Reports of the Academy of Sciences of the USSR

1. Volume 138, No. 4

MATHEMATICS

S. P. NOVIKOV

ON THE EMBEDDING OF SIMPLY CONNECTED MANIFOLDS IN EUCLIDEAN SPACE

(Presented by Academician P. S. Aleksandrov on 23 I 1961)

We shall consider smooth closed manifolds and their smooth mappings. As usual, we shall call a mapping \(f: M^n \to W^m\) regular if its Jacobian matrix has rank \(n\) at every point, and completely regular if the preimage of any point \(f^{-1}(w)\), where \(w \in W^m\), contains no more than two points. Our aim is to investigate the possibility of a smooth embedding \(M^n \subset E^{2n-1}\). The method generalizes the method of Whitney’s work \((^2)\), devoted to embeddings \(M^n \subset E^{2n}\), on the basis of ideas of L. S. Pontryagin concerning the homotopy groups of spheres \((^1)\).

Theorem 1. Every simply connected odd-dimensional manifold \(M^n\) with \(n > 6\) can be smoothly embedded in Euclidean space \(E^{2n-1}\).

The proof of this theorem is based on a series of lemmas devoted to the study of regular mappings as a whole. With the help of well-known techniques (developed by Whitney) it is easily proved:

Lemma 1*. For every regular mapping \(f: M^n \to E^{2n-k}\) with \(k < \left[\dfrac{n}{2}\right]\), there exists a \(C^1\)-close regular mapping \(g: M^n \to E^{2n-k}\) such that:

1) the equation \(g(x)=g(y)\) defines a closed submanifold \(\widetilde M_g^k \subset M^n \times M^n \setminus \Delta(M^n)\), where \(\Delta\) is the diagonal mapping;

2) the projection \(p: M^n \times M^n \to M^n\), considered on the submanifold \(\widetilde M_g^k\), is a smooth homeomorphism;

3) the mapping \(g\) is completely regular; on the special submanifold \(M_g^k = p(\widetilde M_g^k) \subset M^n\) it is a two-sheeted covering.

It follows from the lemma that the special manifold \(M_g^k\) decomposes into some number \(s\) of special pairs of mutually homeomorphic connected components
\[ \bigcup_{i=1}^{s} \left(M_{g,1}^{k,i} \cup M_{g,2}^{k,i}\right) \]
such that \(g\left(M_{g,1}^{k,i}\right)=g\left(M_{g,2}^{k,i}\right)\), and some number \(t\) of connected manifolds
\[ \bigcup_{j=1}^{t} M_g^{k,j}, \]
on which the mapping \(g\) is a nontrivial 2-covering. Thus,
\[ M_g^k = \left(\bigcup_{j=1}^{t} M_g^{k,j}\right) \cup \left(\bigcup_{i=1}^{t} \left(M_{g,1}^{k,i} \cup M_{g,2}^{k,i}\right)\right). \]

Definition. We shall call a manifold \(M^n\) \(k\)-parallelizable if the \(\varepsilon\)-neighborhood \(U_\varepsilon^{(k)}\) of the \(k\)-dimensional skeleton of a smooth triangulation of the manifold \(M^n\) is parallelizable for sufficiently small \(\varepsilon\).

* For \(k=1\) this lemma is contained in Whitney’s works; see, for example, \((^3)\).

Obviously, for \(n>2k+2\) our definition does not depend on the triangulation, and a \(k\)-connected manifold is \(k\)-parallelizable. For \(k=1\) this definition means simply orientability. We also note that a \(k\)-parallelizable manifold is \((k-1)\)-parallelizable. In what follows we assume that \(n\geqslant 2k+3\).

Lemma 2. If the manifold \(M^n\) is \(k\)-parallelizable, then the singular submanifold \(M_g^k\subset M^n\) has a trivial normal bundle in the manifold \(M^n\) and is a \(\pi\)-manifold (i.e., the normal bundle under the embedding \(M_g^k\subset E^m\) is trivial for \(m\geqslant 2k+3\)).

The proof of this lemma is based on the fact that the manifold
\(M^n\times M^n\setminus \Delta(M^n)\) is also \(k\)-parallelizable and that the normal bundle of a manifold of dimension at most \(k\) in a \(k\)-parallelizable manifold of large dimension is arranged in the same way as in a Euclidean space of the same dimension.

Let \(n\) be even, \(k=1\), and let the manifold \(M^n\) be orientable.

Lemma 3. The singular submanifold \(M_g^1\subset M^n\) consists only of singular pairs of circles.

Suppose, to the contrary, that the manifold \(M_g^1\) contains a circle \(S_g^1\subset M_g^1\) on which the mapping \(g\) is a connected 2-fold covering. Obviously, \(g(S_g^1)=S^1\subset E^{2n-1}\). Choose a system \((W_1,\ldots,W_{n-1})\) of independent vector fields, transversal to \(S_g^1\subset M^n\) and tangent to the manifold \(M^n\).

