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On the structure of closed 3-manifolds

Jan Mycielski (2003)

Fundamenta Mathematicae

We will show that for every irreducible closed 3-manifold M, other than the real projective space P³, there exists a piecewise linear map f: S → M where S is a non-orientable closed 2-manifold of Euler characteristic χ ≡ 2 (mod 3) such that | f - 1 ( x ) | 2 for all x ∈ M, the closure of the set x M : | f - 1 ( x ) | = 2 is a cubic graph G such that S - f - 1 ( G ) consists of 1/3(2-χ) + 2 simply connected regions, M - f(S) consists of two disjoint open 3-cells such that f(S) is the boundary of each of them, and f has some additional interesting properties....

On the topological structure of compact 5-manifolds

Alberto Cavicchioli, Fulvia Spaggiari (1993)

Commentationes Mathematicae Universitatis Carolinae

We classify the genus one compact (PL) 5-manifolds and prove some results about closed 5-manifolds with free fundamental group. In particular, let M be a closed connected orientable smooth 5 -manifold with free fundamental group. Then we prove that the number of distinct smooth 5 -manifolds homotopy equivalent to M equals the 2 -nd Betti number (mod 2 ) of M .

On universality of countable and weak products of sigma hereditarily disconnected spaces

Taras Banakh, Robert Cauty (2001)

Fundamenta Mathematicae

Suppose a metrizable separable space Y is sigma hereditarily disconnected, i.e., it is a countable union of hereditarily disconnected subspaces. We prove that the countable power X ω of any subspace X ⊂ Y is not universal for the class ₂ of absolute G δ σ -sets; moreover, if Y is an absolute F σ δ -set, then X ω contains no closed topological copy of the Nagata space = W(I,ℙ); if Y is an absolute G δ -set, then X ω contains no closed copy of the Smirnov space σ = W(I,0). On the other hand, the countable power X ω of...

On universality of finite powers of locally path-connected meager spaces

Taras Banakh, Robert Cauty (2005)

Colloquium Mathematicae

It is shown that for every integer n the (2n+1)th power of any locally path-connected metrizable space of the first Baire category is 𝓐₁[n]-universal, i.e., contains a closed topological copy of each at most n-dimensional metrizable σ-compact space. Also a one-dimensional σ-compact absolute retract X is found such that the power X^{n+1} is 𝓐₁[n]-universal for every n.

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