The continuity of Darboux injections between manifolds
aa r X i v : . [ m a t h . GN ] M a r THE CONTINUITY OF DARBOUX INJECTIONS BETWEEN MANIFOLDS
IRYNA BANAKH AND TARAS BANAKH
Abstract.
We prove that an injective map f : X → Y between metrizable spaces X, Y iscontinuous if for every connected subset C ⊂ X the image f ( C ) is connected and one of thefollowing conditions is satisfied: • Y is a 1-manifold and X is compact and connected; • Y is a 2-manifold and X is a closed 2-manifold; • Y is a 3-manifold and X is a rational homology 3-sphere.This gives a partial answer to a problem of Willie Wong, posed on Mathoverflow. Introduction
In [11] Willie Wong asked the following intriguing and still open
Problem 1.1.
Is each bijective Darboux map f : R n → R n a homeomorphism? We recall that a map f : X → Y between topological spaces is Darboux if for any connectedsubspace C ⊂ X the image f ( C ) is connected. Injective (bijective) Darboux functions will becalled Darboux injections (resp.
Darboux bijections ). A bijection of a space onto itself will becalled a self-bijection .Wong’s Problem 1.1 was motivated by the following old result of Pervine and Levine [9] whogeneralized an earlier result of Tanaka [10].
Theorem 1.2 (Tanaka–Pervine–Levine) . A bijective map f : X → Y between semilocally-connected Hausdorff topological spaces is a homeomorphism if and only if both functions f and f − are Darboux. A topological space X is semilocally-connected if each point x ∈ X has a base of neighborhoods O x ⊂ X whose complements X \ O x have only finitely many connected components. Since theEuclidean spaces are semilocally-connected, Theorem 1.2 implies that a bijective map f : R n → R m is a homeomorphism if both functions f and f − are Darboux.We do not know the answer to Wong’s Problem 1.1 even for n = 2, but in order to put thisproblem into a wider perspective, we ask a more general Problem 1.3.
Recognize pairs of topological spaces
X, Y for which every Darboux injection f : X → Y is continuous. In this paper we shall give a partial answer to Problem 1.3 for Darboux injections betweenmanifolds of dimension n ≤ X is called • an n -manifold if every point x ∈ X has an open neighborhood O x , homeomorphic to anopen subset of the half-space R n + := { ( x , . . . , x n ) ∈ R n : x ≥ } ; • a closed n -manifold if X is compact and every point x ∈ X has an open neighborhood O x ,homeomorphic to R n ; • a ( closed ) manifold if it is a (closed) n -manifold for some n ∈ N ; • a rational homology n -sphere if X is a closed manifold with singular homology groups H k ( X ; Q ) ≈ H k ( S n ; Q ) for all k ≥ S n := { x ∈ R n +1 : k x k = 1 } stands for the n -dimensional sphere in the Euclidean space R n +1 , and ≈ denotes the isomorphism of groups. Mathematics Subject Classification.
Key words and phrases.
Connected set, homeomorphism, Darboux function, manifold.
In Proposition 6.5 we prove that a connected closed 3-manifold X is a rational homology 3-sphere if and only if H ( X ; Q ) = 0 if and only if the homology group H ( X ) is finite if andonly if the fundamental group π ( X ) has finite abelianization. According to the Poincar´e Con-jecture (proved to be true by Grigory Perelman), each connected closed 3-manifold with trivialfundamental group π ( X ) is homeomorphic to the 3-dimensional sphere. On the other hand,there are infinitely many topologically non-homeomorphic rational homology 3-spheres, see theMO-discussion at https://mathoverflow.net/q/311063 .The following theorem is a main result of this paper. Theorem 1.4.
A Darboux injection f : X → Y between metrizable spaces is continuous if one ofthe following conditions is satisfied: (1) Y is a -manifold and X is compact and connected; (2) Y is a -manifold and X is a closed -manifold; (3) Y is a -manifold and X is a rational homology -sphere. This theorem follows from Corollaries 2.9, 5.3, and 6.6, proved in Sections 2, 5 and 6, re-spectively. In fact, our results hold more generally for Darboux injections into generalizations ofmanifolds, called varieties. Varieties are introduced and studied in Section 3.Theorem 1.4 has the following corollary, related to Wong’s Problem 1.1.
Corollary 1.5.
Any Darboux self-bijection f : S n → S n of the sphere of dimension n ≤ is ahomeomorphism. Another corollary of Theorem 1.4 establishes a dimension-preserving property of Darboux in-jections between manifolds.
Corollary 1.6.
Let f : X → Y be a Darboux injection between manifolds. If dim( Y ) ≤ , then dim( X ) ≤ dim( Y ) .Proof. Assuming that dim( X ) > dim( Y ), we can find subset S ⊂ X , homeomorphic to thedim( Y )-dimensional sphere. Denote by X ′ the connected component of X containing S . Since f is Darboux, the image f ( X ′ ) is connected and hence is contained in some connected component Y ′ of Y . Applying Theorem 1.4, we conclude that the restriction f ↾ S : S → Y ′ is continuous. By thecompactness of S , the map f ↾ S is a topological embedding and its image f ( S ) is a closed subsetof Y ′ . By the Invariance of Domain [5, 2B.3], the image f ( S ) is open in Y ′ . Being connected, thespace Y ′ coincides with its open-and-closed subset f ( S ). Then f ( S ) ⊂ f ( X ′ ) ⊂ Y ′ = f ( S ) implies X ′ = S (by the injectivity of f ) and hence dim( X ) = dim( X ′ ) = dim( S ) = dim( Y ′ ) = dim( Y ),which contradicts our assumption. (cid:3) We do not know if the triviality of the homology group H ( X ; Q ) is essential in Theorem 1.4(3).Also our technique does not allow to generalize Theorem 1.4 to Darboux injections between n -manifolds of dimension n >
3. So we ask two problems.
Problem 1.7.
