Regular Homotopy Classes Research Articles

Finite order q-invariants of immersions of surfaces into 3-space

Given a surface F, we are interested in ${{\Bbb Z}/2}$ valued invariants of immersions of F into ${{\Bbb R}^3}$ , which are constant on each connected component of the complement of the quadruple point discriminant in ${Imm(F,\E)}{{\Bbb R}^3}$ . Such invariants will be called “q-invariants.” Given a regular homotopy class $A \subseteq {Imm(F,\E)}{{\Bbb R}^3}$ , we denote by $V_n(A)$ the space of all q-invariants on A of order $\leq n$ . We show that ifF is orientable, then for each regular homotopy class A and each n, $\dim (V_n (A) / V_{n-1}(A) ) \leq 1$.

Differential 3-knots in 5-space with and without self-intersections

Regular homotopy classes of immersions S 3→ R 5 constitute an infinite cyclic group. The classes containing embeddings form a subgroup of index 24. The obstruction for a generic immersion to be regularly homotopic to an embedding is described in terms of geometric invariants of its self-intersection. Geometric properties of self-intersections are used to construct two invariants J and St of generic immersions which are analogous to Arnold's invariants of plane curves [1]. We prove that J and St are independent first-order invariants and that any first-order invariant is a linear combination of these. As by-products, some invariants of immersions S 3→ R 4 are obtained. Using them, we find restrictions on the topology of self-intersections.

Willmore two-spheres in the four-sphere

Genus zero Willmore surfaces immersed in the three-sphereS3\mathbb {S}^3correspond via the stereographic projection to minimal surfaces in Euclidean three-space with finite total curvature and embedded planar ends. The critical values of the Willmore functional are4πk4\pi k, wherek∈N∗k\in \mathbb {N}^*, withk≠2,3,5,7k\ne 2,3,5,7. When the ambient space is the four-sphereS4\mathbb {S}^4, the regular homotopy class of immersions of the two-sphereS2\mathbb {S}^2is determined by the self-intersection numberq∈Zq\in \mathbb {Z}; here we shall prove that the possible critical values are4π(|q|+k+1)4\pi (|q|+k+1), wherek∈Nk\in \mathbb {N}. Moreover, ifk=0k=0, the corresponding immersion, or its antipodal, is obtained, via the twistor Penrose fibrationP3→S4\mathbb {P}^3\rightarrow \mathbb {S}^4, from a rational curve inP3\mathbb {P}^3and, ifk≠0k\ne 0, via stereographic projection, from a minimal surface inR4\mathbb {R}^4with finite total curvature and embedded planar ends. An immersion lies in both families when the rational curve is contained in someP2⊂P3\mathbb {P}^2\subset \mathbb {P}^3or (equivalently) when the minimal surface ofR4\mathbb {R}^4is complex with respect to a suitable complex structure ofR4\mathbb {R}^4.

REGULAR HOMOTOPY AND VASSILIEV INVARIANTS OF GENERIC IMMERSIONS Sk → ℝ2k-1, k ≥ 4

The regular homotopy class of a generic immersion Sk → ℝ2k-1 is calculated in terms of its self intersection manifold with natural additional structures. There is a natural notion of Vassiliev invariants of generic immersions. These may take values in any Abelian group G. It is proved that, for any m, the group of mth order G-valued invariants modulo invariants of lower order is isomorphic to G and that the Vassiliev invariants are not sufficient to separate generic immersions, which can not be obtained from each other by a regular homotopy through generic immersions.

Bordism and regular homotopy of low-dimensional immersions

In this paper we study the geometric characteristics of low-dimensional immersions. Smale asked, in his paper on immersions of the /c-sphere in Rn , what are explicit generators for the groups of regular homotopy classes of immersions? We answer this for the 3-sphere in R4 and R5. For S3 in R4 , the answer is: THEOREM. The standard (Froissart-Morin) eversion of S2 in R3 has, as a track, an immersion of S 2 x / in R 4 whose ends are embedded S2s. Each of these bounds a 3-ball in R 4. Capping off the track with these 3-balls yields an immersion K: S 3 —• R4 . Performing the eversion twice and capping off gives an immersion E: S 3 —> R 4 . The immersions E and K generate the group of regular homotopy classes of immersions of S 3 in R 4 .

Immersions with non-zero normal vector fields

Let M be a smooth n-manifold, X be a smooth (2n − 1)-manifold, and g:M → X be a map. It was proved in [6] that g is always homotopic to an immersion. The set of homotopy classes of monomorphisms from TM into g*TX, which is denoted by Sg, may be enumerated either by the method of I. M. James and E. Thomas or by the singularity method of U. Koschorke (see [1] and references therein). When the natural action of π1(XM, g) on Sg is trivial, for example, if X is euclidean, the set Sg is in one-to-one correspondence with the set of regular homotopy classes of immersions homotopic to g (see e.g. [4]).

A proof of the Chern-Lashof conjecture in dimensions greater than five

Chern and Lashof ([1], [2]) conjectured that if a smooth manifoldM m has an immersion intoR w, then the best possible lower bound for its total absolute curvature is the Morse number μ(M). We give a proof of this whenm>5. Under the same dimension restriction, our methods allow us to show that μ(M) is still the best possible lower bound among immersions within a fixed regular homotopy class except in the casew=m+1=even, for which the best lower bound is max {μ(M), 2 |d|}, whered the degree of the Gauss map.

