Regular blocks and Conley index of isolated invariant continua in surfaces ^{1}^{1}1The author is supported by the FPI grant BES2013062675 and by MINECO (MTM201230719).
Abstract
In this paper we study topological and dynamical features of isolated invariant continua of continuous flows defined on surfaces. We show that near an isolated invariant continuum the flow is topologically equivalent to a flow. We deduce that isolated invariant continua in surfaces have the shape of finite polyhedra. We also show the existence of regular isolating blocks of isolated invariant continua and we use them to compute their Conley index provided that we have some knowledge about the truncated unstable manifold. We also see that the ring structure cohomology index of an isolated invariant continuum in a surface determines its Conley index. In addition, we study the dynamics of nonsaddle sets, preservation of topological and dynamical properties by continuation and we give a topological classification of isolated invariant continua which do not contain fixed points and, as a consequence, we also classify isolated minimal sets.
2010 MSC: 34C45, 37G35, 58J20
Keywords: Conley index, Regular isolating block, Unstable manifold, fixed point, Minimal set, Nonsaddle set.
1 Introduction
In this paper we study topological and dynamical features of isolated invariant continua of continuous flows defined on surfaces. By a surface we mean a connected 2manifold without boundary. To avoid trivial cases, when we refer to an isolated invariant continuum , it will be implicit that it is a proper subset of , i.e. .
The paper is structured as follows. In Section 2 we show that near an isolated invariant continuum the flow is topologically equivalent to a flow and, as a consequence, admits a basis of neighborhoods comprised of what we call isolating block manifolds. The main result of this section is Theorem 7 which establishes that an isolated invariant continuum of a flow on a surface must have the shape of a finite polyhedron. Besides, we characterize the initial sections of the truncated unstable manifold introduced in [3]. Section 3 is devoted to prove the main results of the paper which are Theorem 12 where the existence of the socalled regular isolating blocks of isolated invariant continua on surfaces is established and Theorem 16 which establishes a complete classification of the possible values taken by the Conley index of . In particular, it is proven that the Conley of is the pointed homotopy type of a wedge of circumferences if is neither an attractor nor a repeller, the pointed homotopy type of a disjoint union of a wedge of circumferences and an external point (which is the base point) if is an attractor and the pointed homotopy of a wedge of circumferences and a closed surface if is a repeller. Both the number of circumferences in the wedge and the corresponding genus of the surface in the case of repellers are determined by the first Betti number of and the knowledge of an initial section of . The existence of regular isolating blocks plays a key role in our proof of this classification. In Section 4 we prove Theorem 19 which is a classification of the Conley index of in terms of the ring structure of the cohomology index. Finally, Section 5 is devoted to some applications of the previous results. The main results of this section are Theorem 26 and Theorem 29. Theorem 26 studies the preservation of some topological and dynamical properties by continuation. For instance, it is proven that if is a continuation of an attractor (resp. repeller) then, for each , must have a component which is an attractor (resp. repeller) with the same shape of . It is also proven that the property of being saddle is preserved by continuation for small values of the parameter and that if is a continuum for each , the property of being nonsaddle is preserved if and only if the shape is preserved. On the other hand, Theorem 29 establishes that if an isolated invariant continuum in a surface does not have fixed points it must be nonsaddle and either a limit cycle, a closed annulus bounded by two limit cycles or a Möbius strip bounded by a limit cycle. A nice consequence of this result is Corollary 30 which establishes that a minimal isolated invariant continuum in a surface must be either a fixed point or a limit cycle.
We shall use through the paper the standard notation and terminology in the theory of dynamical systems. By the omegalimit of a set we understand the set while the negative omegalimit is the set . The unstable manifold of an invariant compactum is defined as the set . Similarly the stable manifold . For us, an attractor is an asymptotically stable compactum and a repeller is an asymptotically stable compactum for the reverse flow.
We shall assume in the paper some knowledge of the Conley index theory of isolated invariant compacta of flows. These are compact invariant sets which possess a socalled isolating neighborhood, that is, a compact neighborhood such that is the maximal invariant set in , or setting
such that . We shall make use of a special type of isolating neighborhoods, the socalled isolating blocks, which have good topological properties. More precisely, an isolating block is an isolating neighborhood such that there are compact sets , called the entrance and exit sets, satisfying