Is each Darboux self-bijection of the -dimensional torus a homeomorphism? Problem 1.8.
Is each Darboux self-bijection of the -dimensional sphere a homeomorphism? Darboux injections into 1-manifolds
In this section we prove Theorem 1.4(1). This will be done in Corollary 2.9 after long preliminarywork made in Lemmas 2.1–2.8.It will be convenient to identify the circle S and the sphere S with the one-point compactifi-cations ¯ R and ¯ C of the real line R and the complex plane C , respectively.Corollary 2.9 will be derived from a series of lemmas, the first of which is well-known in Con-tinuum Theory as the Boundary Bumping Theorem [8, 5.4]. Lemma 2.1.
Let X be a compact connected space, U ⊂ X be a non-empty open set, not equal to X , and C be a connected component of U . Then the closure ¯ C of C in X intersects X \ U . Lemma 2.2.
Any Darboux injection f : R → R is a topological embedding. HE CONTINUITY OF DARBOUX INJECTIONS BETWEEN MANIFOLDS 3
Proof.
We shall show that the map f is open. Indeed, for any point x ∈ R and any ε > A := ( x − ε, x ), B := ( x, x + ε ) and C := ( x − ε, x + ε ) = ( x − ε, x ) ∪{ x }∪ ( x, x + ε ).By the Darboux property of f , the sets f ( A ), f ( A ) ∪{ f ( x ) } , f ( B ) and f ( B ) ∪{ f ( x ) } are connectedand non-empty. Since f ( A ) and f ( B ) are disjoint, the union f ( A ) ∪ { f ( x ) } ∪ f ( B ) = f ( C ) is aneighborhood of the point f ( x ) in the real line. So, the map f is open and hence the map f − : f ( R ) → R is continuous. The set f ( R ), being open and connected in R , is homeomorphic to R . Then the map f − : f ( R ) → R , being continuous and monotone, is a homeomorphism. (cid:3) Lemma 2.3.
Any Darboux injection f : [0 , → R is a topological embedding.Proof. Consider the open interval I := (0 , f ↾ I is a topologicalembedding. In particular, it is monotone. We lose no generality assuming that f ↾ I is increasing.Observing that for every x ∈ I the sets (0 , x ) and [0 , x ) are connected, we can prove that f (0) =inf f ( I ) and f is continuous at 0. So, f is a topological embedding. (cid:3) By analogy we can prove
Lemma 2.4.
Any Darboux injection f : [0 , → R is a topological embedding. Lemma 2.5.
Any Darboux injection f : X → R of a path-connected Hausdorff space X is atopological embedding.Proof. Consider four possible cases.1. Assume that inf f ( X ) and sup f ( X ) belong to f ( X ). In this case we can find points a, b ∈ X such that f ( a ) = inf f ( X ) and f ( b ) = sup f ( X ). Since the space X is path-connected, thereexists a subspace I ⊂ X that contains the points a, b and is homeomorphic to the closed interval[0 , f ( I ) = [ f ( a ) , f ( b )] = f ( X ) and hence X = I . By Lemma 2.4, f ↾ I is a topologicalembedding.2. Assume that inf f ( X ) ∈ f ( X ) but sup f ( X ) / ∈ f ( X ). In this case we can choose a point a ∈ X with f ( a ) = inf f ( X ) and a sequence of points { b n } n ∈ ω ⊂ X such that ( f ( b n )) n ∈ ω is anincreasing sequence of real numbers with lim n →∞ f ( b n ) = b ∞ := sup f ( X ).Since X is path-connected, for any n ∈ ω there exists a continuous injective map γ n : [0 , → X such that γ n (0) = a and γ n (1) = b n . Put I n := γ n ([0 , f ↾ I n : I n → R is a topological embedding and hence f ( I n ) = [ f ( a ) , f ( b n )]. The continuity of therestrictions f − ↾ [ f ( a ) , f ( b n )] implies the continuity of the map f − : f ( X ) → X . To see that f iscontinuous, take any point x ∈ X and find n ∈ ω such that f ( b n ) > x . The continuity of the map f − implies that the set X \ I n = f − (( b n , b ∞ )) is path-connected. Assuming that this set contains x in its closure, we would conclude that f ( { x } ∪ ( X \ I n )) = { f ( x ) } ∪ ( b n , b ∞ ) is connected, whichis not true. Consequently, I n is a neighborhood of x in X and the continuity of f ↾ I n implies thecontinuity of f at x .3. By analogy we can consider the case inf f ( X ) / ∈ f ( X ) and sup f ( X ) ∈ f ( X ).4. Finally, we consider the case inf f ( X ) / ∈ f ( X ) and sup f ( X ) / ∈ f ( X ). In this case wecan choose two sequences of points ( a n ) n ∈ ω and ( b n ) n ∈ ω in X such that a < b , the sequence( f ( a n )) n ∈ ω decreases to a ∞ := inf f ( X ) and ( f ( b n )) n ∈ ω increases to b ∞ := sup f ( X ). For every n ∈ ω choose an injective continuous map γ n : [0 , → X such that γ n (0) = a n and γ n (1) = b n .By Lemma 2.4, for the interval I n := γ ([0 , f ↾ I n is a topological embedding.Then f ( I n ) = [ a n , b n ]. The continuity of the restrictions f − ↾ [ f ( a n ) , f ( b n )] implies the continuityof the map f − : f ( X ) → X . To see that f is continuous, take any point x ∈ X and find n ∈ ω such that f ( a n ) < x < f ( b n ). The continuity of the map f − implies that the sets f − (( b n , b ∞ ))and f − (( a ∞ , a n )) are path-connected. Assuming that one of these sets contains x in its closure,we would conclude that one of the sets ( a ∞ , a n ) ∪ { f ( x ) } or { f ( x ) } ∪ ( b n , b ω ) is connected, whichis not true. Consequently, I n is a neighborhood of x in X and the continuity of f ↾ I n implies thecontinuity of f at x . (cid:3) Lemma 2.6.
