homogeneous compacta and generalized manifolds · 2013-06-21 · manifolds homogeneous metric...

Homogeneous compacta and generalizedmanifolds

Vesko Valov

Dedicated to Petar Kenderov in occasion of his 70th birthday

Vesko Valov Homogeneous compacta and generalized manifolds

Motivation

Cantor manifolds were introduced by Urysohn as ageneralization of Euclidean manifolds

Euclidean manifolds have a richer structure and that is amotivation for the study of further specifications of Cantormanifolds

Homogeneous metric compacta are Generalized Cantormanifolds (Krupski, Karasev-Krupski-Todorov-Valov,Krupski-Valov)

Bing-Borsuk conjecture: every homogeneous compact metricANR of dimension n is an Euclidean n-manifold

Homology manifolds are supposed to have common propertieswith homogeneous metric compacta because of the modifiedBing-Borsuk conjecture suggested by Bryant: everyhomogeneous separable locally compact metric ANR ofdimension n is a homology n-manifold

Motivation

Some preliminary definitions and results(C is a class of spaces)

Definition

X is a Cantor C-manifold if X cannot be separated by a closedsubset from C, i.e. X 6= U ∪ V ∪ F with U,V disjoint open andnon-empty and dim F ∈ C.

Definition

X is a strong Cantor manifold w.r. to C if X can not be

represented as a union X =∞⋃i=0

Fi with⋃

i 6=j(Fi ∩ Fj) ∈ C, where

all Fi are proper closed subsets of X .

Definition

X is a Mazurkiewicz manifold w.r. to C if for every two closed,disjoint subsets X0,X1 ⊂ X , both having non-empty interiors in X ,and every sequence {Fi} ⊂ C with each Fi closed in X there existsa continuum in X \

⋃∞i=0 Fi joining X0 and X1.

Definition

Fi with⋃

Definition

Fi with⋃

Definition

X is an Alexandroff manifold w.r. to C if for every two closed,disjoint subsets X0,X1 ⊂ X , both having non-empty interiors in X ,there exists an open cover ω of X such that there is no partition Pin X between X0 and X1 admitting an ω-map into a space Y withY ∈ C.

When C is the class D(n − 2) of all spaces of dimension ≤ n − 2,the above notions were introduced by Hadjiivanov (strong Cantorn-manifolds),Hadjiivanov-Todorov (Mazurkiewicz n-manifolds) andAlexandroff (V n-manifolds).

Specifications of C1 Dk

K - at most k-dimensional spaces w.r. to dimension DK,where DK is a dimension unifying dim and dimG ;

2 D<∞K - all spaces represented as a countable union of closedfinite-dimensional subsets w.r. to DK;

3 C - all paracompact C -spaces;4 WID - all weakly infinite-dimensional spaces.

Definition

Theorem (Krupski(1990))

Every n-dimensional metric homogeneous continuum is a Cantorn-manifold.

Theorem (Karasev-Krupski-Todorov-Valov (2012))

Every metrizable homogeneous continuum X /∈ C is a strongCantor manifold with respect to C provided that:

1 C is any of the following three classes: WID, C, Dn−2K (in the

latter case we assume DK(X ) = n);

2 C = D<∞K and X does not contain closed subsets of arbitrarylarge finite dimension DK.

Theorem (Krupski(1990))

Every n-dimensional metric homogeneous continuum is a Cantorn-manifold.

Theorem (Karasev-Krupski-Todorov-Valov (2012))

Every metrizable homogeneous continuum X /∈ C is a strongCantor manifold with respect to C provided that:

1 C is any of the following three classes: WID, C, Dn−2K (in the

latter case we assume DK(X ) = n);

2 C = D<∞K and X does not contain closed subsets of arbitrarylarge finite dimension DK.

Theorem (Krupski-Valov (2011))

Let X be a homogeneous locally compact, locally connected metricspace. Suppose U is a region in X and U /∈ C, where C is one ofthe above four classes. In case C = Dn−2

K assume DK(U) = n.Then U is a Mazurkiewicz manifold with respect to C.

Theorem (VV (2012))

Every homogeneous metric ANR-continuum X with dimG X = nand Hn(X ,G ) 6= 0 is an Alexandroff manifold w.r. to the classDn−2G . Moreover any such X is an (n,G )-bubble ( Hn(A,G ) = 0

for every closed proper subset A ⊂ X ).

Theorem (Krupski-Valov (2011))

Let X be a homogeneous locally compact, locally connected metricspace. Suppose U is a region in X and U /∈ C, where C is one ofthe above four classes. In case C = Dn−2

K assume DK(U) = n.Then U is a Mazurkiewicz manifold with respect to C.

