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Homogeneous compacta and generalizedmanifolds
Vesko Valov
Dedicated to Petar Kenderov in occasion of his 70th birthday
Vesko Valov Homogeneous compacta and generalized manifolds
![Page 2: Homogeneous compacta and generalized manifolds · 2013-06-21 · manifolds Homogeneous metric compacta are Generalized Cantor manifolds (Krupski, Karasev-Krupski-Todorov-Valov, Krupski-Valov)](https://reader033.vdocuments.site/reader033/viewer/2022042920/5f645973f206252da50e6d77/html5/thumbnails/2.jpg)
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
Vesko Valov Homogeneous compacta and generalized manifolds
![Page 3: Homogeneous compacta and generalized manifolds · 2013-06-21 · manifolds Homogeneous metric compacta are Generalized Cantor manifolds (Krupski, Karasev-Krupski-Todorov-Valov, Krupski-Valov)](https://reader033.vdocuments.site/reader033/viewer/2022042920/5f645973f206252da50e6d77/html5/thumbnails/3.jpg)
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
Vesko Valov Homogeneous compacta and generalized manifolds
![Page 4: Homogeneous compacta and generalized manifolds · 2013-06-21 · manifolds Homogeneous metric compacta are Generalized Cantor manifolds (Krupski, Karasev-Krupski-Todorov-Valov, Krupski-Valov)](https://reader033.vdocuments.site/reader033/viewer/2022042920/5f645973f206252da50e6d77/html5/thumbnails/4.jpg)
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
Vesko Valov Homogeneous compacta and generalized manifolds
![Page 5: Homogeneous compacta and generalized manifolds · 2013-06-21 · manifolds Homogeneous metric compacta are Generalized Cantor manifolds (Krupski, Karasev-Krupski-Todorov-Valov, Krupski-Valov)](https://reader033.vdocuments.site/reader033/viewer/2022042920/5f645973f206252da50e6d77/html5/thumbnails/5.jpg)
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
Vesko Valov Homogeneous compacta and generalized manifolds
![Page 6: Homogeneous compacta and generalized manifolds · 2013-06-21 · manifolds Homogeneous metric compacta are Generalized Cantor manifolds (Krupski, Karasev-Krupski-Todorov-Valov, Krupski-Valov)](https://reader033.vdocuments.site/reader033/viewer/2022042920/5f645973f206252da50e6d77/html5/thumbnails/6.jpg)
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
Vesko Valov Homogeneous compacta and generalized manifolds
![Page 7: Homogeneous compacta and generalized manifolds · 2013-06-21 · manifolds Homogeneous metric compacta are Generalized Cantor manifolds (Krupski, Karasev-Krupski-Todorov-Valov, Krupski-Valov)](https://reader033.vdocuments.site/reader033/viewer/2022042920/5f645973f206252da50e6d77/html5/thumbnails/7.jpg)
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
Vesko Valov Homogeneous compacta and generalized manifolds
![Page 8: Homogeneous compacta and generalized manifolds · 2013-06-21 · manifolds Homogeneous metric compacta are Generalized Cantor manifolds (Krupski, Karasev-Krupski-Todorov-Valov, Krupski-Valov)](https://reader033.vdocuments.site/reader033/viewer/2022042920/5f645973f206252da50e6d77/html5/thumbnails/8.jpg)
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.
Vesko Valov Homogeneous compacta and generalized manifolds
![Page 9: Homogeneous compacta and generalized manifolds · 2013-06-21 · manifolds Homogeneous metric compacta are Generalized Cantor manifolds (Krupski, Karasev-Krupski-Todorov-Valov, Krupski-Valov)](https://reader033.vdocuments.site/reader033/viewer/2022042920/5f645973f206252da50e6d77/html5/thumbnails/9.jpg)
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.
Vesko Valov Homogeneous compacta and generalized manifolds
![Page 10: Homogeneous compacta and generalized manifolds · 2013-06-21 · manifolds Homogeneous metric compacta are Generalized Cantor manifolds (Krupski, Karasev-Krupski-Todorov-Valov, Krupski-Valov)](https://reader033.vdocuments.site/reader033/viewer/2022042920/5f645973f206252da50e6d77/html5/thumbnails/10.jpg)
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.
Vesko Valov Homogeneous compacta and generalized manifolds
![Page 11: Homogeneous compacta and generalized manifolds · 2013-06-21 · manifolds Homogeneous metric compacta are Generalized Cantor manifolds (Krupski, Karasev-Krupski-Todorov-Valov, Krupski-Valov)](https://reader033.vdocuments.site/reader033/viewer/2022042920/5f645973f206252da50e6d77/html5/thumbnails/11.jpg)
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.
Vesko Valov Homogeneous compacta and generalized manifolds
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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.
Vesko Valov Homogeneous compacta and generalized manifolds
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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.
Vesko Valov Homogeneous compacta and generalized manifolds
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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.
Vesko Valov Homogeneous compacta and generalized manifolds
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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.
Vesko Valov Homogeneous compacta and generalized manifolds
![Page 16: Homogeneous compacta and generalized manifolds · 2013-06-21 · manifolds Homogeneous metric compacta are Generalized Cantor manifolds (Krupski, Karasev-Krupski-Todorov-Valov, Krupski-Valov)](https://reader033.vdocuments.site/reader033/viewer/2022042920/5f645973f206252da50e6d77/html5/thumbnails/16.jpg)
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 ).