Assume that f : X → R is a Darboux injection of a compact connected space X tothe circle ¯ R . Then for any point x ∈ X the space X \ { x } has at most two connected components.Moreover, for any connected component C of X \ { x } we have x ∈ C and f ( x ) ∈ f ( C ) . IRYNA BANAKH AND TARAS BANAKH
Proof.
Let C be a connected component of X \ { x } . By Lemma 2.1, x ∈ C , which implies thatthe set C ∪ { x } is connected. Since the map f is Darboux, the sets f ( C ) and f ( C ) ∪ f ( x ) areconnected, which implies that f ( x ) ∈ f ( C ).Assuming that X \ { x } contains three pairwise disjoint connected components C , C , C , weconclude that f ( C ), f ( C ), f ( C ) are pairwise disjoint connected sets in the circle ¯ R such that f ( x i ) ∈ T i =1 f ( C i ). But the circle cannot contain such three pairwise disjoint connected sets. (cid:3) Lemma 2.7.
If a connected compact Hausdorff space X admits a Darboux injection f : X → ¯ R to the circle, then X is locally connected.Proof. Without loss of generality, we can assume that the space X contains more than one point.Given any point x ∈ X and open neighborhood O x ⊂ X of x , it suffices to find a closed connectedneighborhood C x ⊂ ¯ O x of x . Replacing O x by a smaller neighborhood, we can assume that itsclosure ¯ O x does not contain some point x ∈ X .Fix any compact neighborhood K x ⊂ O x of x . Let C be the family of connected components of¯ O x that intersect the compact set K x . By Lemma 2.1, each set C ∈ C intersects the set X \ O x .We claim that the family C is finite. To derive a contradiction, assume that C is infinite. ByLemma 2.6, the space X \ { x } ⊃ ¯ O x has at most two connected components. Replacing C bya smaller infinite subfamily, we can assume that each set C ∈ C belongs to the same connectedcomponent X ′ of the space X \ { x } . Since X ′ is closed in X \ { x } and x / ∈ ¯ O x , the space X ′ ∩ ¯ O x is compact.Consider the hyperspace K ( X ′ ∩ ¯ O x ) of non-empty closed subsets of ¯ O x , endowed with theVietoris topology. It is well-known [2, 3.12.27] that the space K ( X ′ ∩ ¯ O x ) is compact and Hausdorff.Then the infinite set C has an accumulation point C ∞ in the compact Hausdorff space K ( X ′ ∩ ¯ O x ).Since each set C ∈ C is connected and meets the sets K x and X \ O x , so does the set C ∞ . Thisimplies that C ∞ is a connected set containing more than two points. Since the injective map f : X → ¯ R is Darboux, the image f ( C ∞ ) is a connected subset of ¯ R \ { f ( x ) } , containing morethan one point. Then there exists a point c ∈ C ∞ such that the set f ( C ∞ ) is a neighborhood ofthe point f ( c ) in ¯ R .Taking into account that f ( C ∞ ) ⊂ f ( X ′ ) ⊂ ¯ R \ { f ( x ) } is a neighborhood of f ( c ), we concludethat the space f ( X ′ ) \ { f ( c ) } is disconnected and so is the space X ′ \ { c } (as f is Darboux).By Lemma 2.6, the space X ′ \ { c } has exactly two connected components U, V such that f ( c ) ∈ f ( U ) ∩ f ( V ). The set f ( C ∞ ), being a neighborhood of f ( c ), intersects both sets f ( U ) and f ( V ).So, we can choose two points u ∈ U ∩ C ∞ and v ∈ V ∩ C ∞ . Since C ∞ is an accumulation pointof the family C , there exists a connected set C ∈ C such that C ∩ U = ∅ and C ∩ V = ∅ . Sincethe point c belongs to at most one set of the disjoint family C , we can additionally assume that c / ∈ C . Then C = ( C ∩ U ) ∪ ( C ∩ V ) is a union of two disjoint non-empty open subsets C ∩ U and C ∩ V of C , which is forbidden by the connectedness of C .This contradiction shows that the family C of connected components of ¯ O x is finite. Then W x = S { C ∈ C : x ∈ C } is a connected neighborhood of x , contained in ¯ O x and containing theneighborhood K x \ S { C ∈ C : x / ∈ C} of x . (cid:3) Lemma 2.8.
