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In algebraic topology, a peripheral subgroup for a space-subspace pair ''X'' ⊃ ''Y'' is a certain subgroup of the fundamental group of the complementary space, π1(''X'' − ''Y''). Its conjugacy class is an invariant of the pair (''X'',''Y''). That is, any homeomorphism (''X'', ''Y'') → (''X''′, ''Y''′) induces an isomorphism π1(''X'' − ''Y'') → π1(''X''′ − ''Y''′) taking peripheral subgroups to peripheral subgroups. A peripheral subgroup consists of loops in ''X'' − ''Y'' which are peripheral to ''Y'', that is, which stay "close to" ''Y'' (except when passing to and from the basepoint). When an ordered set of generators for a peripheral subgroup is specified, the subgroup and generators are collectively called a peripheral system for the pair (''X'', ''Y''). Peripheral systems are used in knot theory as a complete algebraic invariant of knots. There is a systematic way to choose generators for a peripheral subgroup of a knot in 3-space, such that distinct knot types always have algebraically distinct peripheral systems. The generators in this situation are called a longitude and a meridian of the knot complement. == Full definition == Let ''Y'' be a subspace of the path-connected topological space ''X'', whose complement ''X'' − ''Y'' is path-connected. Fix a basepoint ''x'' ∈ ''X'' − ''Y''. For each path component ''V''''i'' of ''X'' − ''Y''∩''Y'', choose a path γi from ''x'' to a point in ''V''''i''. An element () ∈ π1(''X'' − ''Y'', ''x'') is called peripheral with respect to this choice if it is represented by a loop in ''U'' ∪ ∪ ''i''γ''i'' for every neighborhood ''U'' of ''Y''. The set of all peripheral elements with respect to a given choice forms a subgroup of π1(''X'' − ''Y'', ''x''), called a peripheral subgroup. In the diagram, a peripheral loop would start at the basepoint ''x'' and travel down the path γ until it's inside the neighborhood ''U'' of the subspace ''Y''. Then it would move around through ''U'' however it likes (avoiding ''Y''). Finally it would return to the basepoint ''x'' via γ. Since ''U'' can be a very tight envelope around ''Y'', the loop has to stay close to ''Y''. Any two peripheral subgroups of π1(''X'' − ''Y'', ''x''), resulting from different choices of paths γi, are conjugate in π1(''X'' − ''Y'', ''x''). Also, every conjugate of a peripheral subgroup is itself peripheral with respect to some choice of paths γi. Thus the peripheral subgroup's conjugacy class is an invariant of the pair (''X'', ''Y''). A peripheral subgroup, together with an ordered set of generators, is called a peripheral system for the pair (''X'', ''Y''). If a systematic method is specified for selecting these generators, the peripheral system is, in general, a stronger invariant than the peripheral subgroup alone. In fact, it is a complete invariant for knots. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Peripheral subgroup」の詳細全文を読む スポンサード リンク
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