,

for every there exists such that
and for every there exists such that ,

for every there exists such that
and for every there exists such that .
These blocks form a neighborhood basis of in . Associated to an isolating block there are defined two continuous functions
given by
These functions are known as the exit time and the entrance time respectively. We shall also use the notation and . If the flow is differentiable, the isolating blocks can be chosen to be manifolds which contain and as submanifolds of their boundaries and such that . This kind of isolating blocks will be called isolating block manifolds. For flows defined on surfaces, the exit set of an isolating block manifold is the disjoint union of a finite number of intervals and circumferences and the same is true for the entrance set .
We also recall some dynamical concepts introduced in [3]. If is an isolated invariant set, the subset will be referred as the truncated unstable manifold of . A compact section of , i.e., a compact subset of such that for each there exists a unique such that , is said to be initial provided that . Notice that, if and are two different initial sections they are homeomorphic. The subset is the socalled initial part of the truncated unstable manifold and it turns out to be homeomorphic to the product . For instance, if is an isolating block, is an initial section and agrees with . We would like to point out that all this concepts may be dualized for the stable manifold in the obvious way.
We shall also make use of a classical result of C. Gutiérrez about smoothing of 2dimensional flows.
Theorem 1 (Gutiérrez [17]).
Let be a continuous flow on a compact twomanifold . Then there exists a flow on which is topologically equivalent to . Furthermore, the following conditions are equivalent:

any minimal set of is trivial;

is topologically equivalent to a flow;

is topologically equivalent to a flow.
By a trivial minimal set we understand a fixed point, a closed trajectory or the whole manifold if is the dimensional torus and is topologically equivalent to an irrational flow.
We adopt in the paper a topological viewpoint close to the one adopted, for example, in the papers [38, 37, 43]. Homotopy and homology theory play an important role dealing with the Conley index theory. In particular, we shall use through the paper Čech cohomology and singular homology and cohomology , all of them with coefficients. We recall that Čech and singular homology theories agree when working with spaces with good local behaviour such as manifolds, polyhedra and CWcomplexes. We define the dimensional Betti number of a topological space , as the dimension of the vector space . The Euler characteristic , when defined, is the alternated sum of the Betti numbers. The Conley index of an isolated invariant set is defined as the homotopy type of the pair where is any isolating block of . A crucial fact concerning the definition is, of course, that this homotopy type does not depend on the particular choice of . We will also make use of the cohomology index defined as . We refer the reader to [7, 8, 26, 33] for information about the Conley index theory and to [43, 48] to see recent applications of the Conley index techniques to some problems in ecology. There is a form of homotopy which has proved to be the most convenient for the study of the global topological properties of the invariant spaces involved in dynamics, namely the shape theory introduced and studied by Karol Borsuk. We do not use shape theory in this paper. However, it is convenient to know that some topological properties of continua in surfaces have a very nice interpretation in terms of shape. Two compacta are said to be of the same shape if they have the same homotopy type in the homotopy theory of Borsuk (or shape theory). The following results from [36, 34] will be useful in the sequel.
Theorem 2.
Let be a compactum contained in the interior of a compact manifold . If the inclusion induces isomorphisms for , then it is a shape equivalence.
Corollary 3.
Let be a continuum contained in the interior of a manifold . If and is finitely generated, then has the shape of a wedge of circumferences.
Notice that if is a compact and connected 2manifold with boundary and is a subcontinuum contained in its interior, it would be enough to be an isomorphism to meet the assumptions of Theorem 2 and, hence, to ensure that the inclusion is a shape equivalence. On the other hand, if we only consider proper subcontinua contained in the interior of connected 2manifolds, Corollary 3 ensures that , when finitely generated, determines the shape of . These facts can be easily seen using Alexander duality.
We are also going to make use of the fact, proved in [36], that if is a continuum contained in the interior of a 2manifold and and are connected submanifolds of which are neighborhoods of in such that the inclusions are shape equivalences, then and are homeomorphic.
Although we do not make use of shape theory in our proofs, we may occasionally refer to these theorems and to the terminology derived from it to make it clear that some of the results can be interpreted in that context. For a complete treatment of shape theory we refer the reader to [6, 10, 23, 39]. The use of shape in dynamics is illustrated by the papers [14, 42, 18, 21, 30, 31, 35]. For information about basic aspects of dynamical systems we recommend [5, 32, 47]. We also recommend the books written by Hatcher [20] and Spanier [45] for questions regarding algebraic topology and the book [24] and the paper [29] as references about the topology of surfaces.
2 Isolating blocks in surfaces
In this section we study the structure of a flow defined on a surface near an isolated invariant continuum . In particular we will see that admits a neighborhood in which the flow topologically equivalent to a flow. From this fact we will deduce that has the shape of a finite polyhedron.
The next result states some useful properties of isolating blocks which will be exploited through the paper.
Lemma 4.
Suppose that is an isolated invariant continuum of a flow on a manifold and that is a connected isolating block manifold of . Then

