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Complementary Groups of Antoine's Necklaces

Raspat, Tyler Jay (2010) Complementary Groups of Antoine's Necklaces. Master's Thesis, University of Pittsburgh. (Unpublished)

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In 1921 Antoine constructed the first example of a wild Cantor set. A wild Cantor set is a subset of Euclidean 3-space which is homeomorphic to the Cantor set but which is not equivalent in the plane to the standard embedding of the Cantor set. Antoine's example is now called Antoine's necklace.The purpose of this thesis is in to investigate the fundamental group of the complement of Antoine's necklace and other wild Cantor sets.First, a survey of known work on wild Cantor sets and their complementary groups is presented including: the Wirtinger presentation for knots and links, basic results on the complementarygroup of Antoine's necklace, Sher's theorem on canonical defining sequences for Antoine necklaces, and Skora's example of a wild Cantorset with trivial complementary group. Second a complete presentation for the complementary group of Antoine's necklace (and some variants)is calculated. A plausible technique to construct a non--equivalent variant of Antoine's necklace with isomorphic complementary group isshown to fail. The thesis concludes with a survey of open problems on complementary groups and complements of Cantor sets embedded inEuclidean 3-space.


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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Raspat, Tyler
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairGartside, Paulgartside@math.pitt.eduPMG20
Committee MemberEspinoza, Benbee1@pitt.eduBEE1
Committee MemberIon, Bogdanbion@pitt.eduBION
Date: 24 September 2010
Date Type: Completion
Defense Date: 16 August 2010
Approval Date: 24 September 2010
Submission Date: 10 August 2010
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Institution: University of Pittsburgh
Schools and Programs: Dietrich School of Arts and Sciences > Mathematics
Degree: MS - Master of Science
Thesis Type: Master's Thesis
Refereed: Yes
Uncontrolled Keywords: knot group; Seifert van-Kampen Theorem; wild cantor set
Other ID:, etd-08102010-140315
Date Deposited: 10 Nov 2011 19:58
Last Modified: 15 Nov 2016 13:48


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