Inequivalent Cantor sets in 𝑅³ whose complements have the same fundamental group

Garity, Dennis; Repovš, Dušan

doi:10.1090/S0002-9939-2013-11911-8

Inequivalent Cantor sets in $R^{3}$ whose complements have the same fundamental group
HTML articles powered by AMS MathViewer

by Dennis J. Garity and Dušan Repovš PDF

Proc. Amer. Math. Soc. 141 (2013), 2901-2911 Request permission

Abstract:

For each Cantor set $C$ in $R^{3}$, all points of which have bounded local genus, we show that there are infinitely many inequivalent Cantor sets in $R^{3}$ with the complement having the same fundamental group as the complement of $C$. This answers a question from Open Problems in Topology and has as an application a simple construction of nonhomeomorphic open $3$-manifolds with the same fundamental group. The main techniques used are analysis of local genus of points of Cantor sets, a construction for producing rigid Cantor sets with simply connected complement, and manifold decomposition theory. The results presented give an argument that for certain groups $G$, there are uncountably many nonhomeomorphic open $3$-manifolds with fundamental group $G$.

References

Fredric D. Ancel and Michael P. Starbird, The shrinkability of Bing-Whitehead decompositions, Topology 28 (1989), no. 3, 291–304. MR 1014463, DOI 10.1016/0040-9383(89)90010-4
M. L. Antoine, Sur la possibilité d’étendre l’homéomorphie de deux figures à leur voisinages, C. R. Acad. Sci. Paris 171 (1920), 661–663.
Steve Armentrout, Decompostions of $E^{3}$ with a compact $\textrm {O}$-dimensional set of nondegenerate elements, Trans. Amer. Math. Soc. 123 (1966), 165–177. MR 195074, DOI 10.1090/S0002-9947-1966-0195074-4
Amy Babich, Scrawny Cantor sets are not definable by tori, Proc. Amer. Math. Soc. 115 (1992), no. 3, 829–836. MR 1106178, DOI 10.1090/S0002-9939-1992-1106178-4
M. Bestvina and D. Cooper, A wild Cantor set as the limit set of a conformal group action on $S^3$, Proc. Amer. Math. Soc. 99 (1987), no. 4, 623–626. MR 877028, DOI 10.1090/S0002-9939-1987-0877028-4
R. H. Bing, Tame Cantor sets in $E^{3}$, Pacific J. Math. 11 (1961), 435–446. MR 130679
William A. Blankinship, Generalization of a construction of Antoine, Ann. of Math. (2) 53 (1951), 276–297. MR 40659, DOI 10.2307/1969543
Robert J. Daverman, Decompositions of manifolds, Pure and Applied Mathematics, vol. 124, Academic Press, Inc., Orlando, FL, 1986. MR 872468
D. G. DeGryse and R. P. Osborne, A wild Cantor set in $E^{n}$ with simply connected complement, Fund. Math. 86 (1974), 9–27. MR 375323, DOI 10.4064/fm-86-1-9-27
R. F. Dickman Jr., Some characterizations of the Freudenthal compactification of a semicompact space, Proc. Amer. Math. Soc. 19 (1968), 631–633. MR 225290, DOI 10.1090/S0002-9939-1968-0225290-1
Hans Freudenthal, Neuaufbau der Endentheorie, Ann. of Math. (2) 43 (1942), 261–279 (German). MR 6504, DOI 10.2307/1968869
D. J. Garity and D. Repovš, Cantor set problems, Open Problems in Topology. II (E. Pearl, ed.), Elsevier B. V., Amsterdam, 2007, pp. 676–678.
Dennis Garity, Dušan Repovš, and Matjaž Željko, Uncountably many inequivalent Lipschitz homogeneous Cantor sets in ${\Bbb R}^3$, Pacific J. Math. 222 (2005), no. 2, 287–299. MR 2225073, DOI 10.2140/pjm.2005.222.287
Dennis J. Garity, Dušan Repovš, and Matjaž Željko, Rigid Cantor sets in $\textbf {R}^3$ with simply connected complement, Proc. Amer. Math. Soc. 134 (2006), no. 8, 2447–2456. MR 2213719, DOI 10.1090/S0002-9939-06-08459-0
A. Kirkor, Wild $0$-dimensional sets and the fundamental group, Fund. Math. 45 (1958), 228–236. MR 102783, DOI 10.4064/fm-45-1-237-246
J. M. Kister and D. R. McMillan Jr., Locally euclidean factors of $E^{4}$ which cannot be imbedded in $E^{3}$, Ann. of Math. (2) 76 (1962), 541–546. MR 144322, DOI 10.2307/1970374
Joseph Martin, A rigid sphere, Fund. Math. 59 (1966), 117–121. MR 224075, DOI 10.4064/fm-59-2-117-121
D. R. McMillan Jr., Some contractible open $3$-manifolds, Trans. Amer. Math. Soc. 102 (1962), 373–382. MR 137105, DOI 10.1090/S0002-9947-1962-0137105-X
R. P. Osborne, Embedding Cantor sets in a manifold. I. Tame Cantor sets in $E^{n}$, Michigan Math. J. 13 (1966), 57–63. MR 187225
C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69, Springer-Verlag, New York-Heidelberg, 1972. MR 0350744
R. B. Sher, Concerning wild Cantor sets in $E^{3}$, Proc. Amer. Math. Soc. 19 (1968), 1195–1200. MR 234438, DOI 10.1090/S0002-9939-1968-0234438-4
A. C. Shilepsky, A rigid Cantor set in $E^{3}$, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 22 (1974), 223–224 (English, with Russian summary). MR 345110
Laurence Carl Siebenmann, THE OBSTRUCTION TO FINDING A BOUNDARY FOR AN OPEN MANIFOLD OF DIMENSION GREATER THAN FIVE, ProQuest LLC, Ann Arbor, MI, 1965. Thesis (Ph.D.)–Princeton University. MR 2615648
Richard Skora, Cantor sets in $S^3$ with simply connected complements, Topology Appl. 24 (1986), no. 1-3, 181–188. Special volume in honor of R. H. Bing (1914–1986). MR 872489, DOI 10.1016/0166-8641(86)90060-X
David G. Wright, Cantor sets in $3$-manifolds, Rocky Mountain J. Math. 9 (1979), no. 2, 377–383. MR 519952, DOI 10.1216/RMJ-1979-9-2-377
David G. Wright, Rigid sets in manifolds, Geometric and algebraic topology, Banach Center Publ., vol. 18, PWN, Warsaw, 1986, pp. 161–164. MR 925863
David G. Wright, Rigid sets in $E^n$, Pacific J. Math. 121 (1986), no. 1, 245–256. MR 815045
David G. Wright, Bing-Whitehead Cantor sets, Fund. Math. 132 (1989), no. 2, 105–116. MR 1002625, DOI 10.4064/fm-132-2-105-116
M. Željko, On embeddings of Cantor sets into Euclidean spaces, Ph.D. thesis, University of Ljubljana, Ljubljana, Slovenia, 2000.
Matjaž Željko, On defining sequences for Cantor sets, Topology Appl. 113 (2001), no. 1-3, 321–325. Geometric topology: Dubrovnik 1998. MR 1821859, DOI 10.1016/S0166-8641(00)00040-7
Matjaž Željko, Genus of a Cantor set, Rocky Mountain J. Math. 35 (2005), no. 1, 349–366. MR 2117612, DOI 10.1216/rmjm/1181069785