Each component of must contain some component of ,

has a finite number of components, and

if is a point in and a compact neighborhood of in then, the set
is homeomorphic to the product via a homeomorphism which carries each trajectory segment to the fiber .
Proof.
Since the inclusion is a shape equivalence [21], a straightforward application of the five lemma gives that . In addition, the inclusion is also a shape equivalence (see [41]) and, reasoning as before, it follows that . On the other hand, by the strong excision property of Čech cohomology
Since and are connected, and, hence, from the long exact sequence of cohomology of the pair we get that the homomorphism
induced by the inclusion is a monomorphism. This proves a).
Consider the long exact sequence of reduced Čech cohomology of the pair
Since is a manifold, then agrees with and, hence, it is finitely generated. Thus, from the exact sequence we get that is also finitely generated. As a consequence, is finitely generated being isomorphic to . Moreover, since , the long exact sequence of the pair splits into the short exact sequence
where is the coboundary homomorphism. In addition, the groups and are finitely generated since has a finite number of components being a compact manifold and being a subgroup of the finitely generated group . Therefore, is finitely generated. This proves b).
Let and be a compact neighborhood of in . Consider for each the linear homeomorphism given by . We define as which is clearly a bijection. See that is continuous. Let and sequences in and convergent to and respectively. Then, , which by the continuity of converges to and, hence, converges to by the continuity of the flow. Therefore, is continuous. Let us see that is also continuous. Consider a sequence of points in convergent to a certain . Each is of the form and respectively, where, , and , . See that converge to and converge to . Since and are compact, we can choose subsequences and . Besides, the continuity of guarantees . But, on the other hand, . As a consequence we get that , leading to . Then, it follows that and . Indeed, suppose, arguing by contradiction, that , then, assuming that the absolute value of is greater than or equal to we would have that and, since , it follows that either or is point of internal tangency in contradiction with the definition of isolating block. It also follows that since, if not, the trajectory of would be periodic and, thus, would be a point of internal tangency. We have proved that every convergent subsequence of converge to and every convergent subsequence of converge to . As a consequence, since and are compact, and . This proves c). ∎
From now on we will focus on flows defined on surfaces. The next result is a local version of classical Gutiérrez’ Theorem. The proof is a kind of mixture of some ideas from [27] and [35].
Lemma 5.
Let be a flow defined on a surface and be an isolated invariant continuum. Then, is topologically equivalent to a flow near . Moreover, admits a basis of neighborhoods comprised of isolating block manifolds.
Proof.
We will start the proof by showing that admits a neighborhood basis comprised of surfaces with boundary. Indeed, since is a surface, we may assume without loss of generality that is (see [19]). Consider the continuous map . Now, fixed we can find a function such that (see [28, Exercise 36, p. 152]). We choose in such a way that . As a consequence, and by Sard’s Theorem [25] there exists a regular value . Then, is a compact 2manifold with boundary [25]. It is clear that is contained in the interior of since, if , . Therefore, choosing as the component of containing we have found the desired neighborhood. Since the choice of was arbitrary, the claim follows.
On the other hand, since we can find a surface neighborhood of as close to as desired, we can choose it to be an isolating neighborhood. Let be the closed surface obtained by capping each boundary component of with a disk. By the Keesling reformulation of Beck’s Theorem [22] we can obtain a flow on such that is topologically equivalent to in and is stationary in . Then, the restriction flow can be extended to a flow on by keeping all the points in fixed. Besides, the flow is topologically equivalent to a flow by Gutiérrez’ Theorem and, as a consequence, is topologically equivalent to , where is the homeomorphism which realizes the equivalence. Therefore, [8] ensures the existence of a basis of isolating block manifolds of for and, hence, for . ∎
The next proposition gives a topological characterization of the initial sections of the truncated unstable manifold of an isolated invariant continuum of a flow on a surface and, as a consequence, it also characterizes the topology of the initial part of the truncated unstable manifold.
Proposition 6.
Let be a flow defined on a surface, be an isolated invariant continuum and an initial section of the truncated unstable manifold . Then, has a finite number of connected components and each one is either an interval (possibly degenerate) or a circle. Moreover, is homeomorphic to a finite disjoint union of halfopen rays, strips and cylinders.
Proof.
By Lemma 5 we can find a connected isolating block manifold of . Besides, is homeomorphic to . Hence, Lemma 4 guarantees that it has a finite number of components. Moreover, consists of a disjoint union of finite many circumferences and closed intervals. Then, since is a compact subset of this disjoint union, it must be a finite union of points, closed intervals and circumferences as we wanted to prove. Therefore, the result follows being homeomorphic to . ∎
Theorem 7.
Let be an isolated invariant continuum of a flow on a surface. Then, has the shape of a finite polyhedron. Moreover, if is a connected isolating block manifold of ,
Proof.
Let be a connected isolating block manifold of . By Alexander duality
and the latter group must be zero since, if not, there would be a component of not meeting , which means that, given , the trajectory must be contained in since it only can leave through . This fact contradicts to be an isolating neighborhood of .
Consider the long exact sequence of reduced Čech cohomology of the pair
Therefore, the homomorphism is surjective and, since is finitely generated, being a compact manifold, so is . Thus, has the shape of a wedge of circumferences by Corollary 3 and . ∎
Corollary 8.
Let be an isolated invariant continuum of a flow on a surface. Suppose that admits an isolating block which is a disk, then has trivial shape and contains a fixed point.
Proof.
