On a structure of non-wandering set of an $\Omega$-stable 3-diffeomorphism
possessing a hyperbolic attractor

Marina Barinova, Olga Pochinka, Evgeniy Yakovlev, HSE University

###### Abstract

This paper belongs to a series of papers devoted to the study of the structure
of the non-wandering set of an A-diffeomorphism. We study such set $NW(f)$ for
an $\Omega$-stable diffeomorphism $f$, given on a closed connected 3-manifold
$M^{3}$. Namely, we prove that if all basic sets in $NW(f)$ are trivial except
attractors, then every non-trivial attractor is either one-dimensional non-
orientable or two-dimensional expanding.

##  1 Introduction and formulation of results

Let $M^{n}$ be a smooth closed connected $n$-manifold with a Riemannian metric
$d$ and $f:M^{n}\to M^{n}$ be a diffeomorphism. A set $\Lambda\subset M^{n}$
is called an invariant set if $f(\Lambda)=\Lambda$. An invariant compact set
$\Lambda\subset M^{n}$ is called hyperbolic if there is a continuous
$Df$-invariant splitting of the tangent bundle $T_{\Lambda}M^{n}$ into stable
and unstable subbundles $E^{s}_{\Lambda}\oplus E^{u}_{\Lambda}$, $\dim
E^{s}_{x}+\dim E^{u}_{x}=n$ ($x\in\Lambda$) such that for $i>0$ and for some
fixed $C_{s}>0$, $C_{u}>0$, $0<\lambda<1$

$\|Df^{i}(v)\|\leq C_{s}\lambda^{i}\|v\|,\quad v\in E^{s}_{\Lambda},$
$\|Df^{-i}(w)\|\leq C_{u}\lambda^{i}\|w\|,\quad w\in E^{u}_{\Lambda}.$

The hyperbolic structure of $\Lambda$ implies the existence of stable and
unstable manifolds $W^{s}_{x}$, $W^{u}_{x}$ respectively for any point
$x\in\Lambda$:

$W^{s}_{x}=\\{y\in M^{n}:\lim_{j\to+\infty}d(f^{j}(x),f^{j}(y))=0\\},$
$W^{u}_{x}=\\{y\in M^{n}:\lim_{j\to+\infty}d(f^{-j}(x),f^{-j}(y))=0\\},$

which are smooth injective immersions of the $E_{x}^{s}$ and $E_{x}^{u}$ into
$M^{n}$. Moreover, $W^{s}_{x}$, $W^{u}_{x}$ are tangent to $E_{x}^{s}$ and
$E_{x}^{u}$ at $x$ respectively. For $r>0$ we will denote by $W^{s}_{x,r}$,
$W^{u}_{x,r}$ the immersions of discs on the subbundles $E_{x}^{s}$,
$E_{x}^{u}$ of the radius $r$.

Recall that a point $x\in M^{n}$ is non-wandering if for any neighborhood $U$
of $x$ the inequation $f^{n}(U)\cap U\neq\emptyset$ holds for infinitely many
integers $n$. Then $NW(f)$, the non-wandering set of $f$, defined as the set
of all non-wandering points, is an $f$-invariant closed set.

If the non-wandering set $NW(f)$ of $f$ is hyperbolic and periodic points are
dense in $NW(f)$ then $f$ is called an $A$-diffeomorphism [1]. In this case
the non-wandering set is a finite union of pairwise disjoint sets, called
basic sets

$NW(f)=\Lambda_{1}\sqcup\dots\sqcup\Lambda_{m},$

each of which is compact, invariant and topologically transitive. A basic set
$\Lambda_{i}$ of an A-diffeomorphism $f:M^{n}\to M^{n}$ is called trivial if
it coincides with a periodic orbit and non-trivial in the opposite case.

By [2], every non-trivial basic set $\Lambda_{i}$, similarly to a periodic
orbit, is uniquely expressed as a finite union of compact subsets

$\Lambda_{i}=\Lambda_{i_{1}}\sqcup\dots\sqcup\Lambda_{i_{q_{i}}},q_{i}\geqslant
1$

such that
$f^{q_{i}}(\Lambda_{i_{j}})=\Lambda_{i_{j}},f(\Lambda_{i_{j}})=\Lambda_{i_{j+1}},\,j\in\\{1,\dots,q_{i}\\}\,(\Lambda_{i_{q_{i}+1}}=\Lambda_{i_{1}})$.
These subsets $\Lambda_{i_{q_{i}}},\,q_{i}\geqslant 1$ are called periodic
components of the set $\Lambda_{i}$111R. Bowen [2] called these components
$C$-dense.. For every point $x$ of a periodic component $\Lambda_{i_{j}}$ the
set $W^{s}_{x}\cap\Lambda_{i_{j}}$ ($W^{u}_{x}\cap\Lambda_{i_{j}}$) is dense
in $\Lambda_{i_{j}}$.

Without loss of generality, everywhere below we will assume that $\Lambda_{i}$
consists of a unique periodic component and, in addition,
$f|_{W^{u}_{\Lambda_{i}}}$ preserves orientation if $\Lambda_{i}$ is trivial.

A sequence of basic sets $\Lambda_{1},\dots,\Lambda_{l}$ of an
$A$-diffeomorphism $f:M^{n}\to M^{n}$ is called a cycle if
$W^{s}_{\Lambda_{i}}\cap W^{u}_{\Lambda_{i+1}}\neq\emptyset$ for
$i=1,\dots,l$, where $\Lambda_{l+1}=\Lambda_{1}$. A-diffeomorphisms without
cycles form the set of $\Omega$-stable diffeomorphisms; if, in addition, the
stable and the unstable manifolds of every non-wandering point intersect
transversaly then $f$ is structurally stable (see, for example, [3]).

A non-trivial basic set $\Lambda_{i}$ is called orientable if for any point
$x\in\Lambda_{i}$ and any fixed numbers $\alpha>0,\,\beta>0$ the intersection
index222Let $J^{k}:\mathbb{R}^{k}\rightarrow{M}^{3}$ be immersions, $D^{k}$ be
open balls of finite radii in $\mathbb{R}^{k}$, $k=1,2$. Then the restrictions
$J^{k}:D^{k}\rightarrow{M}$ are embeddings and their images
$W^{k}=J^{k}(D^{k})$ are smooth embedded submanifolds of the manifold $M^{3}$.
Let $U^{k}$ be a tubular neighborhood of $W^{k}$, which are images of
embeddings in $M^{3}$ of spaces of $(3-k)$-dimensional vector bundles on
$W^{k}$ [4, Chapter 4, par. 5]. Since the balls $D^{k}$ are contractible, then
these bundles are trivial and, hence, $U^{2}\setminus{W}^{2}$ consists of two
connected components $U^{2}_{+}$ and $U^{2}_{-}$. It allows to define a
function $\sigma:U^{2}_{+}\cup U^{2}_{-}\rightarrow\mathbb{Z}$, such that
$\sigma(x)=1$ if $x\in U^{2}_{+}$ and $\sigma(x)=0$ if $x\in U^{2}_{-}$. If
submanifolds $W^{1}$ and $W^{2}$ intersect transversally at a point
$x=J^{1}(t)$, $t\in D^{1}$ then there exists a number $\delta>0$ such that
$J^{1}(t-2\delta,t+2\delta)\subset U^{2}$. The number  $\mathop{\rm
Ind}\nolimits_{x}(W^{1},W^{2})=\sigma(t+\delta)-\sigma(t-\delta)$ is called an
intersection index of submanifolds $W^{1}$ and $W^{2}$ in the point $x$.
Notice, that this definition does not require orientability of the manifold
$M^{3}$. $W^{u}_{x,\alpha}\cap W^{s}_{x,\beta}$ is the same at all
intersection points ($+1$ or $-1$) [5]. Otherwise, the basic set is called
non-orientable.

A basic set $\Lambda_{i}$ is called an attractor if there exists a compact
neighborhood $U_{\Lambda_{i}}$ (a trapping neighborhood) of $\Lambda_{i}$ such
that $f(U_{\Lambda_{i}})\subset{\rm int}\,U_{\Lambda_{i}}$ and
$\Lambda_{i}=\bigcap\limits^{\infty}_{i=0}f^{i}(U_{\Lambda_{i}})$. Due to [6],
a non-trivial attractor $\Lambda_{i}$ of $f$ is said to be expanding if
$\dim\,\Lambda_{i}=\dim\,W^{u}_{x}$, $x\in\Lambda_{i}$.

The main result of this paper is following.

######  Theorem 1.

Let $f:M^{3}\to M^{3}$ be an $\Omega$-stable diffeomorphism whose basic sets
are trivial except attractors. Then every non-trivial attractor is either one-
dimensional non-orientable or two-dimensional expanding.

Figure 1: $\Omega$-stable diffeomorphism $f:\mathbb{S}^{3}\to\mathbb{S}^{3}$
with the unique non-trivial basic set which is Plykin attractor

Notice, that the attractors of both types described in the Theorem 1 are
realized. In particular, the Figure 1 shows a phase portrait of a structurally
stable diffeomorphism of a 3-sphere, whose non-wandering set consists of a
one-dimensional non-orientable Plykin attractor, four saddle points with a
two-dimensional unstable manifold and two sources. The DA-diffeomorphism of
3-torus on Figure 2 is an example of a combination of an orientable two-
dimensional expanding attractor with a source in the non-wandering set of a
structurally stable diffeomorphism. An example of a diffeomorphism with non-
orientable 2-dimensional expanding attractor will be constructed in section 6.

Figure 2: DA-map on $\mathbb{T}^{3}$

Acknowledgments. This work was supported by grant 22-11-00027, except section
2.3, whose results were supported by the Laboratory of Dynamical Systems and
Applications NRU HSE, by the Ministry of Science and Higher Education of the
Russian Federation (ag. 075-15-2022-1101).

