## Algebraic & Geometric Topology

### Universal nowhere dense subsets of locally compact manifolds

#### Abstract

In each manifold $M$ modeled on a finite- or infinite-dimensional cube $[0,1]n$, $n≤ω$, we construct a closed nowhere dense subset $S⊂M$ (called a spongy set) which is a universal nowhere dense set in $M$ in the sense that for each nowhere dense subset $A⊂M$ there is a homeomorphism $h:M→M$ such that $h(A)⊂S$. The key tool in the construction of spongy sets is a theorem on the topological equivalence of certain decompositions of manifolds. A special case of this theorem says that two vanishing cellular strongly shrinkable decompositions $A,ℬ$ of a Hilbert cube manifold $M$ are topologically equivalent if any two nonsingleton elements $A∈A$ and $B∈ℬ$ of these decompositions are ambiently homeomorphic.

#### Article information

Source
Algebr. Geom. Topol., Volume 13, Number 6 (2013), 3687-3731.

Dates
Accepted: 21 May 2013
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715745

Digital Object Identifier
doi:10.2140/agt.2013.13.3687

Mathematical Reviews number (MathSciNet)
MR3248746

Zentralblatt MATH identifier
1281.57014

#### Citation

Banakh, Taras; Repovš, Dušan. Universal nowhere dense subsets of locally compact manifolds. Algebr. Geom. Topol. 13 (2013), no. 6, 3687--3731. doi:10.2140/agt.2013.13.3687. https://projecteuclid.org/euclid.agt/1513715745

#### References

• T Banakh, D Repovš, Universal meager $F_\sigma$–sets in locally compact manifolds, Comment. Math. Univ. Carolin. 54 (2013) 179–188
• T Banakh, D Repovš, Universal nowhere dense and meager sets in Menger manifolds
• M Barnsley, Fractals everywhere, Academic Press, Boston, MA (1988)
• R Bennett, Countable dense homogeneous spaces, Fund. Math. 74 (1972) 189–194
• J W Cannon, A positional characterization of the $(n-1)$–dimensional Sierpiński curve in $S^{n}(n \not= 4)$, Fund. Math. 79 (1973) 107–112
• Z Čerin, On cellular decompositions of Hilbert cube manifolds, Pacific J. Math. 91 (1980) 47–69
• T A Chapman, Lectures on Hilbert cube manifolds, Regional Conference Series in Mathematics 28, Amer. Math. Soc. (1976)
• A Chigogidze, Infinite dimensional topology and shape theory, from: “Handbook of geometric topology”, (R J Daverman, R B Sher, editors), North-Holland, Amsterdam (2002) 307–371
• C O Christenson, R P Osborne, Pointlike subsets of a manifold, Pacific J. Math. 24 (1968) 431–435
• R J Daverman, Decompositions of manifolds, Pure and Applied Mathematics 124, Academic Press, Orlando, FL (1986)
• R D Edwards, The solution of the $4$–dimensional annulus conjecture (after Frank Quinn), from: “Four-manifold theory”, (C Gordon, R Kirby, editors), Contemp. Math. 35, Amer. Math. Soc. (1984) 211–264
• R Engelking, General topology, 2nd edition, Sigma Series in Pure Mathematics 6, Heldermann, Berlin (1989)
• K Falconer, Fractal geometry, 2nd edition, Wiley, Hoboken, NJ (2003)
• S-t Hu, Theory of retracts, Wayne State University Press, Detroit (1965)
• R C Kirby, Stable homeomorphisms and the annulus conjecture, Ann. of Math. 89 (1969) 575–582
• K Menger, Allgemeine raume und Cartesische raume zweite mitteilung: Uber umfassendste $n$–dimensional mengen, Proc. Akad. Amsterdam 29 (1926) 1125–1128
• E E Moise, Affine structures in $3$–manifolds, V: The triangulation theorem and Hauptvermutung, Ann. of Math. 56 (1952) 96–114
• F Quinn, Ends of maps, III: Dimensions $4$ and $5$., J. Differential Geom. 17 (1982) 503–521
• T Radó, Uber den begriff der Riemannschen fläche, Acta Sci. Math. (Szeged) 2 (1924) 101–121
• D Repovš, P V Semenov, Continuous selections of multivalued mappings, Mathematics and its Applications 455, Kluwer Academic Publishers, Dordrecht (1998)
• H Toruńczyk, On ${\rm CE}$–images of the Hilbert cube and characterization of $Q$–manifolds, Fund. Math. 106 (1980) 31–40
• G T Whyburn, Topological characterization of the Sierpiński curve, Fund. Math. 45 (1958) 320–324