Since , Theorem 7 guarantees that and, hence, Theorem 3 ensures that has trivial shape. Let us see that must contain a fixed point. Since admits an isolating block which is a disk, this disk can be embedded into and, by the arguments presented in the proof of Lemma 5, we may assume, without loss of generality, that the flow restricted to can be extended to a flow on the whole plane. This fact allows us to use PoincaréBendixson Theorem. Choose a point , hence and either it contains a fixed point or it is a limit cycle. If is a limit cycle, it must decompose into two connected components, and, since is an open disk, the bounded component must be contained in . Thus, is an invariant disk contained in and, hence, in , and the Brouwer fixed point theorem combined with the compact character of ensure that must contain a fixed point. ∎
Remark 9.
Theorem 7 does not hold for flows on higherdimensional manifolds. For instance, consider on the vector field
where and is a function which takes the value exactly in those points which belong to the subset
and it takes the value outside a neighborhood of . The flow induced by is depicted in figure 1 and it has the set , which is known as the Hawaaian earring, as an isolated invariant set. It is clear that admits an isolating block which is a ball but, in spite of it, . In particular, does not have polyhedral shape.
This example is a particular instance of a general result from [13] which states that any finite dimensional compactum can be an isolated invariant set of a flow on some . This example also shows that in higherdimensional manifolds, given a connected isolating block manifold of an isolated invariant continuum , may be greater than . In [34] some conditions involving are used to find lower bounds of for flows on 3manifolds.
3 Regular isolating blocks and the Conley index
In this section we will see that the knowledge of the first Betti number of an isolated invariant continuum of a flow on a surface and the topology of an initial section of its truncated unstable manifold allow us to compute its Conley index, extending in this way a result of [3] about planar isolated invariant continua. For this purpose we will make use of a special kind of isolating blocks, the socalled regular isolating blocks. This kind of blocks was first introduced and studied by Easton in [11] and subsequently studied by Gierzkiewicz and Wójcik [12] and J.J. SánchezGabites [34, 36]. However, most of the known results are referred to the 3dimensional case and the more general results, which appear in [12], do not apply to the kind of isolating blocks considered here since we are dealing with a more restrictive definition of isolating block. We will dedicate part of this section to fill this gap and prove that isolated invariant continua of flows on surfaces admit a basis of regular isolating blocks.
Definition 10.
A connected isolating block manifold is said to be regular if the inlcusion is a shape equivalence.
Remark 11.
Notice that the condition for an isolating block to be regular in Definition 10 differs from the one introduced and studied in [11, 12]. However, from the considerations made in the Introduction it follows that for connected isolating block manifolds in surfaces both definitions agree. In addition, it also follows that all regular isolating blocks of the same isolated invariant continuum must be homeomorphic. This facts also hold in 3manifolds [34, 36].
Theorem 12.
If is an isolated invariant continuum of a flow on a surface, it admits a basis of regular isolating blocks.
Proof.
Let be a connected isolating block manifold of . From the proof of Theorem 7 we have that the sequence
is exact and, as a consequence, from Theorem 2, the obstruction for to be a regular block is the existence of nontrivial elements in . On the other hand, as we have seen in the proof of Lemma 4, and, by Alexander duality, we get
Notice that is finitely generated. We will construct the desired block from by cutting from it the leftover information in the following way:
Assume that is a circular component of not contained in . Each component of represents a generator of since it does not contain points of . Choose a point and a compact and connected neighborhood of in disjoint from . Notice that , being a proper nondegerate subcontinuum of the circle must be homeomorphic to the unit interval . Thus, Lemma 4 guarantees that the set
is homeomorphic to the unit square via a homeomorphism which carries each segment of trajectory to , where is a homeomorphism. Now we will perform the following operation: choose in the parabolic segments and depicted in figure 2 and let be the open region between these curves in . Then, if we consider
, it is clear from the construction that it is a connected isolating block manifold. Notice that this operation keeps unaltered. Moreover, the number of boundary components has been reduced by since the component has been joined with a component of , which lies in a different component of . As a consequence, becomes an interval, say , and has one more component than . However, must contain two points of , each one lying in a different component of and, thus, the homology group has exactly one generator less than . After performing this operation to each circular component of not contained in we obtain a connected isolating block manifold such that, all the circular components of are contained in .
We will denote by since it should not lead to confusion. Choose a component of which contains more than one component of . Then, must be an interval. Thus, each component of not containing one of the endpoints represents a generator of . Choose an orientation in and let and be the first and the second components of appeared regarding the chosen orientation. Choose a point in the interval lying between and and perform the previously described operation. We obtain in this way a new isolating block manifold in which the component has been splitted into two disjoint exit intervals, one of them containing and the other containing remaining components of which were contained in the original . Notice that is also connected since, if not, and one of the chosen components of should lie in different components of and this cannot happen. If we perform this operation until we separate all the components of (i.e. a finite number of times) we get the desired block. ∎
Definition 13.
A nonempty continuum contained in a surface is said to be orientable if it admits a basis of neighborhoods comprised of orientable surfaces. Otherwise is said to be nonorientable.
Remark 14.
From the proof of Lemma 5 it follows that any continuum in the interior of a surface has a neighborhood basis comprised of compact and connected 2manifolds with boundary. Combining this with the fact that an orientable 2manifold cannot contain a nonorientable one it follows

