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| {{Technical|date=February 2011}}
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| In [[surgery theory]], a branch of [[mathematics]], the '''stable normal bundle''' of a [[differentiable manifold]] is an invariant which encodes the stable normal (dually, tangential) data. There are analogs for generalizations of manifold, notably [[PL-manifold]]s and [[topological manifold]]s. There is also an analogue in [[homotopy theory]] for [[Poincaré space]]s, the '''Spivak spherical fibration''', named after [[Michael Spivak]] (reference below).
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| ==Construction via embeddings==
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| Given an embedding of a manifold in [[Euclidean space]] (provided by the theorem of [[Hassler_Whitney|Whitney]]), it has a [[normal bundle]]. The embedding is not unique, but for high dimension of the Euclidean space it is unique up to [[Homotopy#Isotopy|isotopy]], thus the (class of the) bundle is unique, and called the ''stable normal bundle''.
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| This construction works for any [[Poincaré space]] ''X'': a finite [[CW-complex]] admits a stably unique (up to homotopy) embedding in [[Euclidean space]], via [[general position]], and this embedding yields a spherical fibration over ''X''. For more restricted spaces (notably PL-manifolds and topological manifolds), one gets stronger data.
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| ===Details===
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| Two embeddings <math>i,i'\colon X \hookrightarrow \mathbf{R}^m</math> are ''isotopic'' if they are [[homotopic]]
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| through embeddings. Given a manifold or other suitable space ''X,'' with two embeddings into Euclidean space <math>i\colon X \hookrightarrow \mathbf{R}^m,</math> <math>j\colon X \hookrightarrow \mathbf{R}^n,</math> these will not in general be isotopic, or even maps into the same space (<math>m</math> need not equal <math>n</math>). However, one can embed these into a larger space <math>\mathbf{R}^N</math> by letting the last <math>N-m</math> coordinates be 0:
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| :<math>i\colon X \hookrightarrow \mathbf{R}^m \cong \mathbf{R}^m \times \left\{(0,\dots,0)\right\} \subset \mathbf{R}^m \times \mathbf{R}^{N-m} \cong \mathbf{R}^N.</math>
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| This process of adjoining trivial copies of Euclidean space is called ''stabilization.''
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| One can thus arrange for any two embeddings into Euclidean space to map into the same Euclidean space (taking <math>N = \max(m,n)</math>), and, further, if <math>N</math> is sufficiently large, these embeddings are isotopic, which is a theorem.
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| Thus there is a unique stable isotopy class of embedding: it is not a particular embedding (as there are many embeddings), nor an isotopy class (as the target space is not fixed: it is just "a sufficiently large Euclidean space"), but rather a stable isotopy class of maps. The normal bundle associated with this (stable class of) embeddings is then the stable normal bundle.
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| One can replace this stable isotopy class with an actual isotopy class by fixing the target space, either by using [[Hilbert space]] as the target space, or (for a fixed dimension of manifold <math>n</math>) using a fixed <math>N</math> sufficiently large, as ''N'' depends only on ''n'', not the manifold in question.
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| More abstractly, rather than stabilizing the embedding, one can take any embedding, and then take a vector bundle direct sum with a sufficient number of trivial line bundles; this corresponds exactly to the normal bundle of the stabilized embedding.
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| ==Construction via [[classifying space]]s==
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| An ''n''-manifold ''M'' has a tangent bundle, which has a classifying map (up to homotopy)
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| :<math>\tau_M\colon M \to BO(n).</math>
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| Composing with the inclusion <math>BO(n) \to BO</math> yields (the homotopy class of a classifying map of) the stable tangent bundle. The normal bundle of an embedding <math>M \subset R^{n+k}</math> (<math>k</math> large) is an inverse <math>\nu_M\colon M \to BO(k)</math> for <math>\tau_M</math>, such that the [[Whitney sum]] <math>\tau_M\oplus \nu_M
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| \colon M \to BO(n+k)</math> is trivial. The homotopy class of the composite
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| <math>\nu_M\colon M \to BO(k) \to BO</math> is independent of the choice of inverse,
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| classifying the stable normal bundle <math>\nu_M</math>.
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| ==Motivation==
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| There is no intrinsic notion of a normal vector to a manifold, unlike tangent or cotangent vectors – for instance, the normal space depends on which dimension one is embedding into – so the stable normal bundle instead provides a notion of a stable normal space: a normal space (and normal vectors) up to trivial summands.
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| Why stable normal, instead of stable tangent? Stable normal data is used instead of unstable tangential data because generalizations of manifolds have natural stable normal-type structures, coming from [[tubular neighborhood]]s and generalizations, but not unstable tangential ones, as the local structure is not smooth.
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| Spherical fibrations over a space ''X'' are classified by the homotopy classes of maps <math>X \to BG</math> to a
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| [[classifying space]] <math>BG</math>, with [[homotopy groups]] the [[stable homotopy groups of spheres]] | |
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| :<math>\pi_*(BG)=\pi_{*-1}^S. </math> | |
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| The forgetful map <math>BO \to BG</math> extends to a [[fibration]] sequence
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| :<math>BO \to BG \to B(G/O).</math>
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| A [[Poincaré space]] ''X'' does not have a tangent bundle, but it does have a well-defined stable spherical [[fibration]], which for a differentiable manifold is the spherical fibration associated to the stable normal bundle; thus a primary obstruction to ''X'' having the homotopy type of a differentiable manifold is that the spherical fibration lifts to a vector bundle, i.e. the Spivak spherical fibration <math>X \to BG</math> must lift to <math>X \to BO</math>, which is equivalent to the map <math>X \to B(G/O)</math> being [[null homotopic]]
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| Thus the bundle obstruction to the existence of a (smooth) manifold structure is the class <math>X \to B(G/O)</math>.
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| The secondary obstruction is the Wall [[surgery obstruction]].
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| ==Applications==
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| The stable normal bundle is fundamental in [[surgery theory]] as a primary obstruction:
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| *For a [[Poincaré space]] ''X'' to have the homotopy type of a smooth manifold, the map <math>X \to B(G/O)</math> must be [[null homotopic]]
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| *For a homotopy equivalence <math>f\colon M \to N</math> between two manifolds to be homotopic to a diffeomorphism, it must pull back the stable normal bundle on ''N'' to the stable normal bundle on ''M''.
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| More generally, its generalizations serve as replacements for the (unstable) tangent bundle.
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| ==References==
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| * {{citation
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| |last=Spivak
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| |first=Michael
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| |author-link=Michael Spivak
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| |title=Spaces satisfying Poincaré duality
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| |journal=[[Topology (journal)|Topology]]
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| |issue=6
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| |year=1967
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| |pages=77–101
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| |postscript=, MR0214071 (35 #4923) 55.50.
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| }}
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| [[Category:Differential geometry]]
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| [[Category:Surgery theory]]
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