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 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

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 Y 1, Y 2,…,Y n of dimension at most 2 such that Y 1  Y 2  …  Y n is homeomorphic to K. Examples: I 5 ≈ M 4  I (Poenaru 1960) I n+1 ≈ M n  I (n≥4) (Curtis 1961) I n ≈ 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  A 1  …  A n ≈ Y  B 1  …  B n where dim A i = dim B i = 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 A i ≈ B σ(i) and X≈ Y if none of A i ’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 M 3, N 3 such that π 1 (M 3 ) ≠π 1 (N 3 ) but M 3  S 1 ≈ N 3  S 1. Theorem 6 (Turaev 1988) Let M 3, N 3 be closed, oriented 3-manifolds (geometric), then M 3  S 1 ≈ N 3  S 1 is equivalent to M 3 ≈ N 3 unless M 3 and N 3 are Seifert fibered 3-manifolds, which are surface bundles over S 1 with periodic monodromy (and the surface genus > 1).

Theorem 7 (Kwasik & R ) Let F g 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 S 1 with as fiber and which become homeomorphic after crossing with S 1. Theorem 8 (Kwasik & R ) Let M 3, N 3 be closed oriented geometric 3-manifolds. Then M 3  S 2k ≈ N 3  S 2k, k ≥ 1, is equivalent to M 3 ≈ N 3. Theorem 9 (Kwasik & R.-2004) Let M 3, N 3 be closed oriented geometric 3-manifolds. Then M 3  S 2k+1 ≈ N 3  S 2k+1, k ≥ 1, is equivalent to a) M 3 ≈ N 3 if M 3 is not a lens space. b) π 1 (M 3 ) ≈ π 1 (N 3 ) if M 3 is a lens space and k=1 c) M 3 N 3 if M 3 is a lens space and k>1.

Theorem 10 (Malesič, Repovš, R., Zastrow ) 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 M 3 and N 3 be closed connected geometric prime and orientable 3-manifolds without decomposition into Cartesian product. Let X, Y be closed connected orientable surfaces. If M 3  X ≈ N 3  Y, then M 3 ≈ N 3 and X ≈ Y unless M 3 and N 3 are Seifert fibered 3-manifolds which are surface bundles over S 1 with periodic monodromy of the surface of genus >1 and X ≈ Y ≈ S 1  S 1 ≈ T 2.

Theorem 13 (Kwasik & R.-2010) Let M 3, N 3 be as in above Theorem, then M 3  T n ≈ N 3  T n is equivalent M 3 ≈ N 3 unless M 3 and N 3 are as above Theorem. Ulam’s problem 1933: Assume that A and B are topological spaces and A 2 = A  A and B 2 =B  B are homeomorphic. Is it true that A and B are homeomorphic? Example: Let I i = [0,1) for i= 1,2,…,n and I i = [0,1] for i>n X n = I i. Then X n 2 ≈ X m 2 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 L n ≈ L’ n  π 1 (L) ≈ π 1 (L’) b)If n is odd then L n ≈ L’ n  L L’.

Theorem 16 (Kwasik & R.-2010) Let M 3, N 3 be connected oriented Seifert fibred 3-manifolds. If M 3  M 3 ≈ N 3  N 3 then M 3 ≈ N 3 unless M 3 and N 3 are lens spaces with isomorphic fundamental groups.

Mycielski’s question: Let K, L be compact connected 2-polyhedra. Is it true that K n ≈ L n  K ≈ L for n>2 ? Theorem 17 (R ) 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 (K n ≈ L n )  (K ≈ L).