Every continuum contained in the interior of an orientable surface must be orientable.

An orientable continuum cannot possess a basis of neighborhoods comprised of nonorientable manifolds.

A nonorientable continuum must admit a basis of neighborhoods comprised of nonorientable surfaces.
However as the next example points out, nonorientable surfaces contain both orientable and nonorientable compact subsets.
Example 15.
Consider as the surface obtained as a connected sum of the torus with the Klein bottle (which is homeomorphic to a connected sum of four projective planes [24]). In this surface we can find two copies of as the 1skeleton of the torus and the Klein bottle summands respectively. It is clear that the one contained in the torus summand is orientable while the other is not.
Now we are ready to prove the main result of this section.
Theorem 16.
Suppose is an isolated invariant continuum of a flow defined on a surface. Let be the number of components of an initial section of the truncated unstable manifold and the number of contractible components of . Then

If is neither an attractor, nor a repeller the Conley index of is the pointed homotopy type of where and is a pointed sphere based on for

If is an attractor, and its Conley index is the pointed homotopy type of where the are pointed spheres based on and denotes a point not belonging to

If is a repeller:

If is orientable its Conley index is the pointed homotopy type of , where is a closed orientable surface of genus . The surface and all the are pointed and based on

If is nonorientable its Conley index is the pointed homotopy type of , where is a closed nonorientable surface of genus . The surface and all the are pointed and based on

