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Whitney embedding theorem
Whitney embedding theorem
Whitney embedding theorem
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In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney:

  • The strong Whitney embedding theorem states that any smooth real m-dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in the real 2m-space, ⁠⁠ if m > 0. This is the best linear bound on the smallest-dimensional Euclidean space that all m-dimensional manifolds embed in, as the real projective spaces of dimension m cannot be embedded into real (2m − 1)-space if m is a power of two (as can be seen from a characteristic class argument, also due to Whitney).
  • The weak Whitney embedding theorem states that any continuous function from an n-dimensional manifold to an m-dimensional manifold may be approximated by a smooth embedding provided m > 2n. Whitney similarly proved that such a map could be approximated by an immersion provided m > 2n − 1. This last result is sometimes called the Whitney immersion theorem.

About the proof

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Weak embedding theorem

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The weak Whitney embedding is proved through a projection argument.

When the manifold is compact, one can first use a covering by finitely many local charts and then reduce the dimension with suitable projections.[1]: Ch. 1 §3 [2]: Ch. 6 [3]: Ch. 5 §3 

Strong embedding theorem

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The general outline of the proof is to start with an immersion ⁠⁠ with transverse self-intersections. These are known to exist from Whitney's earlier work on the weak immersion theorem. Transversality of the double points follows from a general-position argument. The idea is to then somehow remove all the self-intersections. If M has boundary, one can remove the self-intersections simply by isotoping M into itself (the isotopy being in the domain of f), to a submanifold of M that does not contain the double-points. Thus, we are quickly led to the case where M has no boundary. Sometimes it is impossible to remove the double-points via an isotopy—consider for example the figure-8 immersion of the circle in the plane. In this case, one needs to introduce a local double point.

Introducing double-point.

Once one has two opposite double points, one constructs a closed loop connecting the two, giving a closed path in ⁠⁠ Since ⁠⁠ is simply connected, one can assume this path bounds a disc, and provided 2m > 4 one can further assume (by the weak Whitney embedding theorem) that the disc is embedded in ⁠⁠ such that it intersects the image of M only in its boundary. Whitney then uses the disc to create a 1-parameter family of immersions, in effect pushing M across the disc, removing the two double points in the process. In the case of the figure-8 immersion with its introduced double-point, the push across move is quite simple (pictured).

Cancelling opposite double-points.

This process of eliminating opposite sign double-points by pushing the manifold along a disc is called the Whitney Trick.

To introduce a local double point, Whitney created immersions ⁠⁠ which are approximately linear outside of the unit ball, but containing a single double point. For m = 1 such an immersion is given by

Notice that if α is considered as a map to ⁠⁠ like so:

then the double point can be resolved to an embedding:

Notice β(t, 0) = α(t) and for a ≠ 0 then as a function of t, β(t, a) is an embedding.

For higher dimensions m, there are αm that can be similarly resolved in ⁠⁠ For an embedding into ⁠⁠ for example, define

This process ultimately leads one to the definition:

where

The key properties of αm is that it is an embedding except for the double-point αm(1, 0, ... , 0) = αm(−1, 0, ... , 0). Moreover, for |(t1, ... , tm)| large, it is approximately the linear embedding (0, t1, 0, t2, ... , 0, tm).

Eventual consequences of the Whitney trick

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The Whitney trick was used by Stephen Smale to prove the h-cobordism theorem; from which follows the Poincaré conjecture in dimensions m ≥ 5, and the classification of smooth structures on discs (also in dimensions 5 and up). This provides the foundation for surgery theory, which classifies manifolds in dimension 5 and above.

Given two oriented submanifolds of complementary dimensions in a simply connected manifold of dimension ≥ 5, one can apply an isotopy to one of the submanifolds so that all the points of intersection have the same sign.

History

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See also: History of manifolds and varieties

The occasion of the proof by Hassler Whitney of the embedding theorem for smooth manifolds is said (rather surprisingly) to have been the first complete exposition of the manifold concept precisely because it brought together and unified the differing concepts of manifolds at the time: no longer was there any confusion as to whether abstract manifolds, intrinsically defined via charts, were any more or less general than manifolds extrinsically defined as submanifolds of Euclidean space. See also the history of manifolds and varieties for context.

Sharper results

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Although every n-manifold embeds in ⁠⁠ one can frequently do better. Let e(n) denote the smallest integer so that all compact connected n-manifolds embed in ⁠⁠ Whitney's strong embedding theorem states that e(n) ≤ 2n. For n = 1, 2 we have e(n) = 2n, as the circle and the Klein bottle show. More generally, for n = 2k we have e(n) = 2n, as the 2k-dimensional real projective space show. Whitney's result can be improved to e(n) ≤ 2n − 1 unless n is a power of 2. This is a result of André Haefliger and Morris Hirsch (for n > 4) and C. T. C. Wall (for n = 3); these authors used important preliminary results and particular cases proved by Hirsch, William S. Massey, Sergey Novikov and Vladimir Rokhlin.[4] At present the function e is not known in closed-form for all integers (compare to the Whitney immersion theorem, where the analogous number is known).