Then, roughly speaking, there is defined a decomposition of the normal bundle of \(g(S_g^1)\subset E^{2n-1}\) into the sum of plane bundles \(\mu_i^{(2)}\), \(i=1,\ldots,n-1\), generated by the vectors \(W_i\). Each plane bundle \(\mu_i^{(2)}\) is transversal to the circle \(g(S_g^1)\) and has the monodromy matrix
\(A_i=\begin{pmatrix}0&1\\[2pt]1&0\end{pmatrix}\), i.e., each bundle \(\mu_i^{(2)}\) is nonorientable. The Whitney sum of an odd number of such bundles is also nonorientable and, consequently, nontrivial. Since \(n-1\) is odd, whereas the normal bundle to a circle in Euclidean space must be trivial, we arrive at a contradiction. The lemma is proved.

For the general case, when \(k=1\), we have:

Lemma 4. The number \(t(g)\) of connected singular coverings is always even. The mapping \(g\) is regularly homotopic to a mapping \(g_1\) that has no singular nontrivial 2-fold coverings.

The proof of this lemma is of a somewhat different nature and is based on the study of the projection

\[ M^n \xrightarrow{\ \tilde g\ } E^{2n} \xrightarrow{\ \pi\ } E^{2n-1}, \]

where \(g=\pi\circ \tilde g\), and the mapping \(\tilde g\) is completely regular (one can always easily reduce it to this form by a small \(C^1\)-perturbation of the mapping in the space \(E^{2n}\), projecting a small perturbation into \(E^{2n-1}\), which, obviously, preserves the properties of Lemma 1).

The behavior of the projection is described more fully by the following trivial lemmas:

Lemma 4a. If the mapping \(\tilde g:M^n\to E^{2n}\) and \(\pi\tilde g:M^n\to E^{2n-1}\) are regular, then the mapping \(\tilde g\) is regularly homotopic to an embedding and has an even number of pairs of singular points.

Lemma 4b. A connected singular covering of the mapping \(g\) can arise under projection only from an odd number of pairs of singular points of the mapping \(\tilde g\).

A singular pair can arise under projection only from an even number of pairs of singular points of the mapping \(\tilde g\).

Lemma 4c. There exists a regular homotopy \(\widetilde{g}_t\) of the map \(\widetilde{g}=\widetilde{g}_0\) such that:

1) the maps \(\widetilde{g}_t\) and \(\pi \widetilde{g}_t\) are regular for \(t\leqslant 1\) and are completely regular for \(t=1\); the map \(\pi \widetilde{g}_1\) satisfies Lemma 1;

2) under the projection \(\pi\), from one pair of singular points there arise double points, and from nothing—singular pairs of circles of the map \(\pi \widetilde{g}_1\).

In what follows we shall consider only such maps
\[ g:M^n\to E^{2n-1} \]
that have no linked singular double points. We shall also assume that \(\pi_1(M^n)=0\). Following Pontryagin \((^1)\), we shall define the invariant of a singular pair and the invariant of a map \(g\).

Definition of the invariant of a singular pair. Let \(S_1^1\) and \(S_2^1\subset M^n\), \(g(S_1^1)=g(S_2^1)\). Consider a pair of disks \(\sigma_1^2,\sigma_2^2\subset M^n\) such that \(\sigma_1^2\cap\sigma_2^2=\varnothing\) and \(\partial\sigma_1^2=S_1^1,\ \partial\sigma_2^2=S_2^1\). Choose a system of vector fields \(W_j^{(i)}\), \(i=1,2;\ j=1,\ldots,n-2\), tangent to \(M^n\) and orthogonal to \(\sigma_i^2\). Put \(W_{n-1}^{(i)}=\partial\sigma_i^2/\partial t\), where \(t\) are the radii of the films (i.e., transverse to \(S_i^1\) and to \(W_j^{(i)}\), \(j\leqslant n-2\)). Obviously, the vectors \(g(W_j^{(i)})=V_{j+(i-1)i}\), transverse to \(g(S_i^1)\), are defined and independent. They determine an element
\[ \alpha\in \pi_1(GL(2n-2))=Z_2 . \]

Lemma 5. If the generating element of the group \(H^n(M^n,Z_2)\) has the form \(Sq^2(x)\), \(x\in H^{n-2}(M^n,Z_2)\), then the disks \(\sigma_i^2\) and the fields \(W_j^{(i)}\) can be chosen so that \(\alpha=0\).

