Explicit representatives of the regular homotopy classes of immersed spheres, and the classes containing embeddings (Smale's geometric problems of 1959)

Stephen Smale's 1959 Annals paper classified the immersions of a $k$-sphere in Euclidean $n$-space up to regular homotopy by the homotopy group $\pi_k(V_{n,k})$ of a Stiefel manifold, and posed a program of geometric follow-up questions, two of which survive verbatim: find explicit representatives of the regular homotopy classes, and determine which classes have an embedding for representative. The companion manifold-target generalization posed alongside them was resolved by Hirsch's 1959 theorem (immersions of one manifold into another are classified, up to weak homotopy, by vector-bundle monomorphisms of tangent bundles), leaving the geometric realization problem: for each pair $(k,n)$ with $1 \le k < n$, construct explicit immersions $S^k \to \mathbb{R}^n$ realizing every element of $\pi_k(V_{n,k})$, provide an effective geometric characterization of the Smale invariant, and determine the subset of classes containing embeddings. Case-by-case results exist (plane curves; Whitney's self-intersection number for $n = 2k$; codimension one; $S^3 \to \mathbb{R}^4$; $S^{4k-1} \to \mathbb{R}^{4k+1}$; the index-24 embedding subgroup for $S^3 \to \mathbb{R}^5$), and the program remains active through at least 2025, but no general treatment is known.

Background

An immersion of a smooth $k$-manifold into $\mathbb{R}^n$ (with $n > k$) is a smooth map whose differential is injective at every point; an embedding is an immersion that is also a homeomorphism onto its image. Two immersions of the same source are regularly homotopic if they can be connected by a continuous one-parameter family of immersions. Regular homotopy is the natural equivalence for immersed shapes up to smooth deformation: it is weaker than deformation through embeddings (ambient isotopy) and stronger than ordinary homotopy. For instance, an immersed closed plane curve is regularly homotopic to the round circle exactly when its tangent line makes the same net number of full rotations, and this rotation number is a complete invariant in that classical setting.

Smale, after settling the case of the two-sphere, proved in his 1959 Annals paper that the regular homotopy classes of immersions $S^k \to \mathbb{R}^n$ ($1 \le k < n$) are in bijection with $\pi_k(V_{n,k})$, the $k$-th homotopy group of the Stiefel manifold $V_{n,k}$ of orthonormal $k$-frames in $\mathbb{R}^n$ (a homotopy group $\pi_k(X)$ records homotopy classes of maps $S^k \to X$). The bijection sends an immersion to its Gauss map, which records the image of an orthonormal tangent frame; the resulting class is the Smale invariant of the immersion. The theorem translates geometry into homotopy theory, and its most famous corollary is the eversion of the sphere: since $\pi_2(V_{3,2}) = 0$, every immersion $S^2 \to \mathbb{R}^3$ is regularly homotopic to the standard embedding, so the sphere can be turned inside out through immersions, though never through embeddings.

Smale immediately posed the reverse direction. As quoted in Ekholm's paper on differential 3-knots, Smale asked (p. 329 of the 1959 paper) to "Find explicit representatives of regular homotopy classes . . . What regular homotopy classes have an embedding for representative?" Two demands are packed here. First, since the classification theorem is proved by nonconstructive homotopy-theoretic means, it does not exhibit immersions one can write down: "explicit representatives" asks for parametrized or finitely described immersions realizing each class, together with a geometric way to compute the invariant of a given immersion. Second, since every embedding is an immersion, each isotopy class of embeddings $S^k \to \mathbb{R}^n$ occupies a regular homotopy class, and the question asks which elements of $\pi_k(V_{n,k})$ these are.

The classification side was completed almost immediately after Smale's paper. Whitney's self-intersection theory gives a geometric classification in the case $n = 2k$: for $k \ge 2$ the class of a generic immersion is detected by the algebraic number of its self-intersection points (an integer when $k$ is even, a class in $\mathbb{Z}_2$ when $k$ is odd; for $k = 1$ the rotation number classifies and the mod-2 double-point count detects only its parity), and this case is regarded as essentially settled. Hirsch's companion 1959 theorem identifies the space of immersions of any smooth compact manifold $N$ into any manifold $W$ up to weak homotopy (a weak homotopy equivalence induces bijections on path components and on all homotopy groups) with the space of vector-bundle monomorphisms $TN \to TW$; combined with Smale's theorem this classifies immersions of arbitrary sources into arbitrary targets, and both results are special cases of Gromov's h-principle, the general principle that a differential relation satisfiable by tangent-plane data can be integrated to a global solution. These results settle existence and classification, but their proofs produce no representatives and no computable invariant.

The embedding side developed into high-dimensional knot theory. Haefliger classified embeddings $S^k \to S^{k+q}$ of codimension $q > 2$ up to isotopy, in the range where his methods apply, in terms of homotopy groups of spheres and Stiefel manifolds; Skopenkov's survey presents the modern state of embedding and knotting of manifolds in Euclidean spaces. This theory classifies embeddings among themselves; Smale's question is different and additional. It asks how the embedding classes sit inside the full group of regular homotopy classes, which requires computing the Smale invariant of embedding representatives, a geometric step the abstract theory does not provide.