Restrictions on manifolds

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One can strengthen the results by putting additional restrictions on the manifold. For example, the n-sphere always embeds in ⁠⁠ – which is the best possible (closed n-manifolds cannot embed in ⁠⁠). Any compact orientable surface and any compact surface with non-empty boundary embeds in ⁠⁠ though any closed non-orientable surface needs ⁠⁠

If N is a compact orientable n-dimensional manifold, then N embeds in ⁠⁠ (for n not a power of 2 the orientability condition is superfluous). For n a power of 2 this is a result of André Haefliger and Morris Hirsch (for n > 4), and Fuquan Fang (for n = 4); these authors used important preliminary results proved by Jacques Boéchat and Haefliger, Simon Donaldson, Hirsch and William S. Massey.[4] Haefliger proved that if N is a compact n-dimensional k-connected manifold, then N embeds in ⁠⁠ provided 2k + 3 ≤ n.[4]

Isotopy versions

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A relatively 'easy' result is to prove that any two embeddings of a 1-manifold into ⁠⁠ are isotopic (see Knot theory#Higher dimensions). This is proved using general position, which also allows to show that any two embeddings of an n-manifold into ⁠⁠ are isotopic. This result is an isotopy version of the weak Whitney embedding theorem.

Wu proved that for n ≥ 2, any two embeddings of an n-manifold into ⁠⁠ are isotopic. This result is an isotopy version of the strong Whitney embedding theorem.

As an isotopy version of his embedding result, Haefliger proved that if N is a compact n-dimensional k-connected manifold, then any two embeddings of N into ⁠⁠ are isotopic provided 2k + 2 ≤ n. The dimension restriction 2k + 2 ≤ n is sharp: Haefliger went on to give examples of non-trivially embedded 3-spheres in ⁠⁠ (and, more generally, (2d − 1)-spheres in ⁠⁠). See further generalizations Archived 2016-09-30 at the Wayback Machine.

See also

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Notes

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Whitney embedding theorem

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The Whitney embedding theorem is a foundational result in stating that every smooth nn-dimensional manifold admits a smooth embedding into the R2n\mathbb{R}^{2n}. Proved by the American mathematician Hassler Whitney, this theorem demonstrates that the local and global structure of a manifold can be realized without self-intersections in a flat ambient space of twice the dimension, enabling the application of analytic tools from to abstract manifolds. The result holds for Hausdorff, second-countable manifolds and is sharp in the sense that there exist manifolds requiring at least 2n2n dimensions for embedding, such as the in dimension 2. Whitney first established a weaker version of the theorem in 1936, showing that any smooth nn-dimensional manifold can be smoothly embedded into R2n+1\mathbb{R}^{2n+1}. This initial proof relied on triangulations and approximation techniques to construct injective immersions and resolve intersections. The strengthening to R2n\mathbb{R}^{2n}, achieved in 1944, introduced the innovative "Whitney trick"—a method to pair and cancel double points of an immersion using homotopy in higher dimensions, provided the codimension is at least 2. Complementing the embedding theorem, Whitney simultaneously proved an immersion theorem: every smooth nn-dimensional manifold can be immersed into R2n−1\mathbb{R}^{2n-1}, where self-intersections are permitted but the differential is injective everywhere. These theorems revolutionized the study of manifolds by confirming their embeddability in , which facilitates computations in , geometry, and analysis, such as applying for regularity and transversality arguments. For compact manifolds, the embeddings are proper and closed submanifolds; for non-compact ones, proper embeddings ensure the preimage of compact sets is compact. The results extend to finite smoothness classes CkC^k and have analogs in other categories, though the smooth case remains central.