Immersions of surfaces in 3-manifolds

THE problem of the classification of immersions of submanifolds of codimension at least one up to regular homotopy was reduced to a homotopy theoretic problem by the methods of Hirsch and Smale [6]. These techniques however are awkward to apply. James and Thomas [7] used the Hirsch-Smale results to enumerate the immersions of a surface into Iw3. Their technique was nonconstructive. In $1 we will give a classification of immersions of a surface, F, into a 3-manifold, M, in a given homotopy class, up to regular homotopy. It will be shown that there is a correspondence between H’ (F; Z,) and the regular homotopy classes homotopic to any given mapf, : F + M. This correspondence is l-to-l unless both F and M are non-orientable, in which case it may be l-to-2. The classification leads to an explicit construction ofall regular homotopy classes of maps within a given homotopy class of maps. We carry out this construction in $2. In 43 we study exactly what occurs during the process of (generic) regular homotopy. This allows us to define various invariants of the regular homotopy class of an immersion. One such invariant allows us to state precisely which regular homotopy classes can be realized by embeddings in Iw3. We solve the problem of when regular homotopy classes can be realized without triple points. Finally we obtain a new proof of a theorem of Banchoff on the number of triple points of immersed surfaces in Euclidean space. We would like to thank Larry Taylor and the referee for their helpful suggestions.

Regular homotopy classes of immersed surfaces

IN this paper we are concerned with the problem of classifying compact surfaces immersed in Iw” up to regular homotopyt. This subject started in 1958 when Smale classified the immersions of the 2-sphere [17]. For n 2 4 the problem was then completely solved by Hirsch ([8], theorems 8.2 and 8.4): if M2 is a compact surface then for n 2 5 any two immersions f, g : M2 + R” are regularly homotopic, while the immersions f: M2 + R* are completely classified by their normal class. Concerning immersed surfaces in iw3 two results are known:

Note on the Embedding of Manifolds in Euclidean Space

M. Hirsch and independently H. Glover have shown that a closed $k$-connected smooth $n$-manifold $M$ embeds in ${R^{2n - j}}$ if ${M_0}$ immerses in ${R^{2n - j - 1}},j \leqq 2k$ and $2j \leqq n - 3$. Here ${M_0}$ denotes $M$ minus the interior of a smooth disk. In this note we prove the converse and show also that the isotopy classes of embeddings of $M$ in ${R^{2n - j}}$ are in one-one correspondence with the regular homotopy classes of immersions of ${M_0}$ in ${R^{2n - j - 1}},j \leqq 2k - 1$ and $2j \leqq n - 4$.

Homotopy invariants in differential geometry. I

Introduction. In this series of papers we will discuss homotopy invariants of differentiable maps f: M -? N in various situations within the framework of differential geometry. We shall be particularly interested in the case where f is an immersion. Our principle is simple. We use the fact that the pullbackf*W of c is a differentiable homotopy invariant where w is an arbitrary cohomology class always over the real number field, R. We note if F= M x I -1 N is a differentiable map with f=f, thenf*w belongs to the same cohomology class asfJ*w where I is the interval [0, 1] and ft is defined by ft(x)=F(x, t), x E M, t EI L There are many known examples beside various characteristic classes. To quote a few of them, letf be an immersion of the two-torus T2 into the complement of the diagonal set of the six dimensional euclidean space R6 considered as R3 x R3; f: T2 R3 x R3 A, where A is the diagonal set. The space R3 x R 3A is diffeomorphic with R4 x S2 where S2 is the two-sphere. Thus, if a) denotes the volume element of 52, a becomes a 2-dimensional cohomology class of R4 x S2 by pulling back with the projection onto S2 and gives us a homotopy invariantf*co. Consider T2 interpreted as the direct product S1 x S 1 of circles andf as the pair of two closed curves ci: SI --> R3 without intersection. Then f *w is nothing but the linking number of two closed -curves cl and c2 (up to a universal constant multiple) according to Gauss [8]. Another (but somewhat more extraneous) example, due to J. H. C. Whitehead [12], is the Hopf invariant, H(f), for a differentiable map f: S2n-1 _? Sn. Again, denoting the volume element of the n-sphere Sn by c, H(f) is given by the closed form 0 Af*w where 0 is any (n l)-form with dO =f*U. In this paperf will be an isometric immersion of a compact oriented Riemannian manifold M into a euclidean space Rn + v. Since the cohomology groups of N=Rn + v are then trivial, our principle does not apply tof directly, but we construct a manifold B from M and replace f by a certain map fB: M -? B. To do this, let us recall Hirsch's theorem [5] to the effect that the regular homotopy classes of the immersionsf: M -? Nofany manifold Minto another manifold N with dim N> dim M are in a one-to-one correspondence with the homotopy classes of the crosssections fB: M -->B of a certain bundle B over M, where a regular homotopy F= M x I -* N means one for which each ft is an immersion. Despite the triviality of the cohomology groups of N= Rn+ v in the case above, we can expect to obtain homotopy invariants fB*0 corresponding to a cohomology class 0 of B. In ?4, we