
isomorphisms.

$\textstyle{\mathbb{K}_{a}(r)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{}^{\prime}i_{a,b}(r)}$$\textstyle{\mathbb{K}_{b}(r)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ker(j^{a}(n-r))^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ker(j^{b}(n-r))^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{coker}j^{a}(n-r))^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{coker}j^{b}(n-r)^{\ast}}$$\textstyle{\mathbb{K}^{a}(n-r-1)^{\ast}\
\ \ignorespaces\ignorespaces\ignorespaces\ignorespaces\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(^{\prime}i^{b,a}(n-r-1))^{\ast}}$$\textstyle{\
\
\mathbb{K}^{b}(n-r-1))^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{\delta^{b}(n-r)^{\ast}}$
(63)

Let us review the information we have:

  1. (i)

The tameness of $f$ implies that for $a<b,$ $a,b$ critical values and
$\epsilon<\epsilon(f)$ the inclusions