Background Concepts

Smooth Manifolds

A smooth manifold provides the foundational structure for studying differentiable geometry and in higher dimensions. Formally, an nn-dimensional smooth manifold MM is a second-countable Hausdorff that is locally Euclidean of nn, meaning every point in MM has a neighborhood homeomorphic to an open subset of Rn\mathbb{R}^n, together with a maximal atlas of charts where the coordinate transition maps are C∞C^\infty (infinitely differentiable) functions. This atlas defines a on MM, allowing the consistent extension of notions like differentiability and spaces from to the manifold. The key smoothness condition arises from the transition maps: for two charts (U,ϕ)(U, \phi) and (V,ψ)(V, \psi) with U∩V≠∅U \cap V \neq \emptyset, the composition ψ∘ϕ−1:ϕ(U∩V)→ψ(U∩V)\psi \circ \phi^{-1}: \phi(U \cap V) \to \psi(U \cap V) must be a C∞C^\infty diffeomorphism between open subsets of Rn\mathbb{R}^n. These maps ensure that the manifold's geometry is compatible across overlapping local coordinates, enabling global definitions of smooth functions and vector fields on MM. A maximal smooth atlas is obtained by including all charts compatible with a given smooth atlas, guaranteeing that the smooth structure is uniquely determined up to diffeomorphism. Classic examples illustrate these . The Rn\mathbb{R}^n serves as the standard nn-dimensional smooth manifold with the identity atlas. The nn- Sn={x∈Rn+1:∥x∥=1}S^n = \{ x \in \mathbb{R}^{n+1} : \|x\| = 1 \} is a compact smooth manifold of nn, covered by charts. The nn- Tn=S1×⋯×S1T^n = S^1 \times \cdots \times S^1 (nn times) inherits its as a product manifold, while the real projective space RPn\mathbb{RP}^n is obtained by quotienting the SnS^n by antipodal identification, with charts away from the quotient points. In the context of smooth manifolds, second-countability—requiring a countable basis for the —is a standard assumption that implies paracompactness. Paracompactness ensures every open cover admits a locally finite refinement, which is crucial for constructing partitions of unity and supporting the existence of embeddings into Euclidean spaces. This property holds for all second-countable Hausdorff manifolds, facilitating many advanced constructions in .

Embeddings and Immersions

In , smooth manifolds serve as the foundational spaces for studying mappings with . A smooth map f:M→Nf: M \to N between smooth manifolds MmM^m and NnN^n of dimensions m≤nm \leq n is called a smooth immersion if its differential dfp:TpM→Tf(p)Ndf_p: T_pM \to T_{f(p)}N is injective at every point p∈Mp \in M, meaning that the linear map dfpdf_p has full rank mm everywhere. This condition ensures that the map locally preserves the structure without collapsing dimensions, allowing MM to be "locally like" a of NN near each image point. A smooth embedding is a smooth immersion f:M→Nf: M \to N that is also a topological embedding, meaning ff is a from MM onto its image f(M)⊂Nf(M) \subset N. For ff to be a topological embedding, it must be injective and proper (i.e., the preimage of every compact subset of NN is compact in MM), which guarantees that f(M)f(M) inherits the topology of MM without self-intersections or "escaping to infinity." In contrast, a topological embedding is a continuous injective proper map that is a onto its image, but lacks the smoothness or immersion condition; the smooth variant thus combines differential injectivity with topological fidelity. A classic example of a smooth embedding is the standard inclusion i:S1↪R2i: S^1 \hookrightarrow \mathbb{R}^2, where the unit circle is mapped as itself, preserving both and injectivity while being proper due to . Conversely, the figure-eight , parametrized by γ:S1→R2\gamma: S^1 \to \mathbb{R}^2 with γ(θ)=(sin⁡θ,sin⁡2θ)\gamma(\theta) = (\sin \theta, \sin 2\theta), is a smooth immersion because its differential is injective everywhere, but it fails to be an as it is not injective—the map self-intersects at the origin. Whitney's contributions emphasized arguments to achieve transversality in such mappings, allowing generic perturbations of smooth maps to intersect submanifolds transversely, which is crucial for constructing embeddings by avoiding degenerate intersections. This transversality ensures that double points, if present, occur in controlled dimensions, facilitating their resolution in higher ambient spaces.

Statement of the Theorems

Weak Embedding Theorem

The weak Whitney embedding theorem states that every smooth nn-dimensional manifold MM, assumed Hausdorff and second-countable, admits a smooth embedding into R2n+1\mathbb{R}^{2n+1}. This result, established by Hassler Whitney in 1936, guarantees the existence of a smooth map f:M→R2n+1f: M \to \mathbb{R}^{2n+1} that is an immersion and a homeomorphism onto its image, ensuring the image f(M)f(M) is a smooth submanifold of R2n+1\mathbb{R}^{2n+1} without self-intersections and topologically equivalent to MM. A smooth map f:M→R2n+1f: M \to \mathbb{R}^{2n+1} is an embedding if it is an immersion—meaning the differential dfp:TpM→R2n+1df_p: T_p M \to \mathbb{R}^{2n+1} is injective for every p∈Mp \in M—and if ff is a onto its image.
Since dim⁡TpM=n\dim T_p M = n and dim⁡R2n+1=2n+1>n\dim \mathbb{R}^{2n+1} = 2n+1 > n, injectivity is possible, ensuring full rank nn everywhere. This condition implies no singular points, and combined with global injectivity, the image has no self-intersections. The bound 2n+12n+1 arises from early techniques using triangulations and to construct embeddings, providing a higher-dimensional ambient space to avoid intersection issues present in lower dimensions. A key corollary for compact manifolds: every compact smooth nn-manifold embeds into R2n+1\mathbb{R}^{2n+1} as a closed , ensuring properness and realizing MM globally without self-intersections.