In the case when
\[ H^n(M^n,Z_2)/\operatorname{Im} Sq^2=Z_2, \]
the invariant \(\alpha\) of a singular pair does not depend on the choice of the disks \(\sigma_i^2\). In this case we shall regard the sum of the invariants
\[ \sum_k \alpha_k \]
over all singular pairs \(S_k=(S_{g,1}^{1,k}\cup S_{g,2}^{1,k})\) as the invariant of the map
\[ g:M^n\to E^{2n-1}, \]
if it has no linked singular double points.

Lemma 6. For a simply connected odd-dimensional manifold \(M^n\), \(n=4l+3\), the invariant
\[ \sum_k \alpha_k=0 \]
for any regular map
\[ g:M^n\to E^{2n-1}, \]
possessing all the properties of Lemma 1*.

This lemma is an important step, and its direct geometric proof is quite difficult. But from the recent works of Hirsch \((^4)\), devoted to regular maps, it is extracted more simply.

Let \(S_1\) and \(S_2\) be two singular pairs of the map
\[ g:M^n\to E^{2n-1} \]
such that \(\alpha(S_1)=\alpha(S_2)\).

Lemma 7. There exists a regular homotopy \(g_t\) of the map \(g=g_0\) such that the map \(g_1\) satisfies Lemma 1 and has two fewer singular pairs than the map \(g=g_0\).

The proof generalizes the well-known proof of Whitney \((^2)\) for pairs of singular points. We glue into the manifold \(M^n\) the rings
\[ B_1=S_1^1\times I \quad\text{and}\quad B_2=S_2^1\times I \]
so that \(S_i^1\times \varepsilon\) form the pair \(S_1\), and \(S_i^1\times(1-\varepsilon)\) form the pair \(S_2\). It is necessary that \(B_1\cap B_2=\varnothing\). On the rings we prescribe vector fields \(W_j^{(i)}\), \(i=1,2;\ j=1,\ldots,n-2\), extended from the disks \(\sigma_i^2\), determining the invariants \(\alpha(S_i)\). One can easily arrange that the frames
\[ (\tau_i,g(W_j^{(1)}),g(W_j^{(2)})), \]
where \(\tau_i\) are vector fields tangent to \(g(S_i)\), determine opposite orientations for \(i=1,2\). Next we glue in a “Whitney cell”
\[ \psi:\sigma^2\times S^1\to E^{2n-1} \]
such that
\[ \psi(\sigma^2\times S^1)\cap g(M^n)=g(B_1)\cup g(B_2). \]
It is also necessary that on the boundaries

* If \(n=4l+1\), one can assert the existence of maps \(g:M^n\to E^{2n-1}\) with zero invariant, since there exists an immersion \(M^n\to E^{2n-2}\) \((^4)\).

the mapping \(\psi\) has certain compatibility properties. After this, by virtue of the coincidence of the invariants of the pairs \(S_1\) and \(S_2\), one can specify, in a small neighborhood \(U(\psi(\sigma^2\times S^1))\), a suitable coordinate system, one of whose coordinates is a point of the circle, two others are a standard 2-frame on \(\sigma^2\), and the remaining ones satisfy our boundary conditions and are independent of these. Having specified the coordinates, we perform in the neighborhood \(U(\psi(\sigma^2\times S^1))\) a Whitney deformation with the coordinate of the circle held fixed.

Iterating this construction and using Lemma 6, we arrive at a mapping
\[ g_S:M^n\to E^{2n-1}, \]
which has special pairs only with zero invariant.

After this, with the mapping \(g_S\) one can proceed in two ways: either, following (2), glue in additional pairs with zero invariants and apply Lemma 7, or else carry out the direct separation of a pair with zero invariant.* In either case we arrive at an embedding. Thus, the theorem follows from the preceding lemmas.

I note that the lemmas imply the following conditional

Theorem 2. Let \(n=2l\), \(n\geqslant 6\), and \(\pi_1(M^n)=0\). An embedding \(M^n\subset E^{2n-1}\) exists if and only if there exists an immersion \(M^n\to E^{2n-2}\) (see \((^4)\)).*

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
20 I 1961

REFERENCES CITED

1. L. S. Pontryagin, Tr. Matem. inst. im. V. A. Steklova Akademii nauk SSSR, 45 (1955).
2. H. Whitney, Ann. of Math., 37, No. 2 (1944).
3. H. Whitney, Ann. of Math., 37, No. 3 (1944).
4. M. Hirsh, Trans. Am. Math. Soc., 93, No. 2 (1959).

* The method of separating one special pair with zero invariant was suggested to me by D. B. Fuks, who kindly read the present work.

** A more precise investigation shows that, for \(n=4l+2\), the invariant \(\alpha\) defined above is a homotopy invariant of the manifold; it does not depend on the immersion
\[ g:M^n\to E^{2n-1} \]
when \(n\not\equiv 1\pmod 4\).