Transcript On the Uniqueness of the Decomposition of Manifolds, Polyhedra

On the Uniqueness of the
Decomposition of Manifolds,
Polyhedra and Continua into
Cartesian Products
Witold Rosicki (Gdańsk)
6th ECM, Kraków 2012
Example 1:
 I
is homeomorphic to
 I
Example 2:
 I
 I
are homeomorphic
Example 3:
The Cartesian product of a torus with one hole and an
Interval is homeomorphic to the Cartesian product of a disk
with two holes and interval.
I
I
Theorem 1
A decomposition of a finite dimensional
-polyhedron
- ANR
(Borsuk 1938)
(Patkowska 1966)
into Cartesian product of 1 dimensional factors is unique.
Theorem 2 (Borsuk 1945)
n-dimensional closed and connected manifold without boundary has
at most one decomposition into Cartesian product of factors of dimension
≤ 2.
Theorem 3 (R. 1997)
If a connected polyhedron K is homeomorphic to a Cartesian
product of 1-dimensional factors, then there is no other different
system of prime compacta Y1, Y2,…,Yn of dimension at most 2 such
that Y1Y2…Yn is homeomorphic to K.
Examples:
I5≈ M4I
(Poenaru 1960)
In+1≈ MnI (n≥4) (Curtis 1961)
In≈ AB (n≥8)
(Kwun & Raymond 1962)
Theorem 4 (R. 1990)
If a 3-polyhedron has two decompositions into a Cartesian
product then an arc is its topological factor.
Theorem 5 (R. 1997)
If a compact, connected polyhedron K has two decompositions
into Cartesian products
K≈ XA1…An ≈ YB1…Bn
where dim Ai= dim Bi= 1, for i= 1,2,…,n and dim X= dim Y= 2, and the
factors are prime,
then there is i→σ(i), 1-1 correspondence such that Ai≈ Bσ(i) and
X≈ Y if none of Ai’s is an arc.
Example: (R. 2003)
There exist 2-dimensional continuua X,Y and 1-dimensional
continuum Z, such that XZ≈ YZ and Z is not an arc.
Example: (Conner, Raymond 1971)
There exist a Seifert manifolds M3, N3 such that π1(M3) ≠π1(N3)
but M3 S1 ≈ N3 S1.
Theorem 6 (Turaev 1988)
Let M3, N3 be closed, oriented 3-manifolds (geometric), then
M3S1 ≈ N3 S1 is equivalent to M3≈ N3 unless M3 and N3 are Seifert
fibered 3-manifolds, which are surface bundles over S1 with periodic
monodromy (and the surface genus > 1).
Theorem 7 (Kwasik & R.- 2004)
Let Fg fixed closed oriented surface of genus g ≥ 2. Then there
are at least Φ(4g+2) (Euler number) of nonhomeomorphic 3-manifolds
which fiber over S1 with as fiber and which become homeomorphic after
crossing with S1.
Theorem 8 (Kwasik & R.- 2004)
Let M3, N3 be closed oriented geometric 3-manifolds. Then
M3S2k ≈ N3S2k , k ≥ 1, is equivalent to M3 ≈ N3.
Theorem 9 (Kwasik & R.-2004)
Let M3, N3 be closed oriented geometric 3-manifolds. Then
M3S2k+1 ≈ N3S2k+1 , k ≥ 1, is equivalent to
a) M3≈ N3 if M3 is not a lens space.
b) π1(M3) ≈ π1(N3) if M3 is a lens space and k=1
c) M3  N3 if M3 is a lens space and k>1.
Theorem 10 (Malesič, Repovš, R., Zastrow - 2004)
If M, N, M’, N’ are 2-dimensional prime manifolds with boundary
then M  N ≈ M’ N’  M ≈ M’ and N ≈ N’ (or inverse).
Theorem 11 (R.-2004)
If a decomposition of compact connected 4-polyhedron into
Cartesian product of 2-polyhedra is not unique, then in all different
decompositions one of the factors is homeomorphic to the same boundle
of intervals over a graph.
Theorem 12 (Kwasik & R.-2010)
Let M3 and N3 be closed connected geometric prime and
orientable 3-manifolds without decomposition into Cartesian product.
Let X, Y be closed connected orientable surfaces.
If M3 X ≈ N3 Y , then M3≈ N3 and X ≈ Y unless M3 and N3 are
Seifert fibered 3-manifolds which are surface bundles over S1 with
periodic monodromy of the surface of genus >1 and X ≈ Y ≈ S1 S1 ≈ T2.
Theorem 13 (Kwasik & R.-2010)
Let M3, N3 be as in above Theorem, then
M3 Tn ≈ N3 Tn is equivalent M3 ≈ N3 unless M3 and N3 are as above
Theorem.
Ulam’s problem 1933:
Assume that A and B are topological spaces and A2= AA and
B2=BB are homeomorphic.
Is it true that A and B are homeomorphic?
Example:
Let Ii= [0,1) for i= 1,2,…,n and Ii= [0,1] for i>n

Xn=

i 1
Ii
.
Then Xn2 ≈ Xm2 for n≠m.
Theorem 14
The answer for Ulam’s problem is:
Yes- for 2-manifolds with boundary (Fox- 1947)
Yes- for 2-polyhedra (R.-1986)
No- for 2-dimensional continua (R.-2003)
No- for 4-manifolds (Fox 1947).
Theorem 15 (Kwasik , Schultz- 2002)
Let L, L’ be 3-dimensional lens spaces, n≥2,
a) If n is even then Ln ≈ L’n  π1(L) ≈ π1(L’)
b) If n is odd then Ln ≈ L’n  L  L’.
Theorem 16 (Kwasik & R.-2010)
Let M3, N3 be connected oriented Seifert fibred 3-manifolds.
If M3 M3 ≈ N3  N3 then M3 ≈ N3 unless M3 and N3 are lens spaces
with isomorphic fundamental groups.
Mycielski’s question:
Let K, L be compact connected 2-polyhedra. Is it true that
Kn ≈ Ln  K ≈ L for n>2 ?
Theorem 17 (R.- 1990)
Let K and L be compact connected 2-polyhedra and one of the
conditions
1. K is 2-manifold with boundary
2. K has local cut points
3. the non-Euclidean part of K is not a disjoint union of intervals
4. there exist a point xK such that its regular neighborhood is not
homeomorphic to the set cone {1,…,n} I
holds, then
(Kn ≈ Ln)  (K ≈ L) .