##  2 Attractor, index of a hyperbolic point, filtration

###  2.1 Attractors of an A-diffeomorphism $f:M^{3}\to M^{3}$

Let $f:M^{3}\to M^{3}$ be an $A$-diffeomorphism and $\Lambda_{i}$ be its basic
set. Then

$\dim\,W^{u}_{x}+\dim\,W^{s}_{x}=3,\,x\in\Lambda_{i}.$

If $\Lambda_{i}$ is a non-trivial then, moreover,
$\dim\,W^{u}_{x}>0,\,\dim\,W^{s}_{x}>0$.

Now let $\Lambda_{i}$ be a non-trivial attractor. It follows from [7] that

$\Lambda_{i}=\bigcup\limits_{x\in\Lambda_{i}}W^{u}_{x}$

and, hence, $\dim\,\Lambda_{i}>0$.

If $\bf{\dim\,\Lambda_{i}=3}$ then $\Lambda_{i}=M^{3}\cong\mathbb{T}^{3}$ [8].

If ${\bf\dim\,\Lambda_{i}=2}$ then $\Lambda_{i}$ is either expanding (as in
the Figure 2) or an Anosov torus ($f|_{\Lambda_{i}}$ is conjugate to an Anosov
algebraic automorphism of a torus $\mathbb{T}^{2}$) [9], [10]. Herewith, an
expanding attractor $\Lambda_{i}$ is locally homeomorphic to the product of
$\mathbb{R}^{2}$ with a cantor set [11, 12]. There are both type of such
attractor, orientable and non-orientable [13]. By [10] every Anosov torus
$\Lambda_{i}$ is a locally flat (possible non-smoothly [14]) embedded in
$M^{3}$ and, hence, it is always orientable and has a trapping neighborhood
$U_{\Lambda_{i}}$ which is homeomorphic to $\mathbb{T}^{2}\times[-1,1]$.

If ${\bf\dim\,\Lambda_{i}=1}$ then $\Lambda_{i}$ is automatically expanding,
derived from an expansions on a 1-dimensional branched manifold [6] and is the
nested intersections of handlebodies [15]. Thus, any one-dimensional attractor
$\Lambda_{i}$ of an A-diffeomorphism $f:M^{3}\to M^{3}$ has a trapping
neighborhood $U_{\Lambda_{i}}$ which is a handlebody. There are both type of
such attractor, orientable and non-orientable, it is enough to consider
$f=f_{DA}\times f_{NS}$ (see Figure 3) and $f=f_{Pl}\times f_{NS}$, where
$f_{DA}:\mathbb{T}^{2}\to\mathbb{T}^{2}$ is derived from Anosov
diffeomorphism, $f_{NS}:\mathbb{S}^{1}\to\mathbb{S}^{1}$ is a ‘‘source-sink’’
diffeomorphism, $f_{Pl}:\mathbb{S}^{2}\to\mathbb{S}^{2}$ is a diffeomorphism
with the Plykin attractor (as in the Figure 1) and four sources.

Figure 3: A one-dimensional attractor for a diffeomorphism $f_{DA}\times
f_{NS}$

The most famous one-dimensional attractor is Smale solenoid (see Figure 4)
which appears as intersection of the nested tori
$f^{k}(\mathbb{D}^{2}\times\mathbb{S}^{1}),\,k\in\mathbb{N}$ for
$f(d,z)=(d/10,2z)$. An arbitrary one-dimensional attractor is sometimes called
Smale-Williams solenoid.

Figure 4: Smale’s solenoid

It is well known that the presence of an attractor with certain properties in
a non-wandering set of an A-diffeomorphism can determine both the character of
the remaining basic sets and the topology of the ambient manifold.

  * •

If $f:M^{3}\to M^{3}$ is a structurally stable diffeomorphism whose non-
wandering set $NW(f)$ contains a two-dimensional expanding attractor
$\Lambda_{i}$, then it is orientable, $M^{3}\cong\mathbb{T}^{3}$ and the set
$NW(f)\setminus\Lambda_{i}$ consists of a finite number of isolated sources
and saddles [16], [13].

  * •

If $f:M^{3}\to M^{3}$ is an A-diffeomorphism whose every basic set is two-
dimensional then its attractors are either all Anosov tori or all expanding
[17].

  * •

If $f:M^{3}\to M^{3}$ is a structural stable diffeomorphism whose every basic
set is two-dimensional then its attractors are all Anosov tori and $M^{3}$ is
a mapping torus [18].

  * •

An orientable manifold $M^{3}$ admits an A-diffeomorphism $f:M^{3}\to M^{3}$
with the non-wandering set which is a union of finitely many Smale solenoids
if and only if $M^{3}$ is a Lens space $L_{p,q},\,p\neq 0$. Every such a
diffeomorphism is not structurally stable [19].

###  2.2 Orientability of the basic set and index of the hyperbolic point

In this section let $M$ be a compact smooth $n$-manifold $M$ (possibly with a
non-empty boundary) and $f:M\to f(M)$ be a smooth embedding of a compact
$n$-manifold $M$ to itself and $Fix(f)$ be its set of the fixed points.

Let $p\in Fix(f)$ be an isolated hyperbolic point. By [1, Proposition 4.11]
the index $I(p)=I(p,f)$ of $p$ is defined by the formula

$I(p)=(-1)^{\dim\,W^{u}_{p}}\Delta_{p},$

where $\Delta_{p}=+1$ if $f$ preserves orientation on $W^{u}_{p}$ and
$\Delta_{p}=-1$ if $f$ reverses it.

######  Lemma 2.1.

If $\Lambda_{i}$ is an orientable hyperbolic attractor with
$\dim\,W^{u}_{x}=1,x\in\Lambda_{i}$ for $f$ then $I(p)=I(q)$ for any
$p,q\in(Fix(f)\cap\Lambda_{i})$.

###### Proof.

Suppose the contrary: there are different points
$p,q\in(Fix(f)\cap\Lambda_{i})$ such that $I(p)=-I(q)$. As $p,\,q$ belongs to
the same basic set $\Lambda_{i}$ then $\dim\,W^{u}_{p}=\dim\,W^{u}_{q}$ and,
hence, $\Delta_{p}=-\Delta_{q}$. Let us assume for the definiteness that
$\Delta_{q}=-1$ and $\Delta_{p}=+1$. As $\Lambda_{i}$ is an attractor then
$W^{u}_{p},\,W^{u}_{q}\subset\Lambda_{i}$, moreover, ${\rm cl}\,W^{u}_{p}={\rm
cl}\,W^{u}_{p}=\Lambda_{i}$. Denote by
$\ell^{1}_{p},\ell^{2}_{p};\,\ell^{1}_{q},\ell^{2}_{q}$ the connected
components of the sets $W^{u}_{p}\setminus p;\,W^{u}_{q}\setminus q$. By [20]
every such a component is dense in $\Lambda_{i}$. Due to hyperbolicity of
$\Lambda_{i}$ there is a point $x_{1}$ of the transversal intersection
$\ell^{1}_{q}\cap W^{s}_{p}$ (see Figure 5).

Figure 5: Illustration to the proof of Lemma 2.1

As $\Delta_{q}=-1$ then $x_{2}=f(x_{1})$ belongs to $\ell^{2}_{q}$. Let
$(y_{1},z_{1})\subset\ell^{1}_{q}$ be a neighbourhood of the point $x_{1}$ and
$y_{2}=f(y_{1}),\,z_{2}=f(z_{1})$. Then the arc
$(y_{2},z_{2})\subset\ell^{2}_{q}$ be a neighbourhood of the point $x_{2}$. By
the orientability of $\Lambda_{i}$ we get that $y_{1},\,y_{2}$ are separated
by $W^{s}_{p}$. By $\lambda$-lemma (see, for example, [3]) the iteration of
$(y_{1},z_{1}),\,(y_{2},z_{2})$ with respect to $f$ are $C^{1}$-closed to
$W^{u}_{p}$. By continuity of $f$ we conclude that
$f(\ell^{1}_{p})=\ell^{2}_{p}$. Thus, $\Delta_{p}=-1$, that contradicts to the
assumption. ∎

Denote by $f_{*k}:H_{k}(M)\to H_{k}(M),\,k\in\\{0,\dots,n\\}$ the induced
automorphism of the $k$-th homology group $H_{k}(M)$ of $M$ with real
coefficients. The number

$\Lambda(f)=\sum\limits_{k=0}^{n}(-1)^{k}{\rm tr}(f_{*k})$

is called a Lefschetz number of $f$ [21].

Suppose $f$ has only hyperbolic fixed points and their set $Fix(f)$ is finite.
The following equality is named Lefschetz-Hopf theorem.

$\sum\limits_{p\in Fix(f)}I(p)=\Lambda(f).$ (1)

Denote by $N_{m},\,m\in\mathbb{N}$ the number of points in $Fix(f^{m})$. Let
$\lambda_{*k,j},\,j\in\\{1,\dots,\dim\,H_{k}(M)\\}$ be eigenvalues of
$f_{*k}$. If $I(p,f^{m})=I(q,f^{m})$ for any $p,q\in Fix(f^{m})$ then the
Lefschetz-Hopf theorem has the following form

$N_{m}=\left|\sum\limits_{k=0}^{n}(-1)^{k}\left(\sum\limits_{j=1}^{\dim\,H_{k}(M)}\lambda_{*k,j}^{m}\right)\right|.$
(2)

Sometimes it is convenient to pass from homology groups to cohomology groups.
Let us prove the following lemma for this aim.