Theorem (VV (2012))

Every homogeneous metric ANR-continuum X with dimG X = nand Hn(X ,G ) 6= 0 is an Alexandroff manifold w.r. to the classDn−2G . Moreover any such X is an (n,G )-bubble ( Hn(A,G ) = 0

for every closed proper subset A ⊂ X ).

KnG -manifolds: examples and properties

Everywhere below G is a given topological group and Hn(X ,A; G )denotes the nth Cech cohomology group of the pair (X ,A) withcoefficients from G .

Definition

A pair (X ,A) of a compactum X and its closed subset is said to bea Kn

G -manifold if for every two disjoint open subsets P,Q of Xthere exist an open cover ω of Y = X \ (P ∪ Q) such that thefollowing condition holds for every partition C of X between P andQ: any natural map pωC

: (C ,C ∩ F )→ (|ωC |, |ωC∩F |), where |ωC |is the nerve of ω restricted on C , generates a non-trivialhomomorphism

p∗ωC: Hn−1(|ωC |, |ωC∩F |)→ Hn−1(C ,C ∩ F ).

If, in the above situation, there exists also e ∈ Hn−1(|ω|, |ωF∩Y |)such that p∗ωC

(i∗ωC(e)) 6= 0 for every partition C in X between P

and Q, the pair (X ,F ) is called a strong KnG -manifold.

The above definition is motivated by a result of Kuzminov (1961),which actually states that every compactum X of cohomologicaldimension dimG X = n contains a pair of (Y ,A) closed sets suchthat (Y ,A) is a strong Kn

G -manifold.

Proposition

A compactum X is an Alexandroff Dn−2G -manifold provided (X ,F )

is a KnG -manifold for some closed set F ⊂ X .

Theorem

Let (X ,F ) be a strong KnG -manifold P,Q ⊂ X two open disjoint

sets. If and M be a Lindeloff normally placed subset of X withHn−1(M,M ∩ F ) = 0. Then in each of the following two casesthere exists a continuum K ⊂ X \M connecting P and Q.

(1) M ⊂ X \ (P ∪ Q);

(2) dimG M ≤ n − 1 and F ∩M is a Gδ-set in M.

M is normally placed in X if every two closed disjoint sets in Mhave disjoint open in X neighborhoods (for example, M is Fσ inX ). Vesko Valov Homogeneous compacta and generalized manifolds

The above definition is motivated by a result of Kuzminov (1961),which actually states that every compactum X of cohomologicaldimension dimG X = n contains a pair of (Y ,A) closed sets suchthat (Y ,A) is a strong Kn

G -manifold.

Proposition

A compactum X is an Alexandroff Dn−2G -manifold provided (X ,F )

is a KnG -manifold for some closed set F ⊂ X .

Theorem

Let (X ,F ) be a strong KnG -manifold P,Q ⊂ X two open disjoint

sets. If and M be a Lindeloff normally placed subset of X withHn−1(M,M ∩ F ) = 0. Then in each of the following two casesthere exists a continuum K ⊂ X \M connecting P and Q.

(1) M ⊂ X \ (P ∪ Q);

(2) dimG M ≤ n − 1 and F ∩M is a Gδ-set in M.

M is normally placed in X if every two closed disjoint sets in Mhave disjoint open in X neighborhoods (for example, M is Fσ inX ). Vesko Valov Homogeneous compacta and generalized manifolds

Next corollary from the above theorem is interesting because thefollowing Bing-Borsuk question is still unanswered:

Question

[Bing-Borsuk (1965)] Is it true that Hn−1(M) 6= 0 for any partitionM of a homogeneous metric ANR-space X of dimension n?

Corollary (the case M is a partition was done earlier by VV)

Let X be a homogeneous metric ANR compactum withHn(X ) 6= 0. Then Hn−1(M) 6= 0 for every set M ⊂ X , which iscutting X between two disjoint open subsets of X .

Corollary

Every strong KnG -manifold is a Mazurkiewicz Dn−2

G -manifold.

Question

Corollary

G -manifold.

Question

Corollary

G -manifold.

Another corollary can be compared with the classical Mazurkiewicztheorem:

Theorem

Any region X in the Euclidean space Rn has the following property:if M ⊂ X with dim M ≤ n − 2, then every two points from X \Mcan be joined by a continuum K ⊂ X \M.

Corollary

Let M be a bounded subset of Rn with Hn−1(M;Z) = 0. Thenevery pair of disjoint open sets P,Q ⊂ Rn such that(P ∪ Q) ∩M = ∅ can be joined by a continuum in Rn \M. If, inaddition dim M ≤ n− 1, the requirement (P ∪Q) ∩M = ∅ can beremoved.