Progress on the two geometric questions has been case-by-case. Hughes and Melvin determined the embedding classes for $S^3 \to \mathbb{R}^5$ (exactly every 24th class) via their Smale invariant of a knot. Kaiser treated codimension-one immersions up to regular homotopy. Ekholm gave explicit representatives and a self-intersection obstruction for $S^3 \to \mathbb{R}^5$; Ekholm and Szucs found geometric formulas for the Smale invariants of codimension-two immersions $S^{4k-1} \to \mathbb{R}^{4k+1}$; Ekholm and Takase and Kinjo constructed and computed explicit families for $S^3 \to \mathbb{R}^4$; holomorphic and polynomial representatives were studied by Pinter and by Karolkiewicz, Nowel, and Szafraniec; and the program continued into the 2020s with Gauniyal's study of embeddings modulo immersions and Seidel's invariant for triple-point-free immersed spheres. No uniform treatment over all pairs $(k,n)$ is known.

Problem Statement

Smale's classification reduces immersions of spheres in Euclidean space to the groups $\pi_k(V_{n,k})$, but supplies no geometric handle on the classes. The open problem, in the form surviving verbatim from Smale's 1959 paper (p. 329), has two parts whose common objective is to convert the homotopy-theoretic classification back into geometry:

(1) Explicit representatives and a geometric characterization. For each pair of integers $1 \le k < n$: (a) construct, for every element of $\pi_k(V_{n,k})$, an explicit immersion $S^k \to \mathbb{R}^n$ representing the corresponding regular homotopy class, where explicit means given by parametrization formulas or by finite combinatorial-geometric data from which such parametrizations can be effectively written down, so that the constructed representatives meet every class; and (b) give an effective geometric characterization of the Smale invariant: a formula or finite algorithm computing the class in $\pi_k(V_{n,k})$ of an arbitrary generic immersion (one whose self-intersections are transverse, so that they form a well-behaved subset of the sphere) from geometric data of its normal bundle (the orthogonal complement of the tangent image in $\mathbb{R}^n$) and of its self-intersections.

(2) Embedding classes. For each pair $1 \le k < n$: determine the subset of $\pi_k(V_{n,k})$ consisting of the classes that have an embedding for representative, i.e. the image of the natural map from the path components of the space of embeddings $S^k \to \mathbb{R}^n$ to the group of regular homotopy classes.

For $n \ge 2k+1$ the Stiefel manifold $V_{n,k}$ is at least $k$-connected, i.e. all homotopy groups $\pi_i(V_{n,k})$ with $i \le k$ vanish, so both questions are trivial there: the unique class is represented by the standard embedding. The content lies in the low-codimension range $k < n \le 2k$, where self-intersections are unavoidable and the groups $\pi_k(V_{n,k})$ are rich. The question concerns the general case $1 \le k < n$; a proper sub-range (a single pair, or a family such as fixed codimension) is not the full range.

References

1. Stephen Smale, The Classification of Immersions of Spheres in Euclidean Spaces, The Annals of Mathematics, Second Series 69 (1959), no. 2, 327-344, DOI 10.2307/1970186, https://doi.org/10.2307/1970186
2. Stephen Smale, A classification of immersions of the two-sphere, Transactions of the American Mathematical Society 90 (1959), 281-290, DOI 10.1090/s0002-9947-1959-0104227-9, https://doi.org/10.1090/s0002-9947-1959-0104227-9
3. Morris W. Hirsch, Immersions of manifolds, Transactions of the American Mathematical Society 93 (1959), 242-276, DOI 10.1090/s0002-9947-1959-0119214-4, https://doi.org/10.1090/s0002-9947-1959-0119214-4
4. Hassler Whitney, The Self-Intersections of a Smooth n-Manifold in 2n-Space, The Annals of Mathematics, Second Series 45 (1944), no. 2, 220-246, DOI 10.2307/1969265, https://doi.org/10.2307/1969265
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15. Shumi Kinjo, Immersions of 3-sphere into 4-space associated with Dynkin diagrams of types A and D, Bulletin of the London Mathematical Society 47 (2015), no. 4, 651-662, DOI 10.1112/blms/bdv044, https://doi.org/10.1112/blms/bdv044
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17. Gergő Pintér, On certain complex surface singularities, PhD thesis, Eötvös Loránd University, 2019, arXiv:1904.12778, https://arxiv.org/abs/1904.12778
18. Neeti Gauniyal, Haefliger's approach for spherical knots modulo immersions, Homology, Homotopy and Applications 25 (2023), no. 2, 55-73, DOI 10.4310/hha.2023.v25.n2.a4, https://doi.org/10.4310/hha.2023.v25.n2.a4
19. Jona Seidel, An Invariant for Triple-Point-Free Immersed Spheres, preprint, 2025, arXiv:2506.21130, https://arxiv.org/abs/2506.21130