はめ込みのソースを表示

[[File:Klein bottle.svg|thumb|3次元空間にはめ込まれた[[クラインの壺]]．]]
[[数学]]において，'''はめ込み''' (immersion) は[[可微分多様体]]の間の[[可微分写像]]であって[[微分写像|微分]]がいたるところ[[単射]]であるもののことである<ref>This definition is given by {{harvnb|Bishop|Crittenden|1964|page=185}}, {{harvnb|Darling|1994|page=53}}, {{harvnb|do Carmo|1994|page=11}}, {{harvnb|Frankel|1997|page=169}}, {{harvnb|Gallot|Hulin|Lafontaine|2004|page=12}}, {{harvnb|Kobayashi|Nomizu|1963|page=9}}, {{harvnb|Kosinski|2007|page=27}}, {{harvnb|Szekeres|2004|page=429}}.</ref>．明示的には，{{math|''f'': ''M'' → ''N''}} がはめ込みであるとは，

:<math>D_pf \colon T_p M \to T_{f(p)}N\,</math>

が {{mvar|M}} のすべての点 {{mvar|p}} において単射関数であることをいう（ここで {{mvar|T<sub>p</sub>X}} は多様体 {{mvar|X}} の点 {{mvar|p}} における[[接空間]]を表す）．同じことであるが，{{mvar|f}} がはめ込みであるとは，その微分が {{mvar|M}} の次元に等しい定数{{仮リンク|階数 (微分トポロジー)|label=階数|en|rank (differential topology)}}を持つことである<ref>This definition is given by {{harvnb|Crampin|Pirani|1994|page=243}}, {{harvnb|Spivak|1999|page=46}}.</ref>：

:<math>\operatorname{rank}D_p f = \dim M.</math>

関数 {{mvar|f}} それ自身は単射である必要はない．

関連概念は[[微分位相幾何学|埋め込み]]である．滑らかな埋め込みは位相的な埋め込みでもある単射はめ込み {{math|''f'': ''M'' → ''N''}} であり，したがって {{mvar|M}} は {{mvar|N}} におけるその像に[[微分同相]]である．はめ込みはちょうど局所的な埋め込みである――つまり，任意の点 {{math|''x'' ∈ ''M''}} に対して，{{mvar|x}} のある[[近傍 (位相空間論)|近傍]] {{math|''U'' ⊂ ''M''}} が存在して，{{math|''f'': ''U'' → ''N''}} が埋め込みとなり，逆に局所的な埋め込みははめ込みである<ref>局所微分同相に基づいたこの種の定義は {{harvnb|Bishop|Goldberg|1968|page=40}}, {{harvnb|Lang|1999|page=26}} によって与えられている．</ref>．無限次元多様体に対して，これははめ込みの定義として取られることもある<ref>この種の無限次元の定義は {{harvnb|Lang|1999|page=26}} によって与えられている．</ref>．

[[File:Immersedsubmanifold nonselfintersection.jpg|thumb|埋め込みではない単射に{{仮リンク|はめ込まれた部分多様体|en|Immersed submanifold}}．]]
{{mvar|M}} が[[コンパクト空間|コンパクト]]ならば，単射なはめ込みは埋め込みであるが，{{mvar|M}} がコンパクトでなければ，そうとは限らない；連続全単射と[[同相]]を比較せよ．

==正則ホモトピー==
[[多様体]] {{mvar|M}} から {{mvar|N}} への2つのはめ込み {{mvar|f}} と {{mvar|g}} の間の{{仮リンク|正則ホモトピー|en|regular homotopy}}は次のような可微分関数 {{math|''H'': ''M'' × [0, 1] → ''N''}} と定義される：すべての {{math|''t'' &isin; [0, 1]}} に対して，すべての {{math|''x'' ∈ ''M''}} に対して {{math|1=''H<sub>t</sub>''(''x'') = ''H''(''x'', ''t'')}} によって定義される関数 {{math|''H<sub>t</sub>'': ''M'' → ''N''}} ははめ込みで，{{math|1=''H''<sub>0</sub> = ''f''}}, {{math|1=''H''<sub>1</sub> = ''g''}} である．正則ホモトピーはしたがってはめ込みを通した[[ホモトピー]]である．