Any Darboux injection f : X → ¯ R from a connected compact metrizable space X isa topological embedding.Proof. By Lemma 2.7, the space X is locally connected and by the Hahn-Mazurkiewicz Theorem[8, 8.14], X is locally path-connected and path-connected. If f ( X ) = ¯ R , then f is a topologicalembedding by Lemma 2.5. It remains to consider the case f ( X ) = ¯ R . Since X is locally path-connected, each point x ∈ X has a compact path-connected neighborhood K x ⊂ X , which is notequal to X . Then f ( K x ) = ¯ R and we can apply Lemma 2.5 to conclude that the restriction f ↾ K x is a topological embedding. In particular, f is continuous at the point x . So the map f : X → ¯ R is continuous and bijective. By the compactness of X , the map f is a homeomorphism. (cid:3) Since each connected 1-manifold embeds into the circle, Lemma 2.8 implies the following corol-lary that proves Theorem 1.4(1).
Corollary 2.9.
Any Darboux injection f : X → Y from a connected compact metrizable space toany -manifold Y is a topological embedding. HE CONTINUITY OF DARBOUX INJECTIONS BETWEEN MANIFOLDS 5 Darboux injections into n -varieties In fact, n -manifolds Y in Theorem 1.4 can be replaced by their generalizations, called n -varieties.The definition of an n -variety is inductive, which will allow us to use Theorem 1.4( n ) in the proofof Theorem 1.4( n + 1) (for n ∈ { , } ).First we recall some notions related to separators. Definition 3.1.
We say that a subset S of a topological space X separates a set A ⊂ X if thecomplement A \ S is disconnected. In this case S is called a separator of A . A set S is called a separator of X between points a, b ∈ X if these points belong to different connected componentsof X \ S .Next, we introduce a new notion of a componnectness, which is a common generalization of thenotions of the compactness and the semilocally-connectedness. Definition 3.2.
A topological space X is called componnected if X has a base B of the topologysuch that for any set B ∈ B the complement X \ B can be written as finite union C ∪ · · · ∪ C n ofcompact or connected subsets of X .It is clear that a topological space is componnected if it is compact or semilocally-connected.Now we are able to introduce the notion of an n -variety. Definition 3.3.
A Hausdorff topological space X is defined to be(1) a 1 -variety if each point x ∈ X has a neighborhood homeomorphic to R ;( n ) an ( n + 1) -variety for n ∈ N if • Y is first-countable; • each connected component of Y is componnected; • for any connected set C ⊂ Y that contains more than one point and any sequence { y n } n ∈ ω ⊂ Y that converges to a point y ∈ ¯ C there exists a compact connected n -variety S ⊂ Y that separates the set C and contains infinitely many points y n , n ∈ ω . Remark 3.4.
Each compact connected 1-variety is homeomorphic to the circle, see, e.g., [4].
Remark 3.5.
By induction it can be proved that each n -manifold of dimension n ≥ n -variety. On the other hand, the Sierpi´nski carpet also is a 2-variety but is very far from beinga 2-manifold.In the proofs of Theorems 5.1 and 6.1 we shall exploit the following lemma. Lemma 3.6.
Let X be a Darboux function from a regular topological space X to a topologicalspace Y . If a subset S ⊂ Y separates the image f ( X ) , then the preimage f − ( S ) contains a closednowhere dense separator of X .Proof. Since f is Darboux and the space f ( X ) \ S is disconnected, the subset X \ f − ( S ) isdisconnected, too. So, there exist open subsets U, V in X such that X \ f − ( S ) ⊂ U ∩ V and thesets U \ f − ( S ) and V \ f − ( S ) are disjoint and non-empty. If the set X \ f − ( S ) is dense in X ,then ∅ = ( U \ f − ( S )) ∩ ( V \ f − ( S )) = U ∩ V ∩ ( X \ f − ( S ))implies U ∩ V = ∅ . Then the complement L := X \ ( U ∪ V ) ⊂ f − ( S ) is a nowhere dense closedseparator of X .If the set X \ f − ( S ) is not dense in X , then f − ( S ) has non-empty interior in X . By theregularity of X , this interior contains the closure U of some non-empty open set U in X . Then f − ( S ) contains the closed nowhere dense separator L := ¯ U \ U of X . (cid:3) Fences in manifolds
A metrizable separable space X is called a fence if X is compact and and each connectedcomponent of X is homeomorphic to the segment [ a, b ] ⊂ R for some a ≤ b . So, a singleton andthe Cantor set both are fences. By Corollary 1.9.10 [3], each fence F has dimension dim( F ) ≤ X has trivial ˇCech cohomology groupsˇ H n ( X ; G ) for every n ≥ G . We shall use Huber’s Theorem [7] IRYNA BANAKH AND TARAS BANAKH saying that the group ˇ H n ( X ; G ) is isomorphic to the group [ X, K ( G, n )] of homotopy classes ofcontinuous maps from X to the Eilenberg-MacLane complex K ( G, n ). By definition, K ( G, n )is a CW-complex that has a unique non-trivial homotopy group π n ( K ( G, n )) and this group isisomorphic to G . By a coefficient group we understand any non-trivial countable abelian group G . Lemma 4.1.