######  Lemma 2.2.

Let $M$ be an $n$-dimensional orientable smooth manifold with boundary
$\partial{M}$, $f:M\rightarrow{M}$ be a diffeomorphism,
$k\in\\{0,1,\dots,n\\}$, $f_{*}:H_{k}(M)\rightarrow H_{k}(M)$,
$\tilde{f}_{*}:H_{n-k}(M,\partial{M})\rightarrow H_{n-k}(M,\partial{M})$ and
$f^{*}:H^{k}(M)\rightarrow H^{k}(M)$ be induced automorphisms for groups with
real coefficients. Then:

  * •

if $\lambda$ is an eigenvalue for $f_{*}$, then
$\tilde{\lambda}=\pm\lambda^{-1}$ is an eigenvalue for $\tilde{f}_{*}$;

  * •

if $\tilde{\lambda}$ is an eigenvalue for $\tilde{f}_{*}$, then
$\lambda=\pm\tilde{\lambda}^{-1}$ is an eigenvalue for $f^{*}$.

In the both cases a sign $+$ corresponds to an orientation-preserving
diffeomorphism and a sign $-$ is used in the opposite situation.

###### Proof.

According to the strong part of the Poincare-Lefschetz duality groups
$H_{k}(M)$ and $H_{n-k}(M,\partial{M})$ have bases $e_{1},\dots,e_{m}$ and
$\varepsilon_{1},\dots,\varepsilon_{m}$, dual with respect to the intersection
form $\mathop{\rm Ind}\nolimits:H_{k}(M)\times
H_{n-k}(M,\partial{M})\rightarrow\mathbb{R}$. The duality means that the
following equalities take place

$\mathop{\rm Ind}\nolimits(e_{i},\varepsilon_{j})=\delta_{ij},\quad
i,j=1,\dots,m.$

Let $A$ and $B$ be matrices of automorphisms $f_{*}$ and $\tilde{f}_{*}$ in
the bases $e_{1},\dots,e_{m}$ and $\varepsilon_{1},\dots,\varepsilon_{m}$
correspondingly. Then

$f_{*}(e_{i})=\sum_{s=1}^{m}a_{is}e_{s},\quad\tilde{f}_{*}(\varepsilon_{j})=\sum_{t=1}^{m}b_{jt}\varepsilon_{t}$

Herewith

$\mathop{\rm
Ind}\nolimits(f_{*}(e_{i}),\tilde{f}_{*}(\varepsilon_{j}))=\sum_{s,t=1}^{m}a_{is}b_{jt}\mathop{\rm
Ind}\nolimits(e_{s},\varepsilon_{t})=\sum_{s,t=1}^{m}a_{is}b_{jt}\delta_{st}=\sum_{s=1}^{m}a_{is}b_{js}.$
(3)

On the other hand, since $\deg{f}=\pm 1$, then

$\mathop{\rm
Ind}\nolimits(f_{*}(e_{i}),\tilde{f}_{*}(\varepsilon_{j}))=\pm\mathop{\rm
Ind}\nolimits(e_{i},\varepsilon_{j})=\pm\delta_{ij}.$ (4)

(3) and (4) imply $B^{T}=\pm A^{-1}$. Therefore, the roots of the
characteristic equations $|A-\lambda E|=0$ and $|B-\tilde{\lambda}E|=0$ are
related by the equation $\tilde{\lambda}=\pm\lambda^{-1}$. Thus, the first
statement is proved.

For the Poincare-Lefschetz isomorphism $l:H^{k}(M)\rightarrow
H_{n-k}(M,\partial{M})$ the following diagram is commutative

$\begin{CD}H^{k}(M)@<{f^{*}}<{}<H^{k}(M)\\\ @V{\pm l}V{}V@V{}V{l}V\\\
H_{n-k}(M,\partial{M})@>{\tilde{f}_{*}}>{}>H_{n-k}(M,\partial{M}).\end{CD}$
(5)

Let $v\in H_{n-k}(M,\partial{M})$, $v\neq 0$, $\tilde{\lambda}\in\mathbb{R}$
$\tilde{f}_{*}(v)=\tilde{\lambda}v$. Then
$\tilde{f}_{*}^{-1}(v)=\tilde{\lambda}^{-1}v$. Set $\alpha=l^{-1}(v)$. Since
$l$ is an isomorphism, then $\alpha\neq 0$. According to (5) we have

$f^{*}(\alpha)=\pm l^{-1}\circ\tilde{f}_{*}^{-1}\circ l(\alpha)=\pm
l^{-1}\circ\tilde{f}_{*}^{-1}(v)=\pm
l^{-1}(\tilde{\lambda}^{-1}v)=\pm\tilde{\lambda}^{-1}l^{-1}(v)=\pm\tilde{\lambda}^{-1}\alpha.$

Thus, $\lambda=\tilde{\lambda}^{-1}$ is an eigenvalue of the automorphism
$f^{*}$ corresponding to the eigenvector $\alpha\in H^{k}(M)$. ∎

According to the lemma proved above for the eigenvalues
$\lambda^{*}_{k,j},\,j\in\\{1,\dots,\dim\,H^{k}(M)\\}$ of $f^{*}_{k}$ and
$f^{m}$ such that $I(p,f^{m})=I(q,f^{m})$ for any $p,q\in Fix(f^{m})$ the
following equality takes place

$N_{m}=\left|\sum\limits_{k=0}^{n}(-1)^{n-k}\left(\sum\limits_{j=1}^{\dim\,H^{k}(M)}\lambda_{k,j}^{*m}\right)\right|.$
(6)

###  2.3 Filtration

Let $f:M^{n}\to M^{n}$ be an $\Omega$-stable diffeomorphism. As $f$ has no
cycles then $\prec$ is a partial order relation on the basic sets

$\Lambda_{i}\prec\Lambda_{j}\iff W^{s}_{\Lambda_{i}}\cap
W^{u}_{\Lambda_{j}}\neq\emptyset.$

Intuitively the definition means that ‘‘everything trickles down’’ towards
‘‘smaller elements’’. The partial order $\prec$ extends to the order relation,
i.e. the basic sets can be enumerated $\Lambda_{1},\dots,\Lambda_{m}$ in
accordance with the relation $\prec$:

$\hbox{if}\,\Lambda_{i}\prec\Lambda_{j},\,\hbox{then}\,i\leq j.$

We pick a sequence of nested subsets of the ambient manifold $M^{n}$ in the
following way. Let the first subset of $M^{n}$ be a neighborhood $M_{1}$ of
the basic set $\Lambda_{1}$, let the next subset $M_{2}$ be the union of
$M_{1}$ and some neighborhood of the unstable manifold of the element
$\Lambda_{2}$. If we continue this process we get the entire manifold $M^{n}$.
This construction gives the idea to the following notion of filtration.

A sequence $M_{1},\dots,M_{m-1}$ of compact $n$-submanifolds of $M^{n}$, each
having a smooth boundary, and such that $M^{n}=M_{m}\supset
M_{m-1}\supset\dots\supset M_{1}\supset M_{0}=\emptyset$ is called a
filtration for a diffeomorphism $f$ with its ordered basic sets
$\Lambda_{1}\prec\dots\prec\Lambda_{m}$ if for each $i=1,\dots,m$ the
following holds:

  1. 1.

$f(M_{i})\subset{\rm int}\,M_{i}$;

  2. 2.

$\Lambda_{i}\subset{\rm int}\,(M_{i}\setminus M_{i-1})$;

  3. 3.

$\Lambda_{i}=\bigcap\limits_{l\in\mathbb{Z}}f^{l}(M_{i}\setminus M_{i-1})$;

  4. 4.

$\bigcap\limits_{l\geq 0}f^{l}(M_{i})=\bigcup\limits_{j\leq
i}W^{u}_{\Lambda_{j}}=\bigcup\limits_{j\leq i}cl(W^{u}_{\Lambda_{j}})$.

Below we describe following from [22] interrelations between actions $f$ on
cohomology groups $H^{k}(M^{n})$, $H^{k}(M_{i},M_{i-1})$ and homology group
$H_{k}(M^{n})$ with real coefficients. If an action in these group is
nilpotent then all eigenvalues equal zero and if it is unipotent then it has
only roots of unity as eigenvalues.

######  Proposition 2.1.

Let $f:M^{n}\to M^{n}$ be an $\Omega$-stable diffeomorphism and
$M^{n}=M_{m}\supset M_{m-1}\supset\dots\supset M_{1}\supset M_{0}=\emptyset$
be a filtration for its ordered basic sets
$\Lambda_{1}\prec\dots\prec\Lambda_{m}$. Then

  1. 1.

If $\lambda$ is an eigenvalue of $f^{*}_{k}:H^{k}(M^{n})\to H^{k}(M^{n})$,
then there is an $i\in\\{1,\dots,m\\}$ such that
$f^{*}_{k}:H^{k}(M_{i},M_{i-1})\to H^{k}(M_{i},M_{i-1})$ has $\lambda$ as an
eigenvalue.

  2. 2.

If $\Lambda_{i}$ is a trivial basic set then
$f^{*}_{k}:H^{k}(M_{i},M_{i-1})\to H^{k}(M_{i},M_{i-1})$ is nilpotent unless
$k=\dim\,W^{u}_{x},\,x\in\Lambda_{i}$ and $f^{*}_{k}:H^{k}(M_{i},M_{i-1})\to
H^{k}(M_{i},M_{i-1})$ is unipotent for $k=\dim\,W^{u}_{x},\,x\in\Lambda_{i}$.

##  3 Proof of theorem 1

In this section we prove that if $f:M^{3}\to M^{3}$ is an $\Omega$-stable
diffeomorphism whose basic sets are trivial except attractors, then every non-
trivial attractor is either one-dimensional non-orientable or two-dimensional
expanding. We will use in this proof some results, which will be proven in the
next section. As above, the symbols $H_{k}(X,A)$ and $H^{k}(X,A)$ will denote
homology and cohomology groups with real coefficients. For homology groups
with integer coefficients, the notation $H_{k}(X,A;\mathbb{Z})$ will be used.

