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| In [[mathematics]], specifically [[geometry and topology]], the '''classification of manifolds''' is a basic question, about which much is known, and many open questions remain.
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| ==Main themes==
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| ===Overview===
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| * Low dimensional manifolds are classified by geometric structure; high dimensional manifolds are classified algebraically, by [[surgery theory]].
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| : "Low dimensions" means dimensions up to 4; "high dimensions" means 5 or more dimensions. The case of dimension 4 is somehow a boundary case, as it manifests "low dimentional" behaviour smoothly (but not topologically); see [[Geometric_topology#Dimension|discussion of "low" versus "high" dimension]].
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| * Different categories of manifolds yield different classifications; these are related by the notion of "structure", and more general categories have neater theories.
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| * Positive curvature is constrained, negative curvature is generic.
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| * The abstract classification of high dimensional manifolds is [[Effectively computable|ineffective]]: given two manifolds (presented as [[CW complex]]es, for instance), there is no algorithm to determine if they are isomorphic.
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| ===Different categories and additional structure===
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| {{details|Categories of manifolds}}
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| Formally, classifying [[manifold]]s is classifying objects up to [[isomorphism]].
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| There are many different notions of "manifold", and corresponding notions of
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| "map between manifolds", each of which yields a different [[category (mathematics)|category]] and a different classification question.
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| These categories are related by [[forgetful functor]]s: for instance, a differentiable manifold is also a topological manifold, and a differentiable map is also continuous, so there is a functor <math>\mbox{Diff} \to \mbox{Top}</math>.
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| These functors are in general neither one-to-one nor onto; these failures are generally referred to in terms of "structure", as follows. A topological manifold that is in the image of <math>\mbox{Diff} \to \mbox{Top}</math> is said to "admit a differentiable structure", and the fiber over a given topological manifold is "the different differentiable structures on the given topological manifold".
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| Thus given two categories, the two natural questions are:
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| * Which manifolds of a given type '''admit''' an additional structure?
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| * If it admits an additional structure, how many does it admit?
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| :More precisely, what is the structure of the set of additional structures?
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| In more general categories, this ''structure set'' has more structure: in Diff it is simply a set, but in Top it is a group, and functorially so.
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| Many of these structures are [[G-structure]]s, and the question is [[reduction of the structure group]]. The most familiar example is orientability: some manifolds are orientable, some are not, and orientable manifolds admit 2 orientations.
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| ===Enumeration versus invariants===
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| There are two usual ways to give a classification: explicitly, by an enumeration, or implicitly, in terms of invariants.
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| For instance, for orientable surfaces,
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| the [[classification of surfaces]] enumerates them as the connect sum of <math>n \geq 0</math> tori, and an invariant that classifies them is the [[genus (mathematics)|genus]] or [[Euler characteristic]].
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| Manifolds have a rich set of invariants, including:
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| * [[Point-set topology]]
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| ** [[Compact space|Compactness]]
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| ** [[Connected space|Connectedness]]
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| * Classic [[algebraic topology]]
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| ** [[Euler characteristic]]
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| ** [[Fundamental group]]
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| ** [[Cohomology ring]]
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| * [[Geometric topology]]
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| ** normal invariants ([[orientability]], [[characteristic classes]], and characteristic numbers)
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| ** [[Simple homotopy]] ([[Reidemeister torsion]])
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| ** [[Surgery theory]]
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| Modern algebraic topology (beyond [[cobordism]] theory), such as
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| [[List of cohomology theories|Extraordinary (co)homology]], is little-used
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| in the classification of manifolds, because these invariant are homotopy-invariant, and hence don't help with the finer classifications above homotopy type.
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| Cobordism groups (the bordism groups of a point) are computed, but the bordism groups of a space (such as <math>MO_*(M)</math>) are generally not.
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| ====Point-set====
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| {{details|closed manifold}}
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| The point-set classification is basic—one generally fixes point-set assumptions and then studies that class of manifold.
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| The most frequently classified class of manifolds is closed, connected manifolds.
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| Being homogeneous (away from any boundary), manifolds have no local point-set invariants, other than their dimension and boundary versus interior, and the most used global point-set properties are compactness and connectedness. Conventional names for combinations of these are:
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| * A '''compact manifold''' is a compact manifold, possibly with boundary, and not necessarily connected (but necessarily with finitely many components).
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| * A '''closed manifold''' is a compact manifold without boundary, not necessarily connected.
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| * An '''open manifold''' is a manifold without boundary (not necessarily connected), with no compact component.
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| For instance, <math>[0,1]</math> is a compact manifold, <math>S^1</math> is a closed manifold, and <math>(0,1)</math> is an open manifold, while <math>[0,1)</math> is none of these.
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| ====Computability====
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| The Euler characteristic is a [[Homology (mathematics)|homological]] invariant, and thus can be [[Effectively computable|effectively computed]] given a [[CW complex|CW structure]], so 2-manifolds are classified homologically.