Strong Embedding Theorem

The strong Whitney embedding theorem states that every smooth nn-dimensional manifold MM admits a smooth embedding into R2n\mathbb{R}^{2n}. This result, proved by Hassler Whitney in 1944, guarantees the existence of a smooth injective immersion f:M→R2nf: M \to \mathbb{R}^{2n} that is a homeomorphism onto its image, ensuring the image f(M)f(M) is a smooth submanifold of R2n\mathbb{R}^{2n} without self-intersections and topologically equivalent to MM. The theorem applies to manifolds that are Hausdorff and second-countable, encompassing both compact and non-compact cases, and resolves the global injectivity issues inherent in lower-dimensional immersions. The "strong" designation highlights the theorem's advancement over immersion results, as it achieves a topological in the minimal even $2ndimension,avoidingtheself−intersectionsthatcanplaguemapsintodimension, avoiding the self-intersections that can plague maps into\mathbb{R}^{2n-1}.[](https://www.ias.ac.in/article/fulltext/reso/021/09/0815−0826)Thisboundisoptimalfor.[](https://www.ias.ac.in/article/fulltext/reso/021/09/0815-0826) This bound is optimal for n=1,wherethecircle, where the circle S^1embedssmoothlyintoembeds smoothly into\mathbb{R}^2butcannotembedintobut cannot embed into\mathbb{R} due to topological obstructions.[](https://math.stackexchange.com/questions/1066617/is-the-whitney-embedding-theorem-tight-for-all-n) For higher dimensions, the $2n dimension is sharp, as demonstrated by the real RP2\mathbb{RP}^2, which embeds into R4\mathbb{R}^4 but not into R3\mathbb{R}^3. For compact manifolds, the theorem yields corollaries in piecewise linear (PL) category: the smooth embedding into R2n\mathbb{R}^{2n} implies a PL embedding, from which a triangulation of MM follows, as smooth submanifolds of Euclidean space admit triangulations via approximation techniques. This connection underscores the theorem's role in bridging smooth and combinatorial topology for compact cases. The strong Whitney embedding theorem shares conceptual parallels with the Nash embedding theorem, which analogously guarantees isometric embeddings of Riemannian manifolds into higher-dimensional Euclidean spaces while preserving the given metric.