Another corollary can be compared with the classical Mazurkiewicztheorem:

Theorem

Any region X in the Euclidean space Rn has the following property:if M ⊂ X with dim M ≤ n − 2, then every two points from X \Mcan be joined by a continuum K ⊂ X \M.

Corollary

Let M be a bounded subset of Rn with Hn−1(M;Z) = 0. Thenevery pair of disjoint open sets P,Q ⊂ Rn such that(P ∪ Q) ∩M = ∅ can be joined by a continuum in Rn \M. If, inaddition dim M ≤ n− 1, the requirement (P ∪Q) ∩M = ∅ can beremoved.

Example

(In,Sn−1) as well Sn, n ≥ 1, are strong KnZ-manifolds.

Recall that the Eilenberg-MacLane complexes K (G , n) have thefollowing property: dimG X ≤ n if and only if any mapg : A→ K (G , n) can be extended over X , where A ⊂ X is closedand X compact.

Proposition

Let X be a compactum and F ⊂ X a closed nowhere dense subsetof X . If there exists a map f : F → K (G , n − 1), which is notextendable over X but it is extendable over Y ∪ F for any properclosed subset Y of X , then (X ,F ) is a strong Kn

G -manifold.

Example

(In,Sn−1) as well Sn, n ≥ 1, are strong KnZ-manifolds.

Recall that the Eilenberg-MacLane complexes K (G , n) have thefollowing property: dimG X ≤ n if and only if any mapg : A→ K (G , n) can be extended over X , where A ⊂ X is closedand X compact.

Proposition

Let X be a compactum and F ⊂ X a closed nowhere dense subsetof X . If there exists a map f : F → K (G , n − 1), which is notextendable over X but it is extendable over Y ∪ F for any properclosed subset Y of X , then (X ,F ) is a strong Kn

G -manifold.

Homology manifoldsWe are going to show that some homological properties of a metricspace X imply that X is a Mazurkiewicz arc n-manifold in thefollowing sense:

Definition

For every Fσ-subset of M ⊂ X with dim M ≤ n − 2 and any twomassive disjoint sets A,B ⊂ X there exists an arc in X \M joiningA and B.

Obviously, every Mazurkiewicz arc n-manifold is a Mazurkiewiczmanifold with respect to the class of all spaces with dim ≤ n − 2.

We consider singular homology groups reduced in dimension zerowith coefficients in a given group G (if G is not written then thecoefficients are integers).

Definition

We say that X has the property H(n,G ) at the points of a set M ⊂ Xif Hk(X ,X \ x ; G ) = 0 for all k ≤ n and all x ∈ M. When M = Xin the above definition, X is said to have the H(n,G )-property.

Homology manifoldsWe are going to show that some homological properties of a metricspace X imply that X is a Mazurkiewicz arc n-manifold in thefollowing sense:

Definition

For every Fσ-subset of M ⊂ X with dim M ≤ n − 2 and any twomassive disjoint sets A,B ⊂ X there exists an arc in X \M joiningA and B.

Obviously, every Mazurkiewicz arc n-manifold is a Mazurkiewiczmanifold with respect to the class of all spaces with dim ≤ n − 2.

We consider singular homology groups reduced in dimension zerowith coefficients in a given group G (if G is not written then thecoefficients are integers).

Definition

We say that X has the property H(n,G ) at the points of a set M ⊂ Xif Hk(X ,X \ x ; G ) = 0 for all k ≤ n and all x ∈ M. When M = Xin the above definition, X is said to have the H(n,G )-property.

The results concerning homology manifolds are inspired by thementioned above modified Bing-Borsuk conjecture. Becausehomogeneous metric ANR-spaces are Mazurkiewicz manifolds, it isinteresting to what extent homology manifolds have similarproperties. The following result of Krupski was one of the first inthat direction:

Theorem (Krupski (1993))

Let X be a locally compact locally connected separable metricspace having the H(n− 1,Z)-property. Then every open connectedsubset of U ⊂ X is a Cantor n-manifold.

Here is an extension of the Krupski’ result:

Theorem

Let X be a complete metric space and M be an Fσ-subset of Xsuch that dim ≤ n − 2 and X has the property H(n − 1,G ) at thepoints of M. Suppose P,Q ⊂ X are open sets which can be joinedby an arc in X . Then there is an arc in X \M joining P and Q.So, any arcwise connected open subset of X is a Mazurkiewicz arcn-manifold provided X has the property H(n − 1,G ).