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| [[Characteristic class]]es and characteristic numbers are the corresponding generalized homological invariants, but they do not classify manifolds in higher dimension (they are not a [[complete set of invariants]]): for instance, orientable 3-manifolds are [[parallelizable]] (Steenrod's theorem in [[low-dimensional topology]]), so all characteristic classes vanish. In higher dimensions, characteristic classes do not in general vanish, and provide useful but not complete data.
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| Manifolds in dimension 4 and above cannot be [[effectively computable|effectively]] classified: given two ''n''-manifolds (<math>n \geq 4</math>) presented as [[CW complex]]es or [[handlebody|handlebodies]], there is no algorithm for determining if they are isomorphic (homeomorphic, diffeomorphic). This is due to the unsolvability of the [[word problem for groups]], or more precisely, the triviality problem (given a finite presentation for a group, is it the trivial group?). Any finite presentation of a group can be realized as a 2-complex, and can be realized as the 2-skeleton of a 4-manifold (or higher). Thus one cannot even compute the [[fundamental group]] of a given high dimensional manifold, much less a classification.
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| This ineffectiveness is a fundamental reason why surgery theory does not classify manifolds up to homeomorphism. Instead, for any fixed manifold ''M'' it classifies pairs ''(N,f)'' with ''N'' a manifold and ''f:N-->M'' a ''[[homotopy equivalence]]'', two such pairs ''(N,f)'', ''(N',f')'' being regarded as equivalent if there exist a homeomorphism ''h:N-->N''' and a homotopy ''f'h ~ f:N-->M''.
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| ===Positive curvature is constrained, negative curvature is generic===
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| Many [[Riemannian_geometry#Local_to_global_theorems|classical theorems in Riemannian geometry]] show that manifolds with positive curvature are constrained, most dramatically the [[Sphere theorem|1/4-pinched sphere theorem]]. Conversely, negative curvature is generic: for instance, any manifold of dimension <math>n\geq 3</math> admits a metric with negative Ricci curvature.
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| This phenomenon is evident already for surfaces: there is a single orientable (and a single non-orientable) closed surface with positive curvature (the sphere and [[projective plane]]),
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| and likewise for zero curvature (the [[torus]] and the [[Klein bottle]]), and all surfaces of higher genus admit negative curvature metrics only.
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| Similarly for 3-manifolds: of the [[Geometrization conjecture|8 geometries]],
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| all but hyperbolic are quite constrained.
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| ==Overview by dimension==
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| * Dimensions 0 and 1 are trivial.
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| * Low dimension manifolds (dimensions 2 and 3) admit geometry.
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| * Middle dimension manifolds (dimension 4 differentiably) exhibit exotic phenomena.
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| * High dimension manifolds (dimension 5 and more differentiably, dimension 4 and more topologically) are classified by [[surgery theory]].
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| Thus dimension 4 differentiable manifolds are the most complicated:
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| they are neither geometrizable (as in lower dimension),
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| nor are they classified by surgery (as in higher dimension or topologically),
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| and they exhibit unusual phenomena, most strikingly the uncountably infinitely many [[exotic R4|exotic differentiable structures on '''R'''<sup>4</sup>]]. Notably, differentiable 4-manifolds is the only remaining open case of the [[generalized Poincaré conjecture]].
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| One can take a low dimensional point of view on high dimensional manifolds
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| and ask "Which high dimensional manifolds are geometrizable?",
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| for various notions of geometrizable (cut into geometrizable pieces as in 3 dimensions, into symplectic manifolds, and so forth). In dimension 4 and above not all manifolds
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| are geometrizable, but they are an interesting class.
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| Conversely, one can take a high dimensional point of view on low dimensional manifolds
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| and ask "What does surgery ''predict'' for low dimensional manifolds?",
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| meaning "If surgery worked in low dimensions, what would low dimensional manifolds look like?".
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| One can then compare the actual theory of low dimensional manifolds
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| to the low dimensional analog of high dimensional manifolds,
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| and see if low dimensional manifolds behave "as you would expect":
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| in what ways do they behave like high dimensional manifolds (but for different reasons,
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| or via different proofs)
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| and in what ways are they unusual?
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| ==Dimensions 0 and 1: trivial==
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| {{details|Curve}}
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| There is a unique connected 0-dimensional manifold, namely the point, and disconnected 0-dimensional manifolds are just discrete sets, classified by cardinality. They have no geometry, and their study is combinatorics.
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| A connected 1-dimensional manifold without boundary is either the circle (if compact) or the real line (if not).
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| However, maps of 1-dimensional manifolds are a non-trivial area; see below.
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| ==Dimensions 2 and 3: geometrizable==
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| {{details|Surface}}
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| {{details|3-manifold}}
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| Every closed 2-dimensional manifold (surface) admits a constant curvature metric, by the [[uniformization theorem]]. There are 3 such curvatures (positive, zero, and negative).
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| This is a classical result, and as stated, easy (the full uniformization theorem is subtler). The study of surfaces is deeply connected with [[complex analysis]] and [[algebraic geometry]], as every orientable surface can be considered a [[Riemann surface]] or complex [[algebraic curve]].
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| Every closed 3-dimensional manifold can be cut into pieces that are geometrizable, by the [[geometrization conjecture]], and there are 8 such geometries.