== 分類 ==
[[ハスラー・ホイットニー]]は1940年代にはめ込みと正則ホモトピーの系統的な研究を創始し，{{math|2''m'' < ''n'' + 1}} に対して {{mvar|m}} 次元多様体から {{mvar|n}} 次元多様体へのすべての写像 {{math|''f'': ''M<sup>m</sup>'' → ''N<sup>n</sup>''}} がはめ込みに[[ホモトープ]]であること，そして {{math|2''m'' < ''n''}} に対しては実は[[埋め込み (数学)|埋め込み]]にホモトープであることを証明した．これらが{{仮リンク|ホイットニーのはめ込み定理|en|Whitney immersion theorem}}と{{仮リンク|ホイットニーの埋め込み定理|en|Whitney embedding theorem}}である．

[[スティーブン・スメール]]ははめ込み {{math|''f'': ''M<sup>m</sup>'' → '''R'''<sup>''n''</sup>}} の正則ホモトピー類をある{{仮リンク|スティーフェル多様体|en|Stiefel manifold}}の[[ホモトピー群]]として表した．{{仮リンク|sphere eversion|en|sphere eversion}} は特に著しい結果であった．

{{仮リンク|Morris Hirsch|en|Morris Hirsch}} は Smale の表示を任意の {{mvar|n}} 次元多様体 {{mvar|N<sup>n</sup>}} 内の任意の {{mvar|m}} 次元多様体 {{mvar|M<sup>m</sup>}} のはめ込みの正則ホモトピー類の[[ホモトピー論]]による記述に一般化した．

はめ込みの Hirsch–Smale 分類は[[ミハイル・グロモフ|Michael Gromov]]によって一般化された．

=== 存在 ===
[[File:Moebius Surface 1 Display Small.png|thumb|[[メビウスの帯]]は接束が非自明だから余次元 {{math|0}} にはめ込めない．]]
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The primary obstruction to the existence of an immersion {{nowrap|''i'' : ''M<sup>m</sup>'' → '''R'''<sup>''n''</sup>}} is the [[stable normal bundle]] of ''M'', as detected by its [[characteristic classes]], notably its [[Stiefel–Whitney class]]es. That is, since '''R'''<sup>''n''</sup> is parallelizable, the pullback of its tangent bundle to ''M'' is trivial; since this pullback is the direct sum of the (intrinsically defined) tangent bundle on ''M'', ''TM'', which has dimension ''m'', and of the normal bundle ''ν'' of the immersion ''i'', which has dimension {{nowrap|''n'' − ''m''}}, for there to be a [[codimension]] ''k'' immersion of ''M'', there must be a vector bundle of dimension ''k'', ''ξ''<sup>''k''</sup>, standing in for the normal bundle ''ν'', such that {{nowrap|''TM'' ⊕ ''ξ''<sup>''k''</sup>}} is trivial. Conversely, given such a bundle, an immersion of ''M'' with this normal bundle is equivalent to a codimension 0 immersion of the total space of this bundle, which is an open manifold.

The stable normal bundle is the class of normal bundles plus trivial bundles, and thus if the stable normal bundle has cohomological dimension ''k'', it cannot come from an (unstable) normal bundle of dimension less than ''k''. Thus, the cohomology dimension of the stable normal bundle, as detected by its highest non-vanishing characteristic class, is an obstruction to immersions.

Since characteristic classes multiply under direct sum of vector bundles, this obstruction can be stated intrinsically in terms of the space ''M'' and its tangent bundle and cohomology algebra. This obstruction was stated (in terms of the tangent bundle, not stable normal bundle) by Whitney.