Proof Techniques

Outline of the Weak Proof

The proof of the weak embedding theorem, which states that any smooth nn-dimensional manifold admits a smooth into R2n+1\mathbb{R}^{2n+1}, proceeds in two main stages: first constructing an into a high-dimensional Euclidean space, then reducing the dimension via generic projections. For compact manifolds, begin with a finite atlas {(Ui,ϕi)}i=1m\{(U_i, \phi_i)\}_{i=1}^m covering MM, where each ϕi:Ui→Rn\phi_i: U_i \to \mathbb{R}^n is a onto its , and a subordinate {ρi}\{\rho_i\}. Construct a global embedding F:M→Rm(n+1)F: M \to \mathbb{R}^{m(n+1)} by including, for each pair (i,j)(i,j), the components μij(p)=[ϕj(p)]iρj(p)\mu_{ij}(p) = [\phi_j(p)]_i \rho_j(p) (the ii-th coordinate of ϕj(p)\phi_j(p) scaled by ρj(p)\rho_j(p)), and additional components for the ρk(p)\rho_k(p). This map is smooth due to the and injective because if F(p)=F(q)F(p) = F(q), the supports of the ρk\rho_k and coordinate discrepancies ensure p=qp = q. The differential dFpdF_p is injective as the local charts provide full rank, and overlaps are handled linearly. To see why dFpdF_p is injective (i.e., has full rank nn) at any point p∈Mp \in M: Local charts provide full rank: Each chart map ϕi:Ui→Rn\phi_i: U_i \to \mathbb{R}^n is a local diffeomorphism, so its differential dϕid\phi_i is an isomorphism (invertible, hence full rank nn) on the tangent space TpMT_p M when p∈Uip \in U_i. This preserves the local tangent structure of MM, ensuring no loss of dimension in the differential locally. Overlaps are handled linearly: On overlapping regions (where multiple ρj>0\rho_j > 0), the components of FF are built as linear combinations—specifically, scalar multiplications of coordinates from the ϕj\phi_j by the smooth ρj(p)\rho_j(p), which are non-zero in their supports. The differential dFpdF_p thus involves terms like ρj(p) dϕj+dρj⊗ϕj(p)\rho_j(p) \, d\phi_j + d\rho_j \otimes \phi_j(p), but the high ambient dimension m(n+1)>nm(n+1) > n and the linear nature of these scalings/extractions prevent rank deficiency. The partition of unity ensures smooth transitions without "collapsing" the tangent space, as the contributions from multiple charts add independent directions in the target space, maintaining overall injectivity of dFpdF_p. To reduce the dimension, iteratively apply generic linear projections, realized as orthogonal projections onto hyperplanes, starting from N0=m(n+1)>2n+1N_0 = m(n+1) > 2n+1 down to R2n+1\mathbb{R}^{2n+1}. Let F:M→RNF: M \to \mathbb{R}^N be the initial embedding with N=Nk>2n+1N = N_k > 2n+1. For ∈RPN−1 \in \mathbb{RP}^{N-1}, let Ψ:RN→P\Psi_{}: \mathbb{R}^N \to P_{} be the orthogonal projection onto the hyperplane PP_{} perpendicular to vv. Define F=Ψ∘F:M→RN−1F_{} = \Psi_{} \circ F: M \to \mathbb{R}^{N-1}. The set of 's for which $F_{}$ is not an embedding has measure zero in $\mathbb{RP}^{N-1}$, so it is possible to choose such that FF_{} is an embedding. If FF_{} fails to be an embedding, either it is not injective or not an immersion. For non-injectivity, there exist p1≠p2p_1 \neq p_2 such that F(p1)=F(p2)F_{}(p_1) = F_{}(p_2), i.e., 0≠F(p1)−F(p2)0 \neq F(p_1) - F(p_2) lies in the line $$. Thus, =[F(p1)−F(p2)] = [F(p_1) - F(p_2)], so the bad set is contained in the image of the map α:(M×M)∖ΔM→RPN−1,(p1,p2)↦[F(p1)−F(p2)],\alpha: (M \times M) \setminus \Delta_M \to \mathbb{RP}^{N-1}, \quad (p_1, p_2) \mapsto [F(p_1) - F(p_2)], where ΔM={(p,p)∣p∈M}\Delta_M = \{(p, p) \mid p \in M\}. Since dim⁡((M×M)∖ΔM)=2n<N−1\dim((M \times M) \setminus \Delta_M) = 2n < N-1, Sard's theorem implies that the image of α\alpha has measure zero in RPN−1\mathbb{RP}^{N-1}. For failure of immersion, there exists p∈Mp \in M and 0≠Xp∈TpM0 \neq X_p \in T_p M such that (dF)p(Xp)=0(d F_{})_p(X_p) = 0, i.e., (dΨ)F(p)(dF)p(Xp)=0(d \Psi_{})_{F(p)} (d F)_p (X_p) = 0, meaning dFp(Xp)d F_p (X_p) is parallel to vv, so =[dFp(Xp)] = [d F_p (X_p)]. The bad set is contained in the image of the map from the disjoint union over p∈Mp \in M of the projectivized tangent spaces P(TpM)\mathbb{P}(T_p M) (each of dimension n−1n-1) to RPN−1\mathbb{RP}^{N-1}, with total domain dimension n+(n−1)=2n−1<N−1n + (n-1) = 2n-1 < N-1. By Sard's theorem, this image has measure zero. By , at each step, the set of projections where πk∘F\pi_k \circ F fails to be an immersion (i.e., d(πk∘F)pd(\pi_k \circ F)_p not injective) or creates self-intersections (images of distinct points coincide) has measure zero, as these occur when the projection direction lies in the tangent spaces or joining lines, which are lower-dimensional varieties. The Nk−n>nN_k - n > n ensures generic projections preserve embedding properties until 2n+12n+1. For non-compact manifolds, which are σ\sigma-compact, exhaust MM by an increasing sequence of compact subsets KkK_k with M=⋃KkM = \bigcup K_k. Inductively embed each KkK_k into R2n+1\mathbb{R}^{2n+1} compatibly on overlaps Kk∩Kk−1K_k \cap K_{k-1} via small perturbations in a , ensuring the limit map is a proper on MM.