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| This is a recent result, and quite difficult. The proof (the [[Solution of the Poincaré conjecture]]) is analytic, not topological.
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| While the classification of surfaces is classical, maps of surfaces is an active area; see below.
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| ==Dimension 4: exotic==
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| {{details|4-manifold}}
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| 4-dimensional manifolds are the most unusual:
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| they are not geometrizable (as in lower dimensions),
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| and surgery works topologically, but not differentiably.
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| Since ''topologically'', 4-manifolds are classified by surgery,
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| the differentiable classification question is phrased in terms of
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| "differentiable structures":
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| "which (topological) 4-manifolds admit a differentiable structure,
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| and on those that do, how many differentiable structures are there?"
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| 4-manifolds often admit many unusual differentiable structures, most strikingly the uncountably infinitely many [[exotic R4|exotic differentiable structures on '''R'''<sup>4</sup>]].
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| Similarly, differentiable 4-manifolds is the only remaining open case of the [[generalized Poincaré conjecture]].
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| ==Dimension 5 and more: surgery==
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| {{details|surgery theory}}
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| In dimension 5 and above (and 4 dimensions topologically), manifolds are classified by [[surgery theory]].
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| [[File:Whitneytrickstep2.svg|thumb|The [[Whitney trick]] requires 2+1 dimensions (2 space, 1 time), hence the two Whitney disks of surgery theory require 2+2+1=5 dimensions.]]
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| The reason for dimension 5 is that the [[Whitney trick]] works in the middle dimension in dimension 5 and more: two [[Whitney disk]]s generically don't intersect in dimension 5 and above, by [[general position]] (<math>2+2 < 5</math>).
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| In dimension 4, one can resolve intersections of two Whitney disks via [[Casson handle]]s, which works topologically but not differentiably; see [[Geometric topology#Dimension|Geometric topology: Dimension]] for details on dimension.
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| More subtly, dimension 5 is the cut-off because the middle dimension has [[codimension]] more than 2: when the codimension is 2, one encounters [[knot theory]], but when the codimension is more than 2, embedding theory is tractable, via the [[calculus of functors]]. This is discussed further below.
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| ==Maps between manifolds==
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| From the point of view of [[category theory]], the classification of manifolds is one piece of understanding the category: it's classifying the ''objects''. The other question is classifying ''maps'' of manifolds up to various equivalences, and there are many results and open questions in this area.
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| For maps, the appropriate notion of "low dimension" is for some purposes "self maps of low dimensional manifolds", and for other purposes "low [[codimension]]".
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| ===Low dimensional self-maps===
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| * 1-dimensional: homeomorphisms of the circle
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| * 2-dimensional: [[mapping class group]] and [[Torelli group]]
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| ===Low codimension===
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| Analogously to the classification of manifolds, in high ''co''dimension (meaning more than 2), embeddings are classified by surgery, while in low codimension or in [[relative dimension]], they are rigid and geometric, and in the middle (codimension 2), one has a difficult exotic theory ([[knot theory]]).
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| * In codimension greater than 2, embeddings are classified by surgery theory.
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| * In codimension 2, particularly embeddings of 1-dimensional manifolds in 3-dimensional ones, one has [[knot theory]].
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| * In codimension 1, a codimension 1 embedding separates a manifold, and these are tractable.
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| * In codimension 0, a codimension 0 (proper) immersion is a [[covering space]], which are classified algebraically, and these are more naturally thoughts of as submersions.
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| * In relative dimension, a submersion with compact domain is a fiber bundle (just as in codimension 0 = relative dimension 0), which are classified algebraically.
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| ===High dimensions===
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| Particularly topologically interesting classes of maps include embeddings, immersions, and submersions.
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| Geometrically interesting are [[Isometry (Riemannian geometry)|isometries]] and isometric immersions.
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| Fundamental results in embeddings and immersions include:
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| * [[Whitney embedding theorem]]
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| * [[Whitney immersion theorem]]
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| * [[Nash embedding theorem]]
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| * [[Smale-Hirsch theorem]]
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| Key tools in studying these maps are:
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| * Gromov's [[Homotopy principle|''h''-principles]]
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| * [[Calculus of functors]]
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| One may classify maps up to various equivalences:
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| * [[homotopy]]
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| * [[cobordism]]
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| * [[concordance (mathematics)|concordance]]
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| * [[Homotopy#Isotopy|isotopy]]
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| Diffeomorphisms up to cobordism have been classified by [http://www.mathi.uni-heidelberg.de/~kreck/ Matthias Kreck]:
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| * M. Kreck, [http://projecteuclid.org/euclid.bams/1183538235 Bordism of diffeomorphisms] Bull. Amer. Math. Soc. Volume 82, Number 5 (1976), 759-761.
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| * M. Kreck, Bordism of diffeomorphisms and related topics, Springer Lect. Notes 1069 (1984)
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| ==See also==
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| * [[Holonomy#The_Berger_classification|The Berger classification]] of [[holonomy]] groups.
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| [[Category:Topology]]
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| [[Category:Differential topology]]
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| [[Category:Differential geometry]]
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| [[Category:Geometric topology]]
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| [[Category:Manifolds]]
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