Any fence X has ˇCech cohomology ˇ H n ( X ; G ) ≈ [ X, K ( G, n )] = 0 for any n ≥ and any coefficient group G .Proof. Since the Eilenberg-MacLane complex K ( G, n ) is path-connected for n ≥
1, it suffices toprove that each continuous map f : X → K ( G, n ) is homotopic to a constant map.Denote by I the family of connected components of the fence X . By definition, each connectedcomponent I ∈ I is homeomorphic to the segment [ a, b ] ⊂ R for some a ≤ b . By the Tietze-Urysohn Theorem [2, 2.1.8], there exists a retraction r I : X → I .The subset f ( X ) of the CW-complex K ( G, n ) is compact and hence is contained in a finitesubcomplex Y of K ( G, n ). By Proposition A.4 in [5], the finite CW-complex Y is locally con-tractible. Being a finite CW-complex, the space Y is compact, metrizable, and finite-dimensional.By Theorem V.7.1 in [6], the finite-dimensional locally contractible metrizable space Y is an ab-solute neighborhood retract. By Theorem IV.1.1 in [6], Y has an open cover U of Y such that anymap g : X → Y with ( g, f ) ≺ U is homotopic to f . The notation ( g, f ) ≺ U means that for every x ∈ X the doubleton { g ( x ) , f ( x ) } is contained in some set U ∈ U .For every z ∈ I choose an open neighborhood W z ⊂ X such that f ( W z ) ⊂ U for some U ∈ U .Next, use the continuity of the retraction r I and choose an open neighborhood V z ⊂ W z of z suchthat r I ( V z ) ⊂ W z .By [2, 6.1.23], connected components of compact Hausdorff spaces coincide with quasicompo-nents. Consequently, the open neighborhood S z ∈ I V z of I contains a closed-and-open neighbor-hood V I ⊂ X of I . For every point x ∈ V I we can find a point z ∈ I with x ∈ V z and concludethat { x, r I ( x ) } ⊂ W z and hence { f ( x ) , f ◦ r I ( x ) } ⊂ U for some U ∈ U .By the compactness of X , the open cover { V I : I ∈ I} of X contains a finite subcover { V I , . . . , V I n } of X . Define a map r : X → S nk =1 I k by the formula r ( x ) = r I k ( x ) where k isa unique number such that x ∈ V I k \ S p
Let F be a fence in a G -orientable closed n -manifold X for dimension n ≥ .If for some k ≤ n − the singular homology group H k ( X ; G ) is trivial, then the group H k ( X \ F ; G ) is trivial, too.Proof. By our assumption, the homology group H k ( X ; G ) is trivial. By the Poincar´e Duality,ˇ H n − k ( X, G ) ∼ = H k ( X ; G ) = 0 and by Lemma 4.1, the group ˇ H n − k − ( F ; G ) is trivial (as n − k − ≥ X, F ) for ˇCech cohomology:0 = ˇ H n − k − ( F ; G ) → ˇ H n − k ( X, F ; G ) → ˇ H n − k ( X ; G ) = 0 , we conclude that the group ˇ H n − k ( X, F ; G ) is trivial. By the Duality Theorem 0.3.1 [1], the ˇCechcohomology group ˇ H n − k ( X, F ; G ) is isomorphic to the singular homology group H k ( X \ F ; G ).So, the group H k ( X \ F ; G ) is trivial. (cid:3) HE CONTINUITY OF DARBOUX INJECTIONS BETWEEN MANIFOLDS 7
Proposition 4.3.
For any fence F in a connected closed n -manifold X of dimension n ≥ , thecomplement X \ F is connected.Proof. Since X is connected, its singular homology group H ( X ; Z ) is isomorphic to Z . Sinceeach n -manifold X is Z -orientable (see [1, p.6]), ˇ H n ( X ; Z ) ∼ = Z . By Lemma 4.1, ˇ H n − ( F ; Z ) =ˇ H n ( F ; Z ) = 0. Writing a piece of the long exact sequence of the pair ( X, F ) for ˇCech cohomology:0 = ˇ H n − ( F ; Z ) → ˇ H n ( X, F ; Z ) → ˇ H n ( X ; Z ) → ˇ H n ( F ; G ) = 0 , we conclude that the group ˇ H n ( X, F ; Z ) is isomorphic to ˇ H n ( X ; Z ) ∼ = Z . By the Poincar´e Du-ality Theorem, the ˇCech cohomology group ˇ H n ( X, F ; Z ) is isomorphic to the singular homologygroup H ( X \ F ; Z ). So, the group H ( X \ F ; Z ) is isomorphic to Z , which implies that thespace X \ F is connected and non-empty. (cid:3) Darboux injections to 2-varieties
In this section we prove Theorem 1.4(2) and its more general version:
Theorem 5.1.
Let X be a compact Hausdorff space that cannot be separated by a fence. AnyDarboux injection f : X → Y to a -variety Y is an open topological embedding.Proof. The proof of this theorem relies on the following lemma in which the injection f and thespaces X, Y satisfy the assumptions of Theorem 5.1.