For example, the [[Möbius strip]] has non-trivial tangent bundle, so it cannot immerse in codimension 0 (in '''R'''<sup>2</sup>), though it embeds in codimension 1 (in '''R'''<sup>3</sup>).

{{harvs|authorlink=William S. Massey|first=William S.|last=Massey|year=1960|txt}} showed that these characteristic classes (the Stiefel–Whitney classes of the stable normal bundle) vanish above degree {{nowrap|''n'' − ''α''(''n'')}}, where {{nowrap|''α''(''n'')}} is the number of "1" digits when ''n'' is written in binary; this bound is sharp, as realized by [[real projective space]]. This gave evidence to the ''Immersion Conjecture'', namely that every ''n''-manifold could be immersed in codimension {{nowrap|''n'' − ''α''(''n'')}}, i.e., in '''R'''<sup>2''n''−α(''n'')</sup>. This conjecture was proven by {{harvs|first=Ralph|last=Cohen|authorlink=Ralph Louis Cohen|year=1985|txt}}.

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=== 余次元 0 ===
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Codimension 0 immersions are equivalently [[relative dimension|''relative'' dimension]] 0 ''[[Submersion (mathematics)|submersions]]'', and are better thought of as submersions. A codimension 0 immersion of a [[closed manifold]] is precisely a [[covering map]], i.e., a [[fiber bundle]] with 0-dimensional (discrete) fiber. By [[Ehresmann's theorem]] and Phillips' theorem on submersions, a [[proper map|proper]] submersion of manifolds is a fiber bundle, hence codimension/relative dimension 0 immersions/submersions behave like submersions.

Further, codimenson 0 immersions do not behave like other immersions, which are largely determined by the stable normal bundle: in codimension 0 one has issues of [[fundamental class]] and cover spaces. For instance, there is no codimension 0 immersion {{nowrap|'''S'''<sup>1</sup> → '''R'''<sup>1</sup>}}, despite the circle being parallelizable, which can be proven because the line has no fundamental class, so one does not get the required map on top cohomology. Alternatively, this is by [[invariance of domain]]. Similarly, although '''S'''<sup>3</sup> and the 3-torus '''T'''<sup>3</sup> are both parallelizable, there is no immersion {{nowrap|'''T'''<sup>3</sup> → '''S'''<sup>3</sup>}} – any such cover would have to be ramified at some points, since the sphere is simply connected.

Another way of understanding this is that a codimension ''k'' immersion of a manifold corresponds to a codimension 0 immersion of a ''k''-dimensional vector bundle, which is an [[open manifold|''open'' manifold]] if the codimension is greater than 0, but to a closed manifold in codimension 0 (if the original manifold is closed).

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==多重点==
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A '''''k''-tuple point''' (double, triple, etc.) of an immersion {{nowrap|''f'' : ''M'' → ''N''}} is an unordered set {{nowrap|{''x''<sub>1</sub>, ..., ''x<sub>k</sub>''}{{null}}}} of distinct points {{nowrap|''x<sub>i</sub>'' ∈ ''M''}} with the same image {{nowrap|''f''(''x<sub>i</sub>'') ∈ ''N''}}. If ''M'' is an ''m''-dimensional manifold and ''N'' is an ''n''-dimensional manifold then for an immersion {{nowrap|''f'' : ''M'' → ''N''}} in [[general position]] the set of ''k''-tuple points is an {{nowrap|(''n'' − ''k''(''n'' − ''m''))}}-dimensional manifold.  Every embedding is an immersion without multiple points (where {{nowrap|''k'' > 1}}).  Note, however, that the converse is false: there are injective immersions that are not embeddings.

The nature of the multiple points classifies immersions; for example, immersions of a circle in the plane are classified up to regular homotopy by the number of double points.

