on the uniqueness of the decomposition of manifolds, polyhedra and continua into cartesian products...

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) .