###### Proof.

Suppose the contrary: $NW(f)$ contains a non-trivial attractor $A$ such that
$A$ is either one-dimensional orientable or two-dimensional Anosov torus.
Without loss of generality we can assume that in the order $\prec$, first
positions occupied by attractors and $A$ is the last of them. Let
$M^{n}=M_{k}\supset M_{k-1}\supset\dots\supset M_{1}\supset M_{0}=\emptyset$
be a filtration for the ordered basic sets
$\Lambda_{1}\prec\dots\prec\Lambda_{k}$. Then
$\tilde{M}_{i}=M^{n}\setminus{\rm int}\,M_{k-i}$ is the filtration for the
basic sets $\tilde{\Lambda}_{i}=\Lambda_{k-i}$ of the diffeomorphism
$g=f^{-1}$. Let $A=\tilde{\Lambda}_{i_{0}}$. Without loss of generality we can
assume that the manifold $\tilde{M}_{i_{0}}$ is connected (in the opposite
case let us consider its connected component containing $A$). Then
$g(\tilde{M}_{i_{0}})\subset{\rm int}\,\tilde{M}_{i_{0}}$. Notice, that
$i_{0}>1$ since any $\Omega$-stable diffeomorphism has non-empty sets of
attractors and repellers.

Let $N_{m}$ be a number of points in $Fix(g^{m})$. As the non-trivial basic
set $A$ belongs to $\tilde{M}_{i_{0}}$ then
$\lim\limits_{m\to\infty}N_{m}=\infty$. Since $A$ is orientable then the Lemma
2.1 and the formula (6) gives the existence of an eigenvalue $\lambda$ with
absolute value greater than 1 for $g^{*}_{k}:H^{k}(\tilde{M}_{i_{0}})\to
H^{k}(\tilde{M}_{i_{0}})$ for some $k\in\\{0,\dots,3\\}$.

First of all, let us show that it is impossible for orientable $M^{3}$. We
will prove it separately for each dimension $k=0,1,2,3$.

a) $k=0$. Eigenvalues of the automorphism $g^{*}:H^{0}(\tilde{M}_{i_{0}})\to
H^{0}(\tilde{M}_{i_{0}})$ are roots of unity by the lemma 4.5.

b) $k=3$. The group $H_{3}(\tilde{M}_{i_{0}};\mathbb{Z})$ is trivial when
$\partial\tilde{M}_{i_{0}}\neq\emptyset$ and is isomorphic to $\mathbb{Z}$
when $\partial\tilde{M}_{i_{0}}=\emptyset$. In the first case we have
$H^{3}(\tilde{M}_{i_{0}})=0$ and so $g^{*}:H^{3}(\tilde{M}_{i_{0}})\to
H^{3}(\tilde{M}_{i_{0}})$ does not have eigenvalues. In the second case,
$g^{*}=\pm\mathop{\rm id}\nolimits$ by the lemma 4.4.

c) $k=1$. Suppose, that the automorphism $g^{*}:H^{1}(\tilde{M}_{i_{0}})\to
H^{1}(\tilde{M}_{i_{0}})$ has an eigenvalue $\lambda$, for which
$\lambda^{2}\neq 1$. Then it follows from the item 1 of the proposition 2.1,
that there exists a number $i$, $1\leqslant{i}\leqslant{i_{0}}$ such that the
automorphism $g^{*}:H^{1}(M_{i},M_{i-1})\rightarrow H^{1}(M_{i},M_{i-1})$ also
has the eigenvalue $\lambda$.

As all basic sets of $g$ before $A$ in the Smale order $\prec$ are trivial
then by the item 2 of proposition 2.1 for $i<i_{0}$ we get that the
automorphisms $g^{*}$ on $H^{1}(M_{i},M_{i-1})$ are either nilpotent or
uniponent. Hence, it is precisely the automorphism
$g^{*}:H^{1}(\tilde{M}_{i_{0}},\tilde{M}_{i_{0}-1})\rightarrow
H^{1}(\tilde{M}_{i_{0}},\tilde{M}_{i_{0}-1})$ must have the eigenvalue
$\lambda$.

Let $\dim\,A=1$. In this case
$\tilde{M}_{i_{0}}=Q_{g}\cup\tilde{M}_{i_{0}-1}$, where $Q_{g}$ is a
handlebody of a genus $g\geqslant 0$ such that
$Q_{g}\cap\tilde{M}_{i_{0}-1}=\partial Q_{g}$. By lemma 4.2
$H_{1}(\tilde{M}_{i_{0}},\tilde{M}_{i_{0}-1};\mathbb{Z})=0$. Then
$H^{1}(\tilde{M}_{i_{0}},\tilde{M}_{i_{0}-1})=0$ and therefore $\lambda$
cannot be an eigenvalue of the automorphism $g^{*}$.

If $\dim\,A=2$, then $\tilde{M}_{i_{0}}=Q\cup\tilde{M}_{i_{0}-1}$, where
$Q\cong\mathbb{T}^{2}\times[0,1]$ and $Q\cap\tilde{M}_{i_{0}-1}=\partial Q$.
In this situation by the lemma 4.3
$H_{1}(\tilde{M}_{i_{0}},\tilde{M}_{i_{0}-1};\mathbb{Z})\cong\mathbb{Z}$. From
here and from the lemma 4.4 it follows, that $g^{*}=\pm\mathop{\rm
id}\nolimits$. Thus, we obtain a contradiction for $k=1$ as well.

d) $k=2$. Let us finally assume that $g^{*}:H^{2}(\tilde{M}_{i_{0}})\to
H^{2}(\tilde{M}_{i_{0}})$ has as eigenvalue $\lambda$, for which
$\lambda^{2}\neq 1$. Due to lemma 2.2, in such a situation the automorphism
$g^{*}:H^{1}(\tilde{M}_{i_{0}},\partial\tilde{M}_{i_{0}})\to
H^{1}(\tilde{M}_{i_{0}},\partial\tilde{M}_{i_{0}})$ has an eigenvalue
$\tilde{\lambda}=\pm\lambda^{-1}$.

Consider the following diagram

$\setcounter{MaxMatrixCols}{11}\begin{CD}\dots
@>{}>{}>H^{0}(\partial\tilde{M}_{i_{0}})@>{\delta^{*}}>{}>H^{1}(\tilde{M}_{i_{0}},\partial\tilde{M}_{i_{0}})@>{j^{*}}>{}>H^{1}(\tilde{M}_{i_{0}})@>{}>{}>\dots\\\
@V{}V{g^{*}}V@V{}V{g^{*}}V@V{}V{g^{*}}V\\\ \dots
@>{}>{}>H^{0}(\partial\tilde{M}_{i_{0}})@>{\delta^{*}}>{}>H^{1}(\tilde{M}_{i_{0}},\partial\tilde{M}_{i_{0}})@>{j^{*}}>{}>H^{1}(\tilde{M}_{i_{0}})@>{}>{}>\dots,\end{CD}$
(7)

where the rows are taken from the cohomological sequence of the pair
$(\tilde{M}_{i_{0}},\partial\tilde{M}_{i_{0}})$ and the vertical arrows denote
the mappings induced by the diffeomorphism $g$. All squares of the diagram are
commutative, and the middle automorphism $g^{*}$ from (7) has an eigenvalue
$\tilde{\lambda}$. From this, by [22, Lemma 3] it follows that for one of the
extreme vertical automorphisms of the diagram (7) $\tilde{\lambda}$ is also an
eigenvalue. Since $\tilde{\lambda}^{2}\neq 1$, then for the automorphism
$g^{*}:H^{1}(\tilde{M}_{i_{0}})\to H^{1}(\tilde{M}_{i_{0}})$ this is
impossible according to proven in c). Since the manifold $\tilde{M}_{i_{0}}$
is compact, its boundary $\partial\tilde{M}_{i_{0}}$ consists of a finite set
of connected components. Then by Lemma 4.5 all eigenvalues of the automorphism
$g^{*}:H^{0}(\partial\tilde{M}_{i_{0}})\to H^{0}(\partial\tilde{M}_{i_{0}})$
are roots of unity. Thus, in this case we also obtain a contradiction.

If $M^{n}$ is non-orientable then, by lemma 5.2, there is an oriented two-fold
covering $p:\bar{M}^{n}\to M^{n}$ and a lift
$\bar{g}:\bar{M}^{n}\to\bar{M}^{n}$ of the diffeomorphism333We have not found
a reference for this fact, so we prove it in the section 5 below. $g$.
Herewith, by lemma 5.3, $\bar{A}=p^{-1}(A)$ is orientable, like $A$. So we can
apply all arguments from an orientable case to $\bar{g}$ and get a
contradiction. ∎

##  4 Homology and induced automorphisms

In this section, we calculate the homology groups of some topological pairs
and study the properties of automorphisms of cohomology groups induced by
homeomorphisms.

###  4.1 Calculations

In this section we calculate relative homology for the following situation.
Let $M$ and $N$ be smooth 3-manifolds with boundaries such that $P=M\cup{N}$
is connected, $M\cap{N}=\partial{M}$ and connected components of $\partial{M}$
are some of connected components of $\partial{N}$. Let us calculate the
relative homology groups of the pair $(P,N)$.

Firstly notice that $H_{0}(P,N;\mathbb{Z})=0$ as the manifold $P$ is connected
and $N$ is not empty. For the calculation of other relative homology groups we
need the following fact.

