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 (Borsuk 1938)

- ANR (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 Y1Y2…Yn is homeomorphic to K.

Examples:

I5≈ M4I (Poenaru 1960)

In+1≈ MnI (n≥4) (Curtis 1961)

In≈ AB (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≈ XA1…An ≈ YB1…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 XZ≈ YZ 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 M3S1 ≈ 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 M3S2k ≈ N3S2k , k ≥ 1, is equivalent to M3 ≈ N3.

Theorem 9 (Kwasik & R.-2004) Let M3, N3 be closed oriented geometric 3-manifolds. Then M3S2k+1 ≈ N3S2k+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= AA and B2=BB 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= Ii .

Then Xn2 ≈ Xm

2 for n≠m.

1i

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 thatKn ≈ 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 xK such that its regular neighborhood is not homeomorphic to the set cone {1,…,n} I

holds, then (Kn ≈ Ln) (K ≈ L) .