Outline of the Strong Proof

The proof of the strong Whitney embedding theorem begins with the of a smooth immersion of an n-dimensional manifold MM into R2n−1\mathbb{R}^{2n-1}, as established by prior results. To achieve an into R2n\mathbb{R}^{2n}, the immersion is lifted to R2n\mathbb{R}^{2n} by appending an additional coordinate, providing extra room to resolve self-intersections without altering the local immersion properties. Next, general position arguments are applied to perturb the map slightly so that all self-intersection points become transverse double points, meaning the images of distinct points in MM intersect transversely where they coincide. This perturbation ensures no triple or higher intersections occur, and the set of double points forms a discrete collection. The double points are then organized into a graph, where vertices represent the intersection points and edges connect those that cannot be simultaneously resolved due to topological obstructions. Resolution proceeds by constructing a small tubular neighborhood around the immersed manifold in R2n\mathbb{R}^{2n}. Finger moves—localized isotopies resembling pushing a finger through the tube—are used iteratively to separate pairs of double points along non-obstructing paths in the , eliminating intersections one by one while preserving the immersion elsewhere. This process relies on the extra to avoid creating new intersections during the moves. Finally, the resolved map in R2n+1\mathbb{R}^{2n+1} (temporarily used for the separation) is projected orthogonally back to R2n\mathbb{R}^{2n} by omitting the auxiliary coordinate. This projection maintains the embedding property, as the separations ensure no self-intersections remain, yielding a smooth of MM into R2n\mathbb{R}^{2n}. The entire construction is detailed in Whitney's 1944 analysis of self-intersections.

Role of the Whitney Trick

The Whitney trick is a geometric technique introduced by Hassler Whitney to resolve transverse double points in immersions of smooth manifolds, playing a pivotal role in the proof of the strong Whitney embedding theorem. For an immersed n-dimensional manifold in R2n\mathbb{R}^{2n}, arguments ensure that self-intersections occur only as isolated double points with local orientations. The trick targets pairs of such points with opposite orientations, allowing their elimination via an that preserves the immersion elsewhere. This process iteratively reduces the number of double points until none remain, yielding an . In dimensions n≥5n \geq 5, the method exploits the high of the ambient to perform the separation. Consider two double points pp and qq connected by an arc γ\gamma in the manifold and a path δ\delta in R2n\mathbb{R}^{2n} such that γ\gamma and δ\delta bound a disk in the product . The geometric involves constructing a "tube" around δ\delta, which is a , and a "finger" move that pushes the manifold along γ\gamma through a small in the ambient of n≥5n \geq 5. This hole arises because the pair (R2n,δ)(\mathbb{R}^{2n}, \delta) embeds unknottedly, as the 1-dimensional δ\delta has 2n−1≥9>22n-1 \geq 9 > 2, permitting the disk to be embedded in the complement without additional intersections. The required for the separating disk (2-dimensional) in the complement of the n-manifold is 2n−n−2=n−2≥32n - n - 2 = n - 2 \geq 3, ensuring the embedding is possible without topological obstructions. The trick fails in lower dimensions n<5n < 5 primarily due to knotting phenomena that prevent the unknotted embedding of the connecting structures. For instance, in codimension less than 3, paths or disks may link nontrivially with the manifold, creating new intersections or impossibly knotted configurations that cannot be isotoped away. This limitation highlights the metastable range in embedding theory, where additional invariants are needed below dimension 5. Beyond embeddings, the Whitney trick has profound consequences for higher-dimensional topology, enabling key results in surgery theory and cobordism. It allows the cancellation of handles in h-cobordisms by resolving dual spheres' intersections, as utilized by in his proof of the h-cobordism theorem for dimensions at least 5. This, in turn, facilitates the classification of simply connected manifolds and links to the in high dimensions via handlebody decompositions.

Historical Development

Early Contributions

In the mid-19th century, Bernhard Riemann laid foundational ideas for the study of higher-dimensional manifolds during his 1854 habilitation lecture, "On the Hypotheses Which Lie at the Foundations of Geometry." He conceptualized n-dimensional manifolds as spaces that locally resemble Euclidean space and can be equipped with a metric structure analogous to surfaces, emphasizing intrinsic geometry over concrete embeddings in Euclidean space. This abstract perspective for manifolds influenced subsequent work, though global realizations in Euclidean space remained challenging and were later refined by others like Schläfli in 1873, who conjectured local isometric embeddings in R^{n(n+1)/2}. Riemann's framework shifted focus from concrete embeddings to abstract metric spaces, setting the stage for topological investigations. At the turn of the 20th century, Henri Poincaré advanced the study of 3-manifolds in his series of papers on analysis situs from 1895 to 1905. His work on homology invariants and the fundamental group addressed dimension-related questions in classifying such manifolds. However, Poincaré's constructions highlighted open issues, such as topological obstructions to embeddings in low dimensions, and his efforts began to reveal barriers through invariants like Betti numbers. This period also saw early explorations of immersions in low dimensions; for instance, Heegaard in 1898 and Dehn around 1910 developed ideas on sphere decompositions and fillings in 3-manifolds, including preliminary concepts for "everting" spheres through immersions that anticipated later regular homotopy results. A pivotal obstruction to lower-dimensional embeddings emerged in 1911 with L.E.J. Brouwer's proof of the invariance of dimension theorem, which established that Euclidean spaces of different dimensions are not homeomorphic and extended to show that no n-manifold can be embedded into R^{m} for m < n. This result blocked attempts at embeddings below the manifold's dimension and solidified topological barriers. Concurrently, specific examples underscored these limitations: in 1901, Werner Boy constructed an immersion of the real projective plane RP^2 into R^3 via what is now known as Boy's surface, demonstrating that immersions could exist where embeddings could not. Indeed, no smooth embedding of RP^2 into R^3 is possible, as it would require the non-orientable surface to separate R^3 into two components with inconsistent homology characteristics, a fact confirmed through early applications of duality arguments.