######  Lemma 4.1.

For every natural $k$ the following isomorphism takes place

$H_{k}(P,N;\mathbb{Z})\cong H_{k}(M,\partial{M};\mathbb{Z}).$

###### Proof.

By [4, Theorem 6.1, Chapter 4] the boundary $\partial{N}$ possesses a collar
in $N$. As connected components of $\partial{M}$ are connected components of
$\partial{N}$ then there is an embedding
$\phi:\partial{M}\times[0,1)\rightarrow{N}$ such that $\phi(a,0)=a$ for every
$a\in\partial{M}$. Let $V=\phi(\partial{M}\times[0,1))$, $B=N\setminus{V}$ and
${\rm cl}\,{B}$ be the closure of $B\subset{P}$ in $P$, ${\rm int}\,{N}$ be
the interior of $N\subset{P}$ in $P$. By the construction ${\rm
cl}\,{B}=B\subset{N}\setminus\partial{M}={\rm int}{N}$. The excision theorem
[23, Corollary 7.4, Chapter III] claims in such case that

$H_{k}(P,N;\mathbb{Z})\cong
H_{k}(P\setminus{B},N\setminus{B};\mathbb{Z})=H_{k}(M\cup{V},V;\mathbb{Z}).$

But the pair $(M\cup{V},V)$ is homotopically equivalent to the pair
$(M,\partial{M})$. Hence, $H_{k}(M\cup{V},V;\mathbb{Z})\cong
H_{k}(M,\partial{M};\mathbb{Z})$ for a natural $k$. ∎

Below we calculate $H_{k}(M,\partial{M};\mathbb{Z})$ in two cases: 1) $M$ is a
handlebody of a genus $g\geqslant 0$, 2) $M\cong\mathbb{T}^{2}\times[0,1]$.

######  Lemma 4.2.

If $M$ is a handlebody of a genus $g\geqslant 0$ then

$H_{3}(M,\partial{M};\mathbb{Z})\cong\mathbb{Z},\quad
H_{2}(M,\partial{M};\mathbb{Z})\cong\mathbb{Z}^{g},\quad
H_{1}(M,\partial{M};\mathbb{Z})=0.$ (8)

###### Proof.

As $H_{3}(M;\mathbb{Z})=0$ and $H_{0}(M,\partial{M};\mathbb{Z})=0$ then the
homological sequence of the pair $(M,\partial M)$ has the following form [23,
Proposition 4.4, Chapter III]:

$0\longrightarrow
H_{3}(M,\partial{M};\mathbb{Z})\stackrel{{\scriptstyle\partial_{*}^{3}}}{{\longrightarrow}}H_{2}(\partial{M};\mathbb{Z})\stackrel{{\scriptstyle\imath_{*}^{2}}}{{\longrightarrow}}H_{2}(M;\mathbb{Z})\stackrel{{\scriptstyle\jmath_{*}^{2}}}{{\longrightarrow}}H_{2}(M,\partial{M};\mathbb{Z})\stackrel{{\scriptstyle\partial_{*}^{2}}}{{\longrightarrow}}\\\
{\longrightarrow}H_{1}(\partial{M};\mathbb{Z})\stackrel{{\scriptstyle\imath_{*}^{1}}}{{\longrightarrow}}H_{1}(M;\mathbb{Z})\stackrel{{\scriptstyle\jmath_{*}^{1}}}{{\longrightarrow}}H_{1}(M,\partial{M};\mathbb{Z})\stackrel{{\scriptstyle\partial_{*}^{1}}}{{\longrightarrow}}\\\
{\longrightarrow}H_{0}(\partial{M};\mathbb{Z})\stackrel{{\scriptstyle\imath_{*}^{0}}}{{\longrightarrow}}H_{0}(M;\mathbb{Z})\stackrel{{\scriptstyle\jmath_{*}^{0}}}{{\longrightarrow}}0.$
(9)

Handlebody $M$ of a genus $g$ is the 3-ball with glued $g$ 3-handles of the
index 1. That is $M$ is homotopically equivalent to the bouquet of $g$
circles. Therefore,

$H_{2}(M;\mathbb{Z})=0,\quad H_{1}(M;\mathbb{Z})\cong\mathbb{Z}^{g},\quad
H_{0}(M;\mathbb{Z})\cong\mathbb{Z}.$

On the other side the boundary $\partial{M}$ is homeomorphic to the surface
$S_{g}$ of the genus $g$. Hence,

$H_{2}(\partial M;\mathbb{Z})\cong\mathbb{Z},\quad H_{1}(\partial
M;\mathbb{Z})\cong\mathbb{Z}^{2g},\quad H_{0}(\partial
M;\mathbb{Z})\cong\mathbb{Z}.$

Substituting the latter in (9), we get the exact sequence

$0\longrightarrow H_{3}(M,\partial
M;\mathbb{Z})\stackrel{{\scriptstyle\partial_{*}^{3}}}{{\longrightarrow}}\mathbb{Z}\stackrel{{\scriptstyle\imath_{*}^{2}}}{{\longrightarrow}}0\stackrel{{\scriptstyle\jmath_{*}^{2}}}{{\longrightarrow}}H_{2}(M,\partial
M;\mathbb{Z})\stackrel{{\scriptstyle\partial_{*}^{2}}}{{\longrightarrow}}\\\
{\longrightarrow}\mathbb{Z}^{2g}\stackrel{{\scriptstyle\imath_{*}^{1}}}{{\longrightarrow}}\mathbb{Z}^{g}\stackrel{{\scriptstyle\jmath_{*}^{1}}}{{\longrightarrow}}H_{1}(M,\partial
M;\mathbb{Z})\stackrel{{\scriptstyle\partial_{*}^{1}}}{{\longrightarrow}}\mathbb{Z}\stackrel{{\scriptstyle\imath_{*}^{0}}}{{\longrightarrow}}\mathbb{Z}\stackrel{{\scriptstyle\jmath_{*}^{0}}}{{\longrightarrow}}0.$
(10)

As $\imath_{*}^{1}$ is an epimorphism then (10) decomposes into short exact
sequences

$\displaystyle 0$ $\displaystyle\longrightarrow H_{3}(M,\partial
M;\mathbb{Z})\stackrel{{\scriptstyle\partial_{*}^{3}}}{{\longrightarrow}}\mathbb{Z}\stackrel{{\scriptstyle\imath_{*}^{2}}}{{\longrightarrow}}0,$
$\displaystyle 0$ $\displaystyle\longrightarrow H_{2}(M,\partial
M;\mathbb{Z})\stackrel{{\scriptstyle\partial_{*}^{2}}}{{\longrightarrow}}\mathbb{Z}^{2g}\stackrel{{\scriptstyle\imath_{*}^{1}}}{{\longrightarrow}}\mathbb{Z}^{g}\longrightarrow
0,$ $\displaystyle 0$ $\displaystyle\longrightarrow H_{1}(M,\partial
M;\mathbb{Z})\stackrel{{\scriptstyle\partial_{*}^{1}}}{{\longrightarrow}}\mathbb{Z}\stackrel{{\scriptstyle\imath_{*}^{0}}}{{\longrightarrow}}\mathbb{Z}\longrightarrow
0,$

from which follows the statement of the lemma. ∎

######  Lemma 4.3.

If $M=\mathbb{T}^{2}\times[0,1]$ then

$H_{3}(M,\partial{M};\mathbb{Z})\cong\mathbb{Z},\quad
H_{2}(M,\partial{M};\mathbb{Z})\cong\mathbb{Z}^{2},\quad
H_{1}(M,\partial{M};\mathbb{Z})=\mathbb{Z}.$ (11)

###### Proof.

As $M$ is homotopically equivalent to $\mathbb{T}^{2}$ and $\partial{M}$ is
homeomorphic to $\mathbb{T}^{2}\times\mathbb{S}^{0}$ then
$H_{k}(M;\mathbb{Z})\cong{H}_{k}(\mathbb{T}^{2};\mathbb{Z})$ and
$H_{k}(\partial{M};\mathbb{Z})\cong{H}_{k}(\mathbb{T}^{2};\mathbb{Z})\times{H}_{k}(\mathbb{T}^{2};\mathbb{Z})$.
In such situation the homological sequence (9) of the pair $(M,\partial M)$
has the following form:

$0\longrightarrow
H_{3}(M,\partial{M};\mathbb{Z})\stackrel{{\scriptstyle\partial_{*}^{3}}}{{\longrightarrow}}\mathbb{Z}^{2}\stackrel{{\scriptstyle\imath_{*}^{2}}}{{\longrightarrow}}\mathbb{Z}\stackrel{{\scriptstyle\jmath_{*}^{2}}}{{\longrightarrow}}H_{2}(M,\partial{M};\mathbb{Z})\stackrel{{\scriptstyle\partial_{*}^{2}}}{{\longrightarrow}}\\\
{\longrightarrow}\mathbb{Z}^{4}\stackrel{{\scriptstyle\imath_{*}^{1}}}{{\longrightarrow}}\mathbb{Z}^{2}\stackrel{{\scriptstyle\jmath_{*}^{1}}}{{\longrightarrow}}H_{1}(M,\partial{M};\mathbb{Z})\stackrel{{\scriptstyle\partial_{*}^{1}}}{{\longrightarrow}}\mathbb{Z}^{2}\stackrel{{\scriptstyle\imath_{*}^{0}}}{{\longrightarrow}}\mathbb{Z}\stackrel{{\scriptstyle\jmath_{*}^{0}}}{{\longrightarrow}}0.$
(12)

As the inclusion of every connected component of $\partial M$ to ${M}$ is a
homotopical equivalence then $\imath_{*}^{2}$ and $\imath_{*}^{1}$ are
epimorphisms. Herewith (12) decomposes into short exact sequences

$\displaystyle 0$ $\displaystyle\longrightarrow
H_{3}(M,\partial{M};\mathbb{Z})\stackrel{{\scriptstyle\partial_{*}^{3}}}{{\longrightarrow}}\mathbb{Z}^{2}\stackrel{{\scriptstyle\imath_{*}^{2}}}{{\longrightarrow}}\mathbb{Z}\longrightarrow
0,$ $\displaystyle 0$ $\displaystyle\longrightarrow
H_{2}(M,\partial{M};\mathbb{Z})\stackrel{{\scriptstyle\partial_{*}^{2}}}{{\longrightarrow}}\mathbb{Z}^{4}\stackrel{{\scriptstyle\imath_{*}^{1}}}{{\longrightarrow}}\mathbb{Z}^{2}\longrightarrow
0,$ $\displaystyle 0$ $\displaystyle\longrightarrow
H_{1}(M,\partial{M};\mathbb{Z})\stackrel{{\scriptstyle\partial_{*}^{1}}}{{\longrightarrow}}\mathbb{Z}^{2}\stackrel{{\scriptstyle\imath_{*}^{0}}}{{\longrightarrow}}\mathbb{Z}\longrightarrow
0,$

from which follows the statement of the lemma. ∎

###  4.2 Eigenvalues of induced automorphisms

In this section we again consider all homology groups $H_{k}(X,A;\mathbb{Z})$
with integer coefficients and cohomology groups $H^{k}(X,A)$ with real
coefficients. Firstly, by the Universal Coefficient Formula [23, Section 7,
Chapter VI], the previous subsection results give the following calculations.