Below by a homology n-manifold over G we mean a metric spaceX such that for every x ∈ X we have Hk(X ,X \ x ; G ) = 0 if k 6= nand Hn(X ,X \ x ; G ) = G .

Corollary

Let X be an arcwise connected complete metric space. In each ofthe following cases any arcwise connected open subset of X is aMazurkiewicz arc n-manifold:

(1) X is a homology n-manifold over a group G ;

(2) X is a product of at least n metric spaces Xi , 1 ≤ i ≤ m.

As it was mentioned above, it is still unknown whether everyhomogeneous locally compact metric ANR of dimension n has theproperty H(n − 1,Z). The next property guarantees thatimplication.

Definition

A metric space X is said to have the local separation property indimension n (written LSn) if for every neighborhood U of x thereexists another neighborhood V ⊂ U of x such that any map f : Sk →V , k ≤ n, can be approximated by maps g : Sn → V such that eachg(Sk) does not separate V .

Theorem

Any homogeneous locally compact metric ANR-space X withX ∈ LSn−2 has the H(n − 1,G )-property.

As it was mentioned above, it is still unknown whether everyhomogeneous locally compact metric ANR of dimension n has theproperty H(n − 1,Z). The next property guarantees thatimplication.

Definition

A metric space X is said to have the local separation property indimension n (written LSn) if for every neighborhood U of x thereexists another neighborhood V ⊂ U of x such that any map f : Sk →V , k ≤ n, can be approximated by maps g : Sn → V such that eachg(Sk) does not separate V .

Theorem

Any homogeneous locally compact metric ANR-space X withX ∈ LSn−2 has the H(n − 1,G )-property.

In particular, we have the following corollary which extends a resultof Mitchell:

Corollary

Let X be a homogeneous locally compact metric ANR-space suchthat dim X ≥ n and X ∈ 4(n − 2). Then X has the propertyH(n − 1,Z) and the product X × R has the disjoint disk property.

Recall the property 4(n) of Borsuk: X ∈ 4(n) if for every x ∈ Xevery neighborhood U of x contains a neighborhood V of x suchthat each compact nonempty set B ⊂ V of dimensiondim B ≤ n − 1 is contractible in a subset of U of dimension≤ n + 1. If U in that definition has a compact closure, thenX ∈ 4(n) implies that every map f : K → U, where K is acompactum of dimension dim K ≤ n, can be approximated bymaps g : K → U such that dim g(K ) ≤ dim K .

In particular, we have the following corollary which extends a resultof Mitchell:

Corollary

Let X be a homogeneous locally compact metric ANR-space suchthat dim X ≥ n and X ∈ 4(n − 2). Then X has the propertyH(n − 1,Z) and the product X × R has the disjoint disk property.

Recall the property 4(n) of Borsuk: X ∈ 4(n) if for every x ∈ Xevery neighborhood U of x contains a neighborhood V of x suchthat each compact nonempty set B ⊂ V of dimensiondim B ≤ n − 1 is contractible in a subset of U of dimension≤ n + 1. If U in that definition has a compact closure, thenX ∈ 4(n) implies that every map f : K → U, where K is acompactum of dimension dim K ≤ n, can be approximated bymaps g : K → U such that dim g(K ) ≤ dim K .

Some open problems

There are many questions about generalized Cantor manifolds andhomogeneous compacta. I will mention only two of them closelyrelated to my talk.

Question.

Is it true that any homogeneous compact metric ANR space X ofdimension n is an Alexandroff Dn−2

Z -manifold?

If the above question has a negative answer, then so does theBing-Borsuk conjecture.

Next question has a positive answer in case G = Z.

Question.

Let X be an Alexandroff manifold with respect to the class Dn−2G .

Is it true that X is a Mazurkiewicz Dn−2G -manifold? What about if

(X ,F ) is a KnG -manifold or X is a V n

G -continuum?

Some open problems

There are many questions about generalized Cantor manifolds andhomogeneous compacta. I will mention only two of them closelyrelated to my talk.

Question.

Is it true that any homogeneous compact metric ANR space X ofdimension n is an Alexandroff Dn−2

Z -manifold?

If the above question has a negative answer, then so does theBing-Borsuk conjecture.

Next question has a positive answer in case G = Z.

Question.

Let X be an Alexandroff manifold with respect to the class Dn−2G .

Is it true that X is a Mazurkiewicz Dn−2G -manifold? What about if

(X ,F ) is a KnG -manifold or X is a V n

G -continuum?

homogeneous compacta and generalized manifolds · 2013-06-21 · manifolds homogeneous metric...

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