Whitney's Work in the 1930s

In the early 1930s, Hassler Whitney, then in his mid-twenties, shifted his focus from graph theory to the emerging field of differential topology, seeking to bridge abstract definitions of manifolds with concrete realizations in Euclidean space. His initial contributions laid the groundwork for understanding immersions and embeddings by developing tools for differentiable functions. A key paper in this period was his 1934 work on analytic extensions of differentiable functions defined in closed sets, which provided essential techniques for approximating and extending maps between manifolds, enabling later constructions of immersions. By 1935, at age 28, Whitney announced preliminary results on embedding theorems in the Proceedings of the National Academy of Sciences, outlining how n-dimensional differentiable manifolds could be realized in higher-dimensional Euclidean spaces. This built on inspirations from early topological studies, including general position arguments and the triangulability of manifolds, which Whitney explored to ensure maps avoided unwanted singularities. His motivation stemmed from a desire to unify combinatorial and analytic approaches to manifold topology, influenced by invariants like homology groups that distinguished manifold structures. The culmination of this research appeared in 1936, when Whitney, aged 29, published his seminal paper "Differentiable Manifolds" in the Annals of Mathematics. In it, he proved that any compact n-dimensional differentiable manifold admits a smooth immersion into R2n\mathbb{R}^{2n} and a smooth embedding into R2n+1\mathbb{R}^{2n+1}, establishing a foundational result that any such manifold is diffeomorphic to a submanifold of Euclidean space. This work not only resolved questions about the embeddability of abstract manifolds but also introduced analytic methods for studying their topological properties, marking a pivotal advancement in the field.

Extensions and Variants

Sharper Dimension Bounds

Following the strong , which guarantees an embedding of an n-dimensional manifold into R2n\mathbb{R}^{2n}, researchers developed sharper bounds for particular classes of manifolds in the decades after Whitney's work. In the late 1950s, Morris Hirsch established that any open (non-compact) n-manifold admits a smooth embedding into R2n−1\mathbb{R}^{2n-1}. For closed orientable n-manifolds with n>1n > 1, Whitney proved that smooth immersions into R2n−1\mathbb{R}^{2n-1} always exist, though embeddings are not guaranteed in general. Additionally, Hirsch showed that parallelizable open n-manifolds can be smoothly immersed into Rn\mathbb{R}^n. Building on this, André Haefliger and Morris Hirsch proved in 1961 that a closed smooth n-manifold embeds into R2n−1\mathbb{R}^{2n-1} its (n-1)th normal Stiefel-Whitney class vanishes; for orientable manifolds, this obstruction often lifts under additional assumptions, such as 1-connectedness or parallelizability, enabling embeddings into R2n−1\mathbb{R}^{2n-1} when n≠4n \neq 4. A concrete illustration occurs for surfaces (n=2n=2): every closed orientable 2-manifold embeds smoothly into R3\mathbb{R}^3, reducing the general bound from 4 to 3 and aligning with the Haefliger-Hirsch criterion since the first normal Stiefel-Whitney class vanishes for orientable surfaces. However, exceptions arise in dimension 4, where the Haefliger-Hirsch obstruction does not always permit embeddings into R7\mathbb{R}^7; additional topological obstructions, such as the Kirby-Siebenmann invariant, prevent certain closed orientable smooth 4-manifolds from embedding.

