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) .