######  Statement 4.1.

If $M$ is a handlebody of a genus $g\geqslant 0$ then

$H^{3}(M,\partial{M})\cong\mathbb{R},\,H^{2}(M,\partial{M})\cong\mathbb{R}^{g},\,H^{1}(M,\partial{M})=0,\,H^{0}(M,\partial{M})=0.$

######  Statement 4.2.

If $M=\mathbb{T}^{2}\times[0,1]$ then

$H^{3}(M,\partial{M})\cong\mathbb{R},\,H^{2}(M,\partial{M})\cong\mathbb{R}^{2},\,H^{1}(M,\partial{M})=\mathbb{R},\,H^{0}(M,\partial{M})=0.$

The groups $H^{k}(X,A)\cong\mathbb{R}^{m}$ admit many automorphisms even for
$m=1$. But in some cases only a small part of them can be induced by
homeomorphisms of the topological space $X$.

######  Lemma 4.4.

Let $X$ be a topological space, $A\subset{X}$ be its subspace,
$f:X\rightarrow{X}$ be homeomorphism, and $f(A)\subset{A}$. Denote by
$H^{\prime}_{k}(X,A;\mathbb{Z})$ a free part of the group of $k$-dimensional
singular homology of the pair $(X,A)$, and by $H^{k}(X,A)$ its $k$-dimensional
cohomology group with real coefficients. It
$H^{\prime}_{k}(X,A;\mathbb{Z})\cong\mathbb{Z}$ for some $k$, then for the
induced automorphism $f^{*}:H^{k}(X,A)\rightarrow H^{k}(X,A)$ the equality
$f^{*}=\pm\mathop{\rm id}\nolimits$ holds.

###### Proof.

Let the automorphism $f_{*}:H^{\prime}_{k}(X,A;\mathbb{Z})\rightarrow
H^{\prime}_{k}(X,A;\mathbb{Z})$ also induced by the homeomorphism $f$. The
formula $f^{*}_{h}(q)=q\circ f_{*}$ defines the automorphism
$f^{*}_{h}:\mathop{\rm
Hom}\nolimits(H^{\prime}_{k}(X,A;\mathbb{Z});\mathbb{R})\rightarrow\mathop{\rm
Hom}\nolimits(H^{\prime}_{k}(X,A;\mathbb{Z});\mathbb{R})$. If
$H^{\prime}_{k}(X,A;\mathbb{Z})\cong\mathbb{Z}$, then $f_{*}=\pm\mathop{\rm
id}\nolimits$. Moreover, $f^{*}_{h}(q)=q\circ(\pm\mathop{\rm
id}\nolimits)=\pm{q}$ for all $q\in\mathop{\rm
Hom}\nolimits(H^{\prime}_{k}(X,A;\mathbb{Z});\mathbb{R})$. Hence
$f^{*}_{h}=\pm\mathop{\rm id}\nolimits$.

It follows for the Universal Coefficient Formula for cohomology [23, Chapter
VI, Section 7] that there exists the natural isomorphism
$\kappa:H^{k}(X,A)\rightarrow\mathop{\rm
Hom}\nolimits(H^{\prime}_{k}(X,A;\mathbb{Z});\mathbb{R})$. The naturalness
means commutativity of the diagram

$\begin{CD}H^{k}(X,A)@>{\kappa}>{}>\mathop{\rm
Hom}\nolimits(H^{\prime}_{k}(X,A;\mathbb{Z});\mathbb{R})\\\
@V{f^{*}}V{}V@V{}V{f^{*}_{h}}V\\\ H^{k}(X,A)@>{\kappa}>{}>\mathop{\rm
Hom}\nolimits(H^{\prime}_{k}(X,A;\mathbb{Z});\mathbb{R}).\end{CD}$ (13)

It follows from (13) and the equation $f^{*}_{h}=\pm\mathop{\rm id}\nolimits$
that $f^{*}=\kappa^{-1}\circ
f^{*}_{h}\circ\kappa=\kappa^{-1}\circ(\pm\mathop{\rm
id}\nolimits)\circ\kappa=\pm\mathop{\rm id}\nolimits$. ∎

######  Lemma 4.5.

Let $X$ be a topological space with a finite number of path-connected
components, $f:X\rightarrow{X}$ be a homeomorphism, and
$f^{*}:H^{0}(X)\rightarrow H^{0}(X)$ be an induced automorphism. Then any
eigenvalue $\lambda$ for $f^{*}$ satisfies the equality $\lambda^{2}=1$.

###### Proof.

Firstly consider a case when $X_{1}$ and $X_{2}$ are path-connected
topological spaces and $f_{2}:X_{1}\rightarrow{X}_{2}$ is a homeomorphism. All
elements of groups $H^{0}(X_{j})$ are constant functions
$c_{j}:X_{j}\rightarrow\mathbb{R}$. Therefore, the formula
$\nu_{j}(c_{j})=\mathop{\rm im}\nolimits{c}_{j}$ defines isomorphisms
$\nu_{j}:H^{0}(X_{j})\rightarrow\mathbb{R}$, $j=1,2$. The induced isomorphism
$f_{2}^{*}:H^{0}(X_{2})\rightarrow{H}^{0}(X_{1})$ is defined by the formula
$f_{2}^{*}(c_{2})=c_{2}\circ{f}_{2}$. Since values of the functions $c_{2}$
and $c_{2}\circ{f}_{2}$ are equal, then

$\nu_{1}\circ{f}^{*}_{2}=\nu_{2}.$ (14)

Now suppose that $X$ consists of path-connected components
$X_{1},\dots,X_{m}$. Then there exists a permutation $\sigma\in{S}_{m}$ such
that $f$ maps the component $X_{j}$ onto the component $X_{\sigma(j)}$
homeomorphically. Thus, setting $f_{\sigma(j)}(x)=f(x)$ for all $x\in{X}_{j}$,
we obtain homeomorphisms $f_{\sigma(j)}:X_{j}\rightarrow{X}_{\sigma(j)}$,
$j=1,\dots,m$. Moreover, the induced homomorphisms
$f_{j}^{*}:H^{0}(X_{j})\rightarrow{H}^{0}(X_{\tau(j)})$ are defined, where
$\tau=\sigma^{-1}$. By virtue of (14)

$\nu_{\tau(j)}\circ{f}_{j}^{*}=\nu_{j},\quad j=1,\dots,m.$ (15)

For each element $c\in{H}^{0}(X)$ we set $c_{j}=c|_{X_{j}}$. Then
$c_{j}\in{H}^{0}(X_{j})$. Define isomorphisms
$\mu:{H}^{0}(X)\rightarrow{H}^{0}(X_{1})\times\dots\times{H}^{0}(X_{m})$ and
$\nu:{H}^{0}(X_{1})\times\dots\times{H}^{0}(X_{m})\rightarrow\mathbb{R}^{m}$
by the formulas $\mu(c)=(c_{1},\dots,c_{m})$ and
$\nu((c_{1},\dots,c_{m}))=(\nu_{1}(c_{1}),\dots,\nu_{m}(c_{m}))$. We construct
the automorphism $p:\mathbb{R}^{m}\rightarrow\mathbb{R}^{m}$ such that the
diagram is commutative

$\begin{CD}H^{0}(X)@>{\mu}>{}>H^{0}(X_{1})\times\dots\times
H^{0}(X_{m})@>{\nu}>{}>\mathbb{R}^{m}\\\
@V{}V{f^{*}}V@V{}V{(f^{*}_{1},\dots,f^{*}_{m})}V@V{}V{p}V\\\
H^{0}(X)@>{\mu}>{}>H^{0}(X_{1})\times\dots\times
<EMAIL_ADDRESS>(16)

For all $y=(y_{1},\dots,y_{m})\in\mathbb{R}^{m}$ we set
$\|y\|=\sqrt{y_{1}^{2}+\dots+y_{m}^{2}}$. Since $f_{j}^{*}$ maps
$H^{0}(X_{j})$ onto ${H}^{0}(X_{\tau(j)})$, then it follows from the equality
(15) and the diagram (16) that $p(y)=(y_{\tau(1)},\dots,y_{\tau(1)})$.
Moreover, $\|{p}(y)\|=\|y\|$.

Finally, let $\lambda\in\mathbb{R}$, $c\in{H}^{0}(X)$, $c\neq 0$ and
$f^{*}(c)=\lambda{c}$. We set $y=\nu\circ\mu(c)$. Then by virtue of (16)
$p(y)=\mu\circ\nu(f^{*}(c))=\mu\circ\nu(\lambda{c})=\lambda{y}$. Hence,
according to what was proved above, we obtain
$\|y\|^{2}=\|{p}(y)\|^{2}=\lambda^{2}\|y\|^{2}$. Hence, $\lambda^{2}=1$. ∎

##  5 On oriented two-fold covering

Let $M$ be a non-orientable connected smooth $n$-manifold, $a\in M$ and
$x:I\rightarrow M$ be a loop based at a point $a$. Let us consider continuous
vector fields $X_{1},\dots,X_{n}$ along $x$ such that
$X_{1}(t),\dots,X_{n}(t)$ linearly independent for each $t\in{I}$. Then there
is a matrix $A=(a_{i}^{j})\in{\rm GL}_{n}(\mathbb{R})$ such that

$X_{i}(1)=a_{i}^{j}X_{j}(0),\quad i,j=1,\dots,n.$ (17)

Let $\omega_{a}(x)={\rm sign}\det{A}$. If $y$ is a loop which based at the
same starting point and $x\sim{y}$ then $\omega_{a}(x)=\omega_{a}(y)$.
Therefore the formula $\omega_{a}([x])=\omega_{a}(x)$ defines a homeomorphism
$\omega_{a}:\pi_{1}(M,a)\rightarrow{G}$, where $G=\\{1,-1\\}$. The manifold
$M$ is orientable if and only if $\ker\omega_{a}=\pi_{1}(M,a)$.