Lemma 5.2.
For any sequence { y n } n ∈ ω ⊂ Y that converges to a point y ∈ f ( X ) there exists aset S ⊂ Y such that (1) S is homeomorphic to the circle ¯ R ; (2) f ( X ) \ S is disconnected; (3) the set { n ∈ ω : y n ∈ S } is infinite; (4) the map f ↾ f − ( S ) : f − ( S ) → S is a homeomorphism.Proof. By definition of a 2-variety, there exists a set S ⊂ Y satisfying the conditions (1)–(3). Since f ( X ) \ S is disconnected, we can apply Lemma 3.6 and conclude that the set f − ( S ) contains aclosed separator L of the set X . By our assumption, L is not a fence and hence some connectedcomponent C of L is not homeomorphic to a subset of [0 , f ↾ C : C → S is a Darboux injection to the space S , which is homeomorphic to ¯ R . By Lemma 2.8,the map f ↾ C : C → ¯ R is a topological embedding. Since C is not homeomorphic to a subset of[0 , f ( C ) = S and hence f − ( S ) = C . (cid:3) Now we shall derive Theorem 5.1 from Lemma 5.2.By our assumption, the space X cannot be separated by a fence, which implies that X isconnected (otherwise it would be separated by the empty set, which is a fence). Since the map f is Darboux, the image f ( X ) is connected. So, f ( X ) is contained in some connected component Y ′ of the space Y . By Definition 3.3, the connected component Y ′ of Y is componnected.Now we prove that the image f ( X ) is closed-and-open in Y . Assuming that f ( X ) is not closedin Y , we can choose a sequence { y n } n ∈ ω ⊂ f ( X ) that converges to some point y ∈ Y \ f ( X ). ByLemma 5.2, there exists a compact subset S ⊂ Y such that S ⊂ f ( X ) and S contains infinitelymany points y n and hence contains lim n →∞ y n = y . But this contradicts the choice of y / ∈ f ( X ).Assuming that f ( X ) is not open in Y , we can choose a sequence { y n } n ∈ ω ⊂ Y \ f ( X ) thatconverges to some point y ∈ f ( X ). By Lemma 5.2, there exists a subset S ⊂ Y such that y n ∈ S ⊂ f ( X ) for infinitely many numbers n ∈ ω . But this contradicts the choice of the sequence { y n } n ∈ ω ⊂ Y \ f ( X ).Therefore, the connected set f ( X ) ⊂ Y ′ is closed-and-open in Y and hence coincides with Y ′ by the connectedness of Y ′ .Now we prove that the map f − : Y ′ → X is continuous. To derive a contradiction, assumethat f − is discontinuous at some point y ∈ Y . Since the 2-variety is first-countable, we can finda neighborhood U ⊂ X of f − ( y ) and a sequence { y n } n ∈ ω ⊂ Y such that lim n →∞ y n = y but f − ( y n ) / ∈ U for all n ∈ ω .By Lemma 5.2, there exists a subset S ⊂ Y such that the set Ω = { n ∈ ω : y n ∈ S } is infiniteand f − ↾ S : S → f − ( S ) is a homeomorphism. By the compactness of S , the limit point y of thesequence { y n } n ∈ Ω ⊂ S belongs to S . Then the sequence { f − ( y n ) } n ∈ Ω ⊂ f − ( S ) converges to IRYNA BANAKH AND TARAS BANAKH f − ( y ) ∈ U and hence f − ( y n ) ∈ U for all but finitely many numbers n ∈ Ω. But this contradictsthe choice of the sequence ( y n ) n ∈ ω . This contradiction completes the proof of the continuity ofthe map f − : Y → X .Finally, we prove that the map f : X → Y ′ ⊂ Y is continuous. To derive a contradiction,assume that f is discontinuous at some point x ∈ X . Then we can find a neighborhood O y ⊂ Y of y = f ( x ) whose preimage f − ( O y ) is not a neighborhood of x . Since the set f ( X ) = Y ′ is open in Y and the (connected) space Y ′ is componnected, we can replace O y by a smaller neighborhoodand assume that the complement Y ′ \ O y can be written as the finite union C , . . . , C n of compactor connected sets. The continuity of the map f − ensures that the sets f − ( C ) , . . . , f − ( C n ) arecompact or connected. Since f − ( O y ) is not a neighborhood of x , x ∈ X \ f − ( O y ) = f − ( Y \ O y ) = n [ i =1 f − ( C i )and hence x ∈ f − ( C i ) for some i . Observe that the set C i is not compact (otherwise, f − ( C i )would be compact and x / ∈ f − ( C i ) = f − ( C i )).Then C i is connected and so is its preimage f − ( C i ) under the continuous map f − . Since x ∈ f − ( C i ), the subspace C := { x } ∪ f − ( C i ) of X is connected but its image f ( C ) = { f ( x ) } ∪ C i is not (as the singleton { y } = O y ∩ C i is clopen in f ( C )). But this contradicts the Darbouxproperty of f . This contradiction implies that f : X → Y ′ ⊂ Y is continuous and hence an opentopological embedding. (cid:3) Theorem 5.1 and Proposition 4.3 imply
Corollary 5.3.
