#### Glasnik Matematicki, Vol. 41, No.1 (2006), 165-176.

### ON LINKING OF CANTOR SETS

### Matjaž Željko

Faculty of Mathematics and Physics,
University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia

*e-mail:* `matjaz.zeljko@fmf.uni-lj.si`

**Abstract.** We introduce a property **L** for a subset of a manifold which
enables us to pass the geometric linking property from the
manifold to this subset. We prove that cubes with handles *M* and
*N* are linked if and only if subsets *X*
Int *M* and
*Y* Int *N*
having property **L** are linked. We
present a criterion which shows that many known Cantor sets
explicitly given by defining sequences have this property. As an
application of the property **L** we extend the theorem on
rigid Cantor sets thus allowing slightly more complicated terms in
their defining sequences.

**2000 Mathematics Subject Classification.**
57M30.

**Key words and phrases.** Geometric linking, Cantor set, defining sequence,
rigid Cantor set.

**Full text (PDF)** (free access)
DOI: 10.3336/gm.41.1.14

**References:**

- S. Armentrout,
*Decompositions of E*^{3} with a compact
0-dimensional set of nondegenerate elements,
Trans. Amer. Math. Soc. **123** (1966), 165-177.

MathSciNet
CrossRef

- H. G. Böthe,
*Eine fixierte Kurve in E*^{3},
General Topology and Its Relations to Modern Analysis and Algebra. II
(Proc. Second Prague Topological Symposium 1966), Academia,
Prague, 1967, 68-73.

MathSciNet

- R. J. Daverman,
*On the scarcity of tame disks in certain
wild cells*, Fund. Math. **79** (1973), 63-77.

MathSciNet

- R. J. Daverman and R. D. Edwards,
*Wild Cantor sets as
approximations to codimension two manifolds*,
Topologly Appl. **26** (1987), 207-218.

MathSciNet
CrossRef

- D. Garity, D. Repovs and D. Zeljko,
*Rigid Cantor sets in ***R**^{3} with simply connected complement,
Preprint Series IMFM Ljubljana Vol. **42** (2004) No. 943.

- J. M. Martin,
*A rigid sphere*,
Fund. Math. **59** (1966), 117-121.

MathSciNet

- R. B. Sher,
*Concerning wild Cantor sets in E*^{3},
Proc. Amer. Math. Soc. **19** (1968), 1195-1200.

MathSciNet
CrossRef

- A. C. Shilepsky,
*A rigid Cantor set in E*^{3},
Bull. Acad. Polon. Sci. **22** (1974), 223-224.

MathSciNet

- D. G. Wright,
*Rigid sets in E*^{n},
Pacific J. Math. **121** (1986), 245-256.

MathSciNet

- D. G. Wright,
*Rigid sets in manifolds*,
Geometric and Algebraic Topology,
Banach Center Publ. **18**, PWN, Warsaw, 1986, 161-164.

MathSciNet

*Glasnik Matematicki* Home Page