Let $a,b\in{M}$, $z:I\rightarrow M$ be a path which starts in $z(0)=a$ and
ends in $z(1)=b$ and $T_{z}:\pi_{1}(M,a)\rightarrow\pi_{1}(M,b)$ be the
isomorphism defined by the formula $T_{z}([x])=[z^{-1}xz]$. Then $zz^{-1}\sim
1_{a}$ and $z^{-1}z\sim 1_{b}$ implies commutativity of the diagram

$\begin{CD}\pi_{1}(M,a)@>{\omega_{a}}>{}>\mathbb{R}\\\ @V{T_{z}}V{}V@V{}V{{\rm
id}}V\\\<EMAIL_ADDRESS>(18)

######  Lemma 5.1.

Let $M,N$ be connected smooth manifolds, $f:M\rightarrow N$ be a local
diffeomorphism, $a\in M$, $b=f(a)$ and
$f_{*}:\pi_{1}(M,a)\rightarrow\pi_{1}(N,b)$ be an induced homeomorphism. Then
the following diagram is commutative

$\begin{CD}\pi_{1}(M,a)@>{\omega_{a}}>{}>\mathbb{R}\\\ @V{f_{*}}V{}V@V{}V{{\rm
id}}V\\\<EMAIL_ADDRESS>(19)

###### Proof.

Let $[x]\in\pi_{1}(M,a)$, $X_{1},\dots,X_{n}$ be continuous vector fields
along $x$, linearly independent at each point $x(t)$, and the equality (17) is
satisfied. Let $y=f\circ{x}$ and $Y_{i}(t)=df_{x(t)}(X_{i}(t))$ for every
$i=1,\dots,n$ and $t\in{I}$. Then $[y]\in\pi_{1}(N,b)$, $[y]=f_{*}([x])$ and
$Y_{1},\dots,Y_{n}$ are continuous vector fields along the loop $y$. According
to the condition, $df_{x(t)}:T_{x(t)}M\rightarrow T_{y(t)}N$ are isomorphisms.
Therefore $Y_{1}(t),\dots,Y_{n}(t)$ are linearly dependent for all $t\in{I}$.
But

$Y_{i}(1)=df_{a}(X_{i}(1))=df_{a}(a_{i}^{j}X_{j}(0))=a_{i}^{j}df_{a}(X_{j}(0))=a_{i}^{j}Y_{j}(0).$

from (17) and the linearity of the differential $df_{a}:T_{a}M\rightarrow
T_{b}N$. Thus, $\omega_{b}([y])={\rm sign}\det{A}=\omega_{a}([x])$. ∎

######  Lemma 5.2.

Let $M$ be a non-orientable connected smooth manifold and $f:M\rightarrow M$
be a diffeomorphism. Then there exists a connected smooth orientable manifold
$\bar{M}$, a smooth two-fold cover $p:\bar{M}\rightarrow M$ and a
diffeomorphism $\bar{f}:\bar{M}\rightarrow\bar{M}$ for which the diagram is
commutative

$\begin{CD}\bar{M}@>{\bar{f}}>{}>\bar{M}\\\ @V{p}V{}V@V{}V{p}V\\\
<EMAIL_ADDRESS>(20)

###### Proof.

Let $a\in M$. Then ${\rm ker}\,\omega_{a}$ is the normal divisor of the group
$\pi_{1}(M,a)$. By the theorem of the existence of covers, there will be a
connected smooth manifold $\bar{M}$, a regular smooth cover
$p:\bar{M}\rightarrow M$ and a point $u\in\bar{M}$ such that $p(u)=a$ and the
induced homomorphism $p_{*}^{u}:\pi_{1}(\bar{M},u)\rightarrow\pi_{1}(M,a)$ has
the image ${\rm im}\,{p}_{*}^{u}={\rm ker}\,\omega_{a}$. As the manifold $M$
is non-orientable then $\pi_{1}(M,a)/{\rm ker}\,\omega_{a}\cong{G}$. Therefore
$p$ is a two-fold covering. As
$p_{*}^{u}:\pi_{1}(\bar{M},u)\rightarrow\ker\omega_{a}$ is an isomorphism then
by Lemma 5.1 we get $\ker\omega_{u}=\pi_{1}(\bar{M},u)$. That means $\bar{M}$
is an orientable manifold.

Let $b=f(a)$ and $v\in{p}^{-1}(b)$. As the manifold $\bar{M}$ is connected
then there is a path $\bar{z}:I\rightarrow\bar{M}$ with the starting in
$\bar{z}(0)=u$ and the end in $\bar{z}(1)=v$. Let $z=p\circ\bar{z}$. Then
$z(0)=a$, $z(1)=b$ and

${\rm im}\,{p}_{*}^{v}=T_{z}({\rm im}\,{p}_{*}^{u}).$ (21)

As $T_{z}:\pi_{1}(M,a)\rightarrow\pi_{1}(M,b)$ is an isomorphism then (18)
implies

$\ker\omega_{b}=T_{z}(\ker\omega_{a}).$ (22)

Finitely, as $f_{*}:\pi_{1}(M,a)\rightarrow\pi_{1}(M,b)$ is an isomorphism,
(19) implies the equality

$\ker\omega_{b}=f_{*}(\ker\omega_{a}).$ (23)

If follows from (21), (22), (23) and the equality ${\rm im}\,{p}_{*}^{u}={\rm
ker}\,\omega_{a}$ that

${\rm im}\,(f\circ{p})_{*}^{u}=f_{*}({\rm im}\,{p}_{*}^{u})=f_{*}({\rm
ker}\,\omega_{a})={\rm ker}\,\omega_{b}=T_{z}({\rm
ker}\,\omega_{a})=T_{z}({\rm im}\,{p}_{*}^{u})={\rm im}\,{p}_{*}^{v}.$

According to a theorem from the theory of covering, in such a situation there
is a map $\bar{f}:\bar{M}\rightarrow\bar{M}$ such that $\bar{f}(u)=v$ and the
diagram (20) is commutative. This mapping is uniquely defined and is smooth.
Similarly, it is proved that for the inverse diffeomorphism
$f^{-1}:M\rightarrow{M}$ there is a smooth map
$\overline{f^{-1}}:\bar{M}\rightarrow\bar{M}$ such that
$\overline{f^{-1}}(v)=u$ and the following diagram is commutative

$\begin{CD}\bar{M}@>{\overline{f^{-1}}}>{}>\bar{M}\\\ @V{p}V{}V@V{}V{p}V\\\
<EMAIL_ADDRESS>(24)

Adding (24) to (20) on the right and on the left, we get the equality
$\overline{f^{-1}}\circ\bar{f}={\rm id}$ and
$\bar{f}\circ\overline{f^{-1}}={\rm id}$. Therefore
$\overline{f^{-1}}=\bar{f}^{-1}$ and $\bar{f}$ is a diffeomorphism. ∎

######  Lemma 5.3.

Let $M$ be a smooth closed non-orientable connected 3-manifold and
$W^{1},\,W^{2}\subset M$ be immersions of open balls $D^{1},\,D^{2}$
accordingly, such that $\mathop{\rm Ind}\nolimits_{x}(W^{1},W^{2})=\mathop{\rm
Ind}\nolimits_{y}({W}^{1},{W}^{2})$ for every points $x,\,y\in(W^{1}\cap
W^{2})$. If $p:\bar{M}\rightarrow{M}$ is an oriented double covering then
$\bar{W}^{1}=p^{-1}(W^{1}),\,\bar{W}^{2}=p^{-1}(W^{2})$ be immersions of two
copies of open balls $D^{1},\,D^{2}$ accordingly,
$\bar{W}^{1}=\bar{W}^{1}_{1}\sqcup\bar{W}^{1}_{2}$,
$\bar{W}^{2}=\bar{W}^{2}_{1}\sqcup\bar{W}^{2}_{2}$, and $\mathop{\rm
Ind}\nolimits_{\bar{x}}(\bar{W}^{1}_{i},\bar{W}^{2}_{j})=\mathop{\rm
Ind}\nolimits_{\bar{y}}(\bar{W}^{1}_{i},\bar{W}^{2}_{j})$ for every points
$\bar{x},\,\bar{y}\in(\bar{W}^{1}_{i}\cap\bar{W}^{2}_{j})$, $i,\,j=1,2$.

###### Proof.

Consider a tubular neighborhood $U^{k}$ of the submanifolds $W^{k}$. Since the
open subsets $U^{k}\subset{M}$, $k=1,2$, are contractible, they are regular
covered neighborhoods. That is
$p^{-1}(U^{k})=\bar{U}^{k}_{1}\cup\bar{U}^{k}_{2}$, where
$\bar{U}^{k}_{1}\cap\bar{U}^{k}_{2}=\emptyset$ and
$p|_{\bar{U}^{k}_{i}}:\bar{U}^{k}_{i}\rightarrow{U^{k}}$ are diffeomorphisms,
$i=1,2$. Then the sets $\bar{U}^{k}_{i}$ are tubular neighborhoods of smooth
submanifolds $\bar{W}^{k}_{i}\subset\bar{M}$, and the differences
$\bar{U}^{2}_{i}\setminus\bar{W}^{2}_{i}$ consist of the connected components
$\bar{U}^{2}_{i+}$ $\bar{U}^{2}_{i-}$.