At a key point in [[surgery theory]] it is necessary to decide if an immersion {{nowrap|''f'' : '''S'''<sup>''m''</sup> → ''N''<sup>2''m''</sup>}} of an ''m''-sphere in a 2''m''-dimensional manifold is regular homotopic to an embedding, in which case it can be killed by surgery. [[C.T.C. Wall|Wall]] associated to ''f'' an invariant ''μ''(''f'') in a quotient of the [[fundamental group]] ring '''Z'''[π<sub>1</sub>(''N'')] which counts the double points of ''f'' in the [[universal cover]] of ''N''. For {{nowrap|''m'' > 2}}, ''f'' is regular homotopic to an embedding if and only if {{nowrap|1=''μ''(''f'') = 0}} by the [[Hassler Whitney|Whitney]] trick.

One can study embeddings as "immersions without multiple points", since immersions are easier to classify. Thus, one can start from immersions and try to eliminate multiple points, seeing if one can do this without introducing other singularities – studying "multiple disjunctions". This was first done by [[André Haefliger]], and this approach is fruitful in codimension 3 or more – from the point of view of surgery theory, this is "high (co)dimension", unlike codimension 2 which is the knotting dimension, as in [[knot theory]]. It is studied categorically via the "[[calculus of functors]]" by [http://www.math.brown.edu/faculty/goodwillie.html Thomas Goodwillie], [http://www.math.wayne.edu/~klein/ John Klein], and [http://www.maths.abdn.ac.uk/staff/display.php?key=m.weiss Michael S. Weiss].

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==例と性質==
* [[クラインの壺]]や，すべての他の向き付け不可能な閉曲面は，3次元空間にはめ込むことができるが，埋め込むことはできない．
[[File:Quadrifolium.svg|thumb|{{仮リンク|四葉 (数学)|label=四葉|en|quadrifolium}}，4弁のバラ．]]
* {{mvar|k}} 弁の[[バラ曲線|バラ]]は円周の平面へのただ1つの {{mvar|k}} 重点を持ったはめ込みである．{{mvar|k}} は任意の奇数でよいが，偶数なら4の倍数で，8の字はバラでない．
* {{仮リンク|Whitney–Graustein の定理|en|Whitney–Graustein theorem}}により，円周の平面へのはめ込みの正則ホモトピー類は[[回転数 (数学)|回転数]]によって分類され，この数は代数的に（すなわち符号付きで）数えた二重点の個数でもある．
* {{仮リンク|sphere eversion|en|sphere eversion|label=球面は表裏をひっくり返すことができる}}：標準的な埋め込み {{math|''f'': '''S'''<sup>2</sup> → '''R'''<sup>3</sup>}} ははめ込みの正則ホモトピー {{math|''f<sub>t</sub>'': '''S'''<sup>2</sup> → '''R'''<sup>3</sup>}} によって {{math|1=''f''<sub>1</sub> = −''f''<sub>0</sub>: '''S'''<sup>2</sup> → '''R'''<sup>3</sup>}} と結ばれる．
* [[ボーイ曲面]]は[[実射影平面]]の3次元空間へのはめ込みである；したがって球面の2対1のはめ込みでもある．
* {{仮リンク|モラン曲面|en|Morin surface}}は球面のはめ込みである；これとボーイ曲面はともに sphere eversion の途中のモデルとして生じる．

<gallery widths="300px" heights="300px">
File:BoysSurfaceTopView.PNG|[[ボーイ曲面]]
File:MorinSurfaceAsSphere'sInsideVersusOutside.PNG|{{仮リンク|モラン曲面|en|Morin surface}}
</gallery>