Any Darboux injection f : X → Y from a connected closed n -manifold X ofdimension n ≥ to a -variety Y is an open topological embedding. Since each 2-manifold is a 2-variety, Corollary 5.3 implies the following corollary that impliesTheorem 5.1(2).
Corollary 5.4.
Any Darboux injection f : X → Y from a connected closed n -manifold X ofdimension n ≥ to a -manifold Y is an open topological embedding. Darboux injections into 3-varieties
We recall that a
Peano continuum is a connected locally connected compact metrizable space.By the Hahn-Mazurkiewicz Theorem [8, 8.14], each Peano continuuum is a continuous image ofthe unit interval [0 , Theorem 6.1.
Let X be a Peano continuum such that for every fence K ⊂ X the space X \ K has trivial first homology group H ( X \ K ; G ) for some coefficient group G . Any Darboux injection f : X → Y to a -variety is an open topological embedding. By analogy with the proof of Theorem 5.1, Theorem 6.1 can be derived from the followinglemma in which the function f : X → Y and spaces X, Y satisfy the assumptions of Theorem 6.1.
Lemma 6.2.
For any sequence { y n } n ∈ ω ⊂ Y that converges to a point y ∈ f ( X ) there exists aset S ⊂ Y such that (1) S is a compact connected -variety; (2) f ( X ) \ S is disconnected; (3) the set { n ∈ ω : y n ∈ S } is infinite; (4) the map f ↾ f − ( S ) : f − ( S ) → S is a homeomorphism.Proof. By the definition of a 3-variety, there exists a set S ⊂ Y satisfying the conditions (1)–(3).By Lemma 3.6, the preimage f − ( S ) contains a closed separator C of the set X . Fix any distinctpoints a, b ∈ X that belong to different connected components of X \ C . So, C is a separator of X between the points a and b . A closed separator M of X between a, b is called an irreducibleseparator between a, b if M coincides with each closed separator F ⊂ M of X between a and b . Claim 6.3.
The separator C of X contains a closed irreducible separator M ⊂ C between thepoints a and b . HE CONTINUITY OF DARBOUX INJECTIONS BETWEEN MANIFOLDS 9
Proof.
Let S be the family of closed subsets D of C that separate X between the points a and b .The family S is partially ordered by the inclusion relation.Let us show that for any linearly ordered subfamily L ⊂ S the intersection T L belongs to S .First observe that T L is a closed subset of C , being the intersection of closed sets in L .Assuming that T L / ∈ S , we conclude that the points a, b belong to the same connected com-ponent of X \ T L . Since the space X is locally path-connected, so is its open subset X \ T L .Then we can find a compact connected subset P ⊂ X \ T L containing the points a, b . Taking intoaccount that for every L ∈ L the points a, b lie in different connected components of X \ L , weconclude that the intersection L ∩ P is not empty. By the compactness of P the linearly orderedfamily { L ∩ P : L ∈ L} of non-empty closed subsets of P has non-empty intersection, whichcoincides with the intersection P ∩ T L = ∅ . The obtained contradiction completes the proof ofthe inclusion T L ∈ L .By the Zorn Lemma, the family S contains a minimal element M ∈ S . It is clear that M isirreducible separator of X between the points a, b . (cid:3) Claim 6.4.
The space M cannot be separated by a fence.Proof. To derive a contradiction, assume that M is separated by some fence J ⊂ M .Then M \ J = U ∪ U for some disjojnt non-empty open subspaces U , U of M . Considerthe closed subsets M := M \ U and M := M \ U in M and observe that M ∩ M = J and M ∪ M = M .Fix any non-zero element g ∈ G and consider the singular 0-cycle c := g · ( a − b ) in X . Theminimality of M ensures that for every i ∈ { , } the points a, b belong to the same connectedcomponent of the locally path-connected space X \ M i . Consequently, the points a, b can belinked by a path in X \ M i , which implies that the cycle c represents zero in the homology groups H ( X \ M ; G ) and H ( X \ M ; G ). By our assumption, H ( X \ J ; G ) = 0.Writing the initial piece of the Mayer-Vietoris exact sequence of the pair ( X \ M , X \ M ), weobtain the exact sequence of Abelian groups:0 = H ( X \ J ; G ) ∂ ∗ / / H ( X \ M ; G ) ( e ,e ) / / H ( X \ M ; G ) ⊕ H ( X \ M ; G ) , where e i : H ( X \ M ; G ) → H ( X \ M i ; G ) is the homomorphism induced by the identity inclusion X \ M ֒ → X \ M i for i ∈ { , } .Since the 0-cycle c represents zero in the homology groups H ( X \ M ; G ) and H ( X \ M ; G ), itshomology class [ c ] ∈ H ( X \ M ; G ) is annulated by the homomorphism ( e , e ). By the exactnessof the Mayer-Vietoris sequence, [ c ] = 0 in H ( X \ M ; G ), which is not true as a and b are containedin different connected components of X \ M . (cid:3) By Claim 6.4, the space M cannot be separated by a fence. Since S is a 2-variety, we can applyTheorem 5.1 and conclude that f ↾ M : M → S is an open topological embedding. The compactnessof M ensures that f ( M ) is closed-and-open subset in S and hence f ( M ) = S by the connectednessof S . Since f is injective, M = f − ( S ) and f ↾ f − ( S ) : f − ( S ) → S is a homeomorphism. Thiscompletes the proofs of Lemma 6.2 and Theorem 6.1. (cid:3) The following proposition characterizes rational homology 3-spheres.