Let
$\bar{\sigma}_{i}:\bar{U}^{2}_{i+}\cup\bar{U}^{2}_{i-}\rightarrow\mathbb{Z}$
be a function such that $\bar{\sigma}(\bar{x})=1$ for
$\bar{x}\in\bar{U}^{2}_{i+}$ and $\bar{\sigma}(\bar{x})=0$ for
$\bar{x}\in\bar{U}^{2}_{i-}$. As
$\bar{W}^{1}_{i}=(p|_{\bar{W}^{1}_{i}})^{-1}(J^{1}(D^{1}))$ then the
intersection index in $\bar{x}\in(\bar{W}^{1}_{i}\cap\bar{W}^{2}_{j})$ is
equal to $\mathop{\rm
Ind}\nolimits_{\bar{x}}(\bar{W}^{1}_{i},\bar{W}^{2}_{j})=\bar{\sigma}(t+\delta)-\bar{\sigma}(t-\delta)$,
where $\delta$ is a small enough positive number. Then $\mathop{\rm
Ind}\nolimits_{x}(W^{1},W^{2})=\mathop{\rm
Ind}\nolimits_{\bar{x}}(\bar{W}^{1}_{i},\bar{W}^{2}_{j})$ and $\mathop{\rm
Ind}\nolimits_{y}(W^{1},W^{2})=\mathop{\rm
Ind}\nolimits_{\bar{y}}(\bar{W}^{1}_{i},\bar{W}^{2}_{j})$. So if $\mathop{\rm
Ind}\nolimits_{x}(W^{1},W^{2})=\mathop{\rm Ind}\nolimits_{y}({W}^{1},{W}^{2})$
for every points $x,\,y\in(W^{1}\cap W^{2})$ then $\mathop{\rm
Ind}\nolimits_{\bar{x}}(\bar{W}^{1}_{i},\bar{W}^{2}_{j})=\mathop{\rm
Ind}\nolimits_{\bar{y}}(\bar{W}^{1}_{i},\bar{W}^{2}_{j})$ for every points
$\bar{x},\,\bar{y}\in(\bar{W}^{1}_{i}\cap\bar{W}^{2}_{j})$, $i,\,j=1,2$. ∎

##  6 Example of a diffeomorphism with a non-orientable expanding
2-dimensional attractor

Let us construct an example of an $\Omega$-stable diffeomorphism of a closed
connected 3-manifold $M^{3}$ the non-wandering set of which consists of
trivial sources, saddles, and a non-orientable expanding 2-dimensional
attractor $\Lambda$.

We will start with hyperbolic toral automorphism
$L_{A}:\mathbb{T}^{3}\to\mathbb{T}^{3}$ induced by linear map of
$\mathbb{R}^{3}$ with a hyperbolic matrix $A\in GL(3,\mathbb{Z})$, eigenvalues
$\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$ of which such that
$0<\lambda_{1}<1<\lambda_{2}\leqslant\lambda_{3}$. The involution
$J:\mathbb{T}^{3}\to\mathbb{T}^{3}$ defined by the formula $J(x)=-x\pmod{1}$
has 8 fixed points in the 3-torus of the form $(a,b,c)$, where
$a,b,c\in\\{0,\frac{1}{2}\\}$. Notice that these points are also fixed for
$L_{A}^{k}$ for some $k\in\mathbb{N}$. Let us ‘‘blow up’’ these points like to
the classical Smale surgery and such that the surgery commutes with the
involution. We will obtain generalized DA-diffeomorphism
$f_{GDA}:\mathbb{T}^{3}\to\mathbb{T}^{3}$ with 8 fixed sources $\alpha_{i}$,
$i\in\\{1,2,\ldots,8\\}$ and one 2-dimensional expanding attractor obtained
from the diffeomorphism $L_{A}^{k}$.

After that we will remove all sources and factorize the basin of the attractor
to obtain a new manifold $\tilde{M}$, i.e.
$\tilde{M}=(\mathbb{T}^{3}\setminus\bigcup\limits^{8}_{i=1}{\alpha_{i}})/_{x\sim-x}$.
The natural projection
$p:\mathbb{T}^{3}\setminus\bigcup\limits^{8}_{i=1}{\alpha_{i}}\to\tilde{M}$ is
a 2-fold cover. As $f_{GDA}J=Jf_{GDA}$ then $f_{GDA}$ is projected to
$\tilde{M}$ by the diffeomorphism
$\tilde{f}=pf_{GDA}p^{-1}:\tilde{M}\to\tilde{M}$ with one 2-dimensional
expanding attractor $\Lambda$ and $\tilde{M}$ is its basin. The set
$\tilde{M}\setminus\Lambda$ consists of 8 connected components $\tilde{N}_{i}$
each of which is diffeomorphic to $\mathbb{R}P^{2}\times\mathbb{R}$, where
$\mathbb{R}P^{2}$ is the real projective plane.

To obtain a fundamental domain $\tilde{D}_{i}$ of $\tilde{f}|_{\tilde{N}_{i}}$
we can consider a local coordinates $(x,y,z):U_{i}\to\mathbb{R}^{3}$ in a
neighborhood $U_{i}$ of $\alpha_{i}$ in which the diffeomorphism $f_{GDA}$ has
a form $f_{GDA}(x,y,z)=(2x,2y,2z)$. A fundamental domain of
$f_{GDA}|_{W^{u}_{\alpha_{i}}\setminus{\\{\alpha_{i}\\}}}$ is
$D_{i}=\\{(x,y,z)\in\mathbb{R}^{3}\,|\,1\leqslant x^{2}+y^{2}+z^{2}\leqslant
4\\}$ and then the desired fundamental domain $\tilde{D}_{i}=p(D_{i})$. By the
construction it is homeomorphic to $RP^{2}\times[0,1]$. The orbit space of
$f_{GDA}|_{W^{u}_{\alpha_{i}}\setminus{\\{\alpha_{i}\\}}}$ is homeomorphic to
$S^{2}\times S^{1}$ since each orientation preserving diffeomorphism of
$S^{2}$ is homotopic to identity. Then the orbit space
${\tilde{N}}_{i}/{\tilde{f}}$ can be obtained as $S^{2}\times
S^{1}|_{\tilde{J}}$, where $\tilde{J}$ is involution of $S^{2}\times S^{1}$
induced by $J$. Since ${\tilde{N}}_{i}/{\tilde{f}}$ is non-orientable, it
follows from [24] that ${\tilde{N}}_{i}/{\tilde{f}}$ is either
$S^{2}\tilde{\times}S^{1}$, $RP^{2}\times S^{1}$, or $RP^{3}\\#RP^{3}$. The
orbit space ${\tilde{N}}_{i}/{\tilde{f}}$ can also be obtained from the
fundamental domain $\tilde{D}_{i}$ as a mapping torus
$RP^{2}\times[0,1]|_{(x,0)\sim(\tilde{f}(x),1)}$. Hence a fundamental group of
the orbit space
$\pi_{1}({\tilde{N}}_{i}/{\tilde{f}})=\mathbb{Z}_{2}\rtimes_{\tilde{f}}\mathbb{Z}$
and then it can be only $RP^{2}\times S^{1}$.

Consider a gradient-like diffeomorphism
$g_{1}:\mathbb{R}P^{2}\to\mathbb{R}P^{2}$ with exactly 3 fixed points: a
source $\alpha$, a sink $\omega$ and a saddle $\sigma$ (see Fig. 6).

Figure 6: Diffeomorphism $g$ on the projective plane

Let $g_{2}:\mathbb{R}\to\mathbb{R}$ be a diffeomorphism given by the formula
$g_{2}(x)=2x$ and
$g(w,x)=(g_{1}(w),g_{2}(x)):\mathbb{R}P^{2}\times\mathbb{R}\to\mathbb{R}P^{2}\times\mathbb{R}$.
Let us denote $N_{1},N_{2}$ the connected components of
$\mathbb{R}P^{2}\times(\mathbb{R}\setminus\\{0\\})$. Analogically with cases
with $\tilde{N}_{i}$ the orbit spaces ${N_{j}}/g$ are diffeomorphic to
$\mathbb{R}P^{2}\times\mathbb{S}^{1}$.

As ${\tilde{N}_{i}}/\tilde{f}$ are diffeomorphic to ${N_{j}}/g$ then there is
a diffeomorphism $h:\tilde{N}_{i}\to N_{j}$ conjugating $\tilde{f}$ with $g$.
Let $h_{i}:\tilde{N}_{i}\to N_{1},\,i=1,3,5,7$ and $h_{i}:\tilde{N}_{i}\to
N_{2},\,i=2,4,6,8$ be such diffeomorphisms. For
$\tilde{N}=\bigcup\limits_{i=1}^{8}\tilde{N}_{i}$ denote by
$h:\tilde{N}\to(N_{1}\sqcup N_{2})\times\mathbb{Z}_{4}$ a diffeomorphism
composed by $h_{i},\,i\in\\{1,\dots,8\\}$. Let
$\tilde{P}=\mathbb{R}P^{2}\times\mathbb{R}\times\mathbb{Z}_{4}$ and
$G:\tilde{P}\to\tilde{P}$ be a diffeomorphism composed by $g$ on every copy of
$\mathbb{R}P^{2}\times\mathbb{R}$. Finitely, let
$M^{3}=\tilde{M}\cup_{h}\tilde{P}$. Denote by $q:\tilde{M}\sqcup\tilde{P}\to
M^{3}$ the natural projection. Then the desired diffeomorphism $f:M^{3}\to
M^{3}$ coincides with the diffeomorphism $q\tilde{f}q^{-1}|_{q(\tilde{M})}$ on
$q(\tilde{M})$ and with the diffeomorphism $qGq^{-1}|_{q(\tilde{P})}$ on
$q(\tilde{P})$.

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