=== はめ込まれた平面曲線 ===
[[File:Winding Number Around Point.svg|thumb|300px|この曲線の{{仮リンク|全曲率|en|total curvature}} は {{math|6''π''}} で，[[回転数 (数学)|turning number]] は {{math|3}} だが，{{mvar|p}} についての[[回転数 (数学)|回転数]] は {{math|2}} である．]]
{{main|{{仮リンク|Whitney–Graustein の定理|en|Whitney–Graustein theorem}}|{{仮リンク|全曲率|en|Total curvature}}|[[回転数 (数学)|Turning number]]}}
はめ込まれた平面曲線は well-defined な [[回転数 (数学)|Turning number]] をもち，{{仮リンク|全曲率|en|total curvature}}を {{math|2''&pi;''}} で割ったものとして定義できる．これは {{仮リンク|Whitney–Graustein の定理|en|Whitney–Graustein theorem}}により正則ホモトピーで不変である――位相幾何学的には，それは[[ガウス写像]]の次数，あるいは同じことであるが，原点についての（消えない）unit tangent の[[回転数 (数学)|回転数]]である．さらに，これは{{仮リンク|完全不変量|en|complete set of invariants}}である――同じ回転数を持つ任意の2つの平面曲線は正則ホモトピックである．

すべてのはめ込まれた平面曲線は交差する点を分離して埋め込まれた空間曲線に持ちあがるが，これは高次元では正しくない．追加の情報（どの紐が上にあるか）により，はめ込まれた平面曲線は {{仮リンク|knot diagram|en|knot diagram}} を生じ，これは[[結び目理論]]において中心的に興味を持たれる．はめ込まれた平面曲線は正則ホモトピーの違いを除いて回転数によって決定されるが，結び目は非常に豊かで複雑な構造を持つ．

=== 3次元空間にはめ込まれた曲面 ===
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3次元空間にはめ込まれた曲面の研究は
The study of immersed surfaces in 3-space is closely connected with the study of knotted (embedded) surfaces in 4-space, by analogy with the theory of [[knot diagram]]s (immersed plane curves (2-space) as projections of knotted curves in 3-space): given a knotted surface in 4-space, one can project it to an immersed surface in 3-space, and conversely, given an immersed surface in 3-space, one may ask if it lifts to 4-space – is it the projection of a knotted surface in 4-space? This allows one to relate questions about these objects.

A basic result, in contrast to the case of plane curves, is that not every immersed surface lifts to a knotted surface.<ref>{{Harvnb|Carter|Saito|1998}}; {{Harvnb|Carter|Kamada|Saito|2004|loc=[https://books.google.co.jp/books?id=erc9fktHqhsC&pg=PA17&redir_esc=y&hl=ja Remark 1.23, p. 17]}}</ref> In some cases the obstruction is 2-torsion, such as in ''[http://www.southalabama.edu/mathstat/personal_pages/carter/nukos.jpg Koschorke's example]'',<ref>{{Harvnb|Koschorke|1979}}</ref> which is an immersed surface (formed from 3 Möbius bands, with a [[Tripoint (disambiguation)|triple point]]) that does not lift to a knotted surface, but it has a double cover that does lift. A detailed analysis is given in {{Harvtxt|Carter|Saito|1998}}, while a more recent survey is given in {{Harvtxt|Carter|Kamada|Saito|2004}}.

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== 一般化 ==
{{main|{{仮リンク|ホモトピー原理|en|Homotopy principle}}}}
はめこみ理論の遠大な一般化は{{仮リンク|ホモトピー原理|en|Homotopy principle}}である：はめこみの条件（微分の階数がつねに {{mvar|k}}）は，関数の偏微分のことばで述べられるから，{{仮リンク|偏微分関係式|en|partial differential relation}} (partial differential relation, PDR) と考えることができる．すると Smale–Hirsch のはめ込み理論はこれがホモトピー論に帰着されるという結果であり，ホモトピー原理は PDR がホモトピー論に帰着する一般の条件や理由を与える．

==関連項目==

*{{仮リンク|はめ込まれた部分多様体|en|Immersed submanifold}}
*[[等長はめ込み]]
*[[沈め込み]]

==脚注==
{{reflist}}

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{{refend}}

==外部リンク==
*[http://www.map.mpim-bonn.mpg.de/Immersion Immersion] at the Manifold Atlas
*[http://www.encyclopediaofmath.org/index.php/Immersion_of_a_manifold Immersion of a manifold] at the Encyclopedia of Mathematics

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