Proposition 6.5.
For a connected closed -manifold X the following conditions are equivalent: (1) X is a rational homology -sphere; (2) the first rational homology group H ( X ; Q ) is trivial; (3) the first integral homology group H ( X ) is finite; (4) the abelianization of the fundamental group π ( X ) is finite.Proof. The implication (1) ⇒ (2) is trivial.(2) ⇒ (1) Assume that H ( X ; Q ) = 0. Since the 3-manifold X is connected, it is path-connectedand hence H ( X ; Q ) ≈ Q ≈ H ( S ; Q ). By Theorem 3.26 in [5], dim H ( X ; Q ) ≤ H k ( X ; Q ) = 0 = H k ( S ; Q ) for all k > dim( X ) = 3. By Corollary 3.37 in [5], closed manifolds of odd dimension have zero Euler characteristic.Consequently, χ ( X ) = 0. By Theorem 2.44 [5],0 = χ ( X ) = dim H ( X ; Q ) − dim H ( X ; Q ) + dim H ( X ; Q ) − dim H ( X ; Q ) == 1 − H ( X ; Q ) − dim H ( X ; Q ) , which implies that dim H ( X ; Q ) = 0 and dim H ( X ; Q ) = 1. Then H ( X ; Q ) = 0 = H ( S ; Q )and H ( X ; Q ) ≈ Q ≈ H ( S ; Q ). Now we see that H k ( X ; Q ) ≈ H k ( S ; Q ) for all k ≥
0, whichmeans that X is a rational homological 3-sphere.(2) ⇒ (3) Assume that H ( X ; Q ) = 0. By Corollaries A.8 and A.9 in [5], the homology group H ( X ) of the compact 3-manifold X is finitely generated. By Corollary 3A.6 in [5], H ( X ) ⊗ Q ≈ H ( X ; Q ) = 0, which implies that the group H ( X ) is a torsion group and being finitely generatedis finite.(3) ⇒ (2) Assuming that the homology group H ( X ) is finite and applying Corollary 3A.6 in[5], we conclude that H ( X ; Q ) ≈ H ( X ) ⊗ Q = 0.The equivalence (3) ⇔ (4) follows from Theorem 2A.1 [5] saying that the homology group H ( X ) of the path-connected space X is isomorphic to the abelianization of the fundamentalgroup π ( X ). (cid:3) Corollary 6.6.
Any Darboux injection f : X → Y from a rational homology -sphere X to a -variety Y is an open topological embedding.Proof. Since H ( X ; Q ) ≈ H ( S ; Q ) ≈ Q , the closed 3-manifold X is Q -orientable (see Theorem3.26 in [5]). By our assumption, X is Q -simply-connected. So, X is path-connected and has trivialhomology group H ( X ; Q ). By Proposition 4.2, for any fence A ⊂ X the complement X \ A alsohas trivial homology group H ( X \ A ; Q ). Applying Theorem 6.1, we conclude that each Darbouxinjection f : X → Y is an open topological embedding. (cid:3) Since each 3-manifold is a 3-variety, Corollary 6.6 implies the following corollary that provesTheorem 1.4(3).
Corollary 6.7.
Any Darboux injection f : X → Y from a rational homology -sphere X to a -manifold Y is an open topological embedding. Acknowledgement
The authors express their sincere thanks to Duˇsan Repovˇs, Chris Gerig and Yves de Cornulierfor their help in understanding complicated techniques of Algebraic Topology that were eventuallyused in some proofs presented in this paper. Also special thanks are due to Alex Ravsky for acareful reading the final version of the paper and many valuable remarks.
References [1] R. Daverman, G. Venema,
Embeddings in manifolds , Amer. Math. Soc., Providence, RI, 2009.[2] R. Engelking,
General Topology , Heldermann Verlag, Berlin, 1989.[3] R. Engelking,
Theory of dimensions finite and infinite , Heldermann Verlag, Lemgo, 1995.[4] D. Gale,
The Classification of 1-Manifolds: A Take-Home Exam , Amer. Math. Monthly, :2 (1987), 170–175.[5] A. Hatcher, Algebraic Topology , Cambridge Univ. Press, 2002.[6] S.-T. Hu,
Theory of Retracts , Wayne State Univ. Press, Detroit, 1965.[7] P. Huber,
Homotopical cohomology and ˇCech cohomology , Math. Annalen (1961), 73–76.[8] S. Nadler,
Continuum Theory. An introduction , Marcel Dekker, Inc., New York, 1992.[9] W.J. Pervin, N. Levine,
Connected mappings of Hausdorff spaces , Proc. Amer. Math. Soc. (1958) 488–496.[10] T. Tanaka, On the family of connected subsets and the topology of spaces , J. Math. Soc. Japan (1955),389–393.[11] W. Wong, Does there exist a bijection of R n to itself such that the forward map is connected but the inverseis not? , https://mathoverflow.net/questions/235893 . I.Banakh: Ya. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of NASU,Naukova 3b, LvivT.Banakh: Ivan Franko National University of Lviv (Ukraine) and Jan Kochanowski University inKielce (Poland)
E-mail address ::