Vesko Valov Homogeneous compacta and generalized manifolds
![Page 17: Homogeneous compacta and generalized manifolds · 2013-06-21 · manifolds Homogeneous metric compacta are Generalized Cantor manifolds (Krupski, Karasev-Krupski-Todorov-Valov, Krupski-Valov)](https://reader033.vdocuments.site/reader033/viewer/2022042920/5f645973f206252da50e6d77/html5/thumbnails/17.jpg)
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 ).
Vesko Valov Homogeneous compacta and generalized manifolds
![Page 18: Homogeneous compacta and generalized manifolds · 2013-06-21 · manifolds Homogeneous metric compacta are Generalized Cantor manifolds (Krupski, Karasev-Krupski-Todorov-Valov, Krupski-Valov)](https://reader033.vdocuments.site/reader033/viewer/2022042920/5f645973f206252da50e6d77/html5/thumbnails/18.jpg)
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.
Vesko Valov Homogeneous compacta and generalized manifolds
![Page 19: Homogeneous compacta and generalized manifolds · 2013-06-21 · manifolds Homogeneous metric compacta are Generalized Cantor manifolds (Krupski, Karasev-Krupski-Todorov-Valov, Krupski-Valov)](https://reader033.vdocuments.site/reader033/viewer/2022042920/5f645973f206252da50e6d77/html5/thumbnails/19.jpg)
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
![Page 20: Homogeneous compacta and generalized manifolds · 2013-06-21 · manifolds Homogeneous metric compacta are Generalized Cantor manifolds (Krupski, Karasev-Krupski-Todorov-Valov, Krupski-Valov)](https://reader033.vdocuments.site/reader033/viewer/2022042920/5f645973f206252da50e6d77/html5/thumbnails/20.jpg)
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
![Page 21: Homogeneous compacta and generalized manifolds · 2013-06-21 · manifolds Homogeneous metric compacta are Generalized Cantor manifolds (Krupski, Karasev-Krupski-Todorov-Valov, Krupski-Valov)](https://reader033.vdocuments.site/reader033/viewer/2022042920/5f645973f206252da50e6d77/html5/thumbnails/21.jpg)
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.
Vesko Valov Homogeneous compacta and generalized manifolds
![Page 22: Homogeneous compacta and generalized manifolds · 2013-06-21 · manifolds Homogeneous metric compacta are Generalized Cantor manifolds (Krupski, Karasev-Krupski-Todorov-Valov, Krupski-Valov)](https://reader033.vdocuments.site/reader033/viewer/2022042920/5f645973f206252da50e6d77/html5/thumbnails/22.jpg)
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.
Vesko Valov Homogeneous compacta and generalized manifolds
![Page 23: Homogeneous compacta and generalized manifolds · 2013-06-21 · manifolds Homogeneous metric compacta are Generalized Cantor manifolds (Krupski, Karasev-Krupski-Todorov-Valov, Krupski-Valov)](https://reader033.vdocuments.site/reader033/viewer/2022042920/5f645973f206252da50e6d77/html5/thumbnails/23.jpg)
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.
Vesko Valov Homogeneous compacta and generalized manifolds
![Page 24: Homogeneous compacta and generalized manifolds · 2013-06-21 · manifolds Homogeneous metric compacta are Generalized Cantor manifolds (Krupski, Karasev-Krupski-Todorov-Valov, Krupski-Valov)](https://reader033.vdocuments.site/reader033/viewer/2022042920/5f645973f206252da50e6d77/html5/thumbnails/24.jpg)
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.
Vesko Valov Homogeneous compacta and generalized manifolds
![Page 25: Homogeneous compacta and generalized manifolds · 2013-06-21 · manifolds Homogeneous metric compacta are Generalized Cantor manifolds (Krupski, Karasev-Krupski-Todorov-Valov, Krupski-Valov)](https://reader033.vdocuments.site/reader033/viewer/2022042920/5f645973f206252da50e6d77/html5/thumbnails/25.jpg)
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.
Vesko Valov Homogeneous compacta and generalized manifolds
![Page 26: Homogeneous compacta and generalized manifolds · 2013-06-21 · manifolds Homogeneous metric compacta are Generalized Cantor manifolds (Krupski, Karasev-Krupski-Todorov-Valov, Krupski-Valov)](https://reader033.vdocuments.site/reader033/viewer/2022042920/5f645973f206252da50e6d77/html5/thumbnails/26.jpg)
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.
Vesko Valov Homogeneous compacta and generalized manifolds
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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.
Vesko Valov Homogeneous compacta and generalized manifolds
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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.
Vesko Valov Homogeneous compacta and generalized manifolds
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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.
Vesko Valov Homogeneous compacta and generalized manifolds
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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:
Vesko Valov Homogeneous compacta and generalized manifolds
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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.
Vesko Valov Homogeneous compacta and generalized manifolds
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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.
Vesko Valov Homogeneous compacta and generalized manifolds
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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.
Vesko Valov Homogeneous compacta and generalized manifolds
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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 .
Vesko Valov Homogeneous compacta and generalized manifolds
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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 .
Vesko Valov Homogeneous compacta and generalized manifolds
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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?
Vesko Valov Homogeneous compacta and generalized manifolds
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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?
Vesko Valov Homogeneous compacta and generalized manifolds