Isotopy and Stability Versions

The isotopy version of the Whitney embedding theorem asserts that every smooth immersion of an nn-dimensional manifold MM into R2n+1\mathbb{R}^{2n+1} is isotopic through a continuous family of smooth immersions to a smooth embedding of MM into R2n+1\mathbb{R}^{2n+1} . This result, established in Whitney's seminal 1936 work, highlights the flexibility of immersions in the weak embedding dimension, allowing the elimination of self-intersections via a smooth deformation without altering the . The Whitney trick plays a brief enabling role here by facilitating the removal of double points in the ambient space during the isotopy . For open manifolds, Gromov's hh-principle provides a powerful extension, asserting that the space of genuine immersions (holonomic sections) of an open nn-dimensional manifold into Rm\mathbb{R}^m with m≥n+1m \geq n+1 is weakly equivalent to the space of formal immersions (monomorphisms of bundles) . This equivalence implies that any formal immersion can be approximated arbitrarily closely by a genuine immersion, and in sufficiently high codimensions (m≥2nm \geq 2n), such approximations yield embeddings of open manifolds . The holonomic approximation theorem underpinning the hh-principle ensures that these solutions satisfy the differential relations defining immersions and embeddings on open domains . In high dimensions, smooth embeddings exhibit stability under C1C^1 perturbations: when the codimension m−n≥nm - n \geq n (the stable range), small C1C^1 changes to an embedding f:Mn→Rmf: M^n \to \mathbb{R}^m preserve the embedding property, as the set of embeddings is open in the C1C^1 topology . This stability follows from transversality theorems, ensuring that generic perturbations avoid self-intersections and maintain injectivity . Smale's work in the 1950s, including his 1958 proof of the existence of sphere eversions, linked immersion theory to embedding isotopies by showing that the standard immersion of S2S^2 into R3\mathbb{R}^3 is regularly homotopic to its antipodal reflection, demonstrating the connectivity of the space of immersions up to regular homotopy . This result, extended via the Smale-Hirsch theorem, equates the homotopy type of immersion spaces to bundle monomorphisms, facilitating isotopies for embeddings in higher dimensions by resolving immersion obstructions . In dimensions n≥5n \geq 5, two smooth embeddings of a compact nn-manifold with boundary into Rm\mathbb{R}^m (m≥2nm \geq 2n) that agree up to on their boundaries are isotopic relative to the boundary . This fact, part of the for embeddings in the stable range, relies on the connectivity of groups relative to boundaries in high dimensions .

Applications to Other Manifolds

The Whitney embedding theorem establishes that every smooth manifold admits a smooth into , which directly implies the triangulability of all smooth manifolds. Specifically, upon embedding an n-dimensional smooth manifold MM into R2n\mathbb{R}^{2n}, a of the ambient can be intersected with the embedded image to yield a homeomorphic to MM, thereby providing a of MM itself. This construction relies on the fact that is triangulable and the embedding is a onto its image. In the topological category, analogs fail in high dimensions, as some topological manifolds are not triangulable, highlighting the theorem's reliance on . In the classification of low-dimensional manifolds, the theorem provides essential bounds on embedding dimensions, aiding the study of spaces and types. For 3-manifolds, Whitney's result guarantees smooth embeddings into R6\mathbb{R}^6, but subsequent refinements, such as Wall's proof that all 3-manifolds embed into R5\mathbb{R}^5, leverage these bounds to classify embeddings up to isotopy and explore topological invariants like fundamental groups and homology. Similarly, for 4-manifolds, where full remains challenging due to exotic smooth structures, the theorem supports analysis of embedding obstructions and aids in distinguishing smooth versus topological categories through ambient space properties. The theorem also connects to Riemannian geometry via Nash's isometric embedding theorem, which extends Whitney's topological embedding to preserve the Riemannian metric. Whitney's embedding into R2n\mathbb{R}^{2n} serves as a starting point, allowing Nash to solve partial differential equations that realize any Riemannian metric on an n-manifold as the induced metric from an embedding into a higher-dimensional Euclidean space, typically Rn(n+1)(3n+11)/2\mathbb{R}^{n(n+1)(3n+11)/2} for the sharp bound. This link underscores the compatibility of intrinsic metric geometry with extrinsic Euclidean realizations. In , embeddings induced by the theorem facilitate computations through . A smooth embedding of a into a manifold (or ) admits a diffeomorphic to the total space of the normal bundle, enabling the application of the Thom isomorphism, which relates the of the ambient space to that of the shifted by the of the normal bundle. This tool is pivotal for computing rings and characteristic classes in embedded settings. A modern application arises in , where the theorem guarantees that Calabi-Yau manifolds—compact 6-dimensional Kähler manifolds with vanishing first Chern class, central to supersymmetric compactifications—can be smoothly embedded into R12\mathbb{R}^{12}. Such embeddings support geometric realizations of flux compactifications and mirror symmetry constructions, allowing theoretical and numerical exploration of string vacua landscapes.

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