Proof.
Let be a regular isolating block of . Then, given an initial section of the truncated unstable manifold , is homotopy equivalent to . Indeed, since the inclusion is a shape equivalence, the cohomology groups . But, as we have seen before and, hence, induces isomorphisms in Čech cohomology. It easily follows that and have the same homotopy type and the claim follows being homeomorphic to .
From this observation we get that has components which are intervals and circular components.
Suppose that is neither an attractor nor a repeller and let be a regular isolating block of . The block is a compact 2manifold with boundary and, since it has the same shape as it must have the homotopy type of a wedge of circumferences. Collapsing to a point an interval component of does not change the homotopy type of . Therefore, the topological space obtained by collapsing all the interval components to a single point is pointed homotopy equivalent to the wedge sum of with copies of . On the other hand, collapsing a circular component of produces the same effect on as capping the boundary component with a disk. Then, the topological space obtained by collapsing to a point all the circle components is pointed homotopy equivalent to a wedge sum of circumferences with the manifold obtained after capping boundary components with disks. Thus, since is neither empty nor the whole the Conley index of must be the pointed homotopy type of a wedge sum of a compact and connected 2manifold with boundary with some circumferences. Hence, it must be pointed homotopy equivalent to a wedge of circumferences. To determine the number of circumferences on the wedge we compute the Euler characteristic of . Since agrees with and is a union of intervals and circumferences it follows
and, hence, . This proves i).
If is an attractor it admits a positively invariant isolating neighborhood and, hence, . Thus, if is a regular isolating block it must have empty exit set. As a consequence, the effect of collapsing its exit set to a point is the same as making the disjoint union of with a singleton not contained in . This proves ii).
Suppose that is a repeller. Then, given a regular isolating block of , must be the whole boundary which is comprised of connected components. The space obtained after collapsing the whole boundary of to a point is pointed homotopy equivalent to the wedge sum of circumferences with the surface obtained after capping all the boundary components of with disks. This surface is orientable if and only if is orientable. Indeed, if is orientable it admits a basis of neighborhoods comprised of orientable 2manifolds with boundary. As a consequence, admits an orientable regular isolating block. If is nonorientable the same argument shows that admits a nonorientable regular block. Let us compute the genus of , the closed surface obtained after capping with a disk each boundary component of . Since, has exactly components, using the fact that
it easily follows that
On the other hand,
This proves iii). ∎
Remark 17.
Notice that in the item iii) of Theorem 16 the genus of the surface which appears as a direct summand must be less than or equal to than the genus of the phase space . This can be easily seen using the MayerVietoris sequence.
4 The cohomology index
The aim of this section is to study the cohomology index of an isolated invariant continuum of a flow on a surface and its relations with the Conley index. Since cohomology groups are easier to compute than homotopy type it is interesting to study to what extent the cohomology index determines the Conley index.
Example 18.
Let be an orientable surface of genus greater than or equal to and consider two flows and on having isolated invariant sets and respectively whose local dynamics are depicted in figures 3 and 4. The Conley indices of and are the pointed homotopy type of and . Then, their cohomology indices agree being
However, these spaces are not homotopy equivalent. This can be seen using the ring structure of and . As rings
Let , be elements of . Then , where is the generator of , . As a consequence, since by the direct sum structure of and , .
On the other hand, if , are the standard generators of , generates . Therefore, the rings and are not isomorphic and .
The previous example shows that the knowledge of the groups which conform the cohomology index is not enough to know the Conley index. We will see that in spite of it, the cohomology ring determines the Conley index.
Given a topological space with it is possible to define a bilinear form
given by . This form determines the cohomology ring when is a closed surface. The rank is defined as the rank of any matrix representing . This number is well defined since two matrices representing must be congruent.
Theorem 19.
Suppose that is an isolated invariant continum of a flow on a surface. Then, the cohomology ring determines its Conley index. In particular,

If , then is neither an attractor nor a repeller and its Conley index is the pointed homotopy type of , where agrees with .

If then is an attractor and its Conley index is the pointed homotopy type of where agrees with . In particular, has the shape of circumferences.

If then is a repeller and:

If for each the Conley index of is the pointed homotopy type of , where and .

If there exists such that the Conley index of is the pointed homotopy type of , where and .
In both cases the number of components of an initial section of the truncated unstable manifold is and has the shape of circumferences.

Proof.
Suppose that , then, must be connected and it cannot contain any closed surface as a wedge summand. Thus, Theorem 16 ensures that it cannot be an attractor or a repeller and must be the homotopy type of a wedge of circumferences. It is clear that the number of circumferences in the wedge is determined by . This proves i).
Let us assume that . Then is not connected and by Theorem 16 it must be an attractor. Moreover, must have the homotopy type of the union of a wedge of circumferences and an exterior point. As before determines the number of circumferences in the wedge.
To prove iii) assume that , then Theorem 16 guarantees that is a repeller. Moreover, must contain a closed connected surface as a wedge summand. This surface is orientable (and hence is orientable) if and only if, given any element , . This is a straightforward consequence of the cohomology ring structure of closed surfaces (See [20]).
Suppose that is orientable. Then is the pointed homotopy type of . Let us show that is exactly . By [20] we have that
(1) 
as rings. Choose the basis of where is the standard basis of and each is the generator of for each . Let be the generator of , then
and , for each . Besides, (1) ensures that for each and each . Therefore, the matrix associated to the bilinear form with respect to the chosen basis takes the form