Mathematics

A. M. Vinogradov

Some homotopic properties of the space of embeddings of a circle in a sphere or ball

(Presented by Academician P. S. Aleksandrov, January 20, 1964)

Let \(M^m, N^n\) be smooth manifolds. Denote by \(B\lambda(M^m, N^n)\) the space of all smooth embeddings of the first manifold into the second, endowed with the \(C^1\)-topology. The problem of classifying embeddings \(M^m \to N^n\) consists in describing the set \(\pi_0(B\lambda(M^m, N^n))\). In the present work we attempt to go further, i.e., to compute the groups \(\pi_i(B\lambda(M^m, N^n))\), \(i > 0\), for the case when \(M^m\) is the one-dimensional sphere \(S^1\), and \(N^n\) is either \(S^n\) (the \(n\)-sphere), or \(D^n\) (the \(n\)-ball) (or, what is the same, the Euclidean space \(R^n\)).

Theorem 1. For \(i \leq 2n - 7\) there is an isomorphism
\[ \pi_i(B\lambda(S^1, S^n)) = \pi_i(T^1(S^n)), \]
where \(T^1(S^n)\) is the space of nonzero tangent vectors of the sphere \(S^n\).

For example, the space \(B\lambda(S^1, S^4)\) is simply connected (although, since the space \(B\lambda(S^1, S^3)\) is disconnected, it would be more natural to suppose that it is only connected).

Theorem 2. The group \(\pi_i(B\lambda(S^1, R^n))\) contains as a direct summand the group \(\pi_i(T^1(S^{n-1}))\). If \(i \leq 2n - 7\), then
\[ \pi_i(B\lambda(S^1, R^n)) = = \pi_i(T^1(S^{n-1})). \]

Comparison of Theorems 1 and 2 indicates a great difference between the spaces \(B\lambda(S^1, R^n)\) and \(B\lambda(S^1, S^n)\). On the other hand, the natural embedding
\[ B\lambda(S^1, S^{n-1}) \to B\lambda(S^1, R^n), \]
generated by the embedding \(S^{n-1} \to R^n\), induces a monomorphism of homotopy groups which, in dimensions \(\leq 2n - 7\), is also an epimorphism.

In connection with Theorem 1 the question naturally arises: what are the groups \(\pi_i(B\lambda(S^1, S^n))\) for \(i \geq 2n - 6\)? One may think that these groups, in particular \(\pi_{2n-6}(B\lambda(S^1, S^n))\), are not even of finite type (cf. \(\pi_0(B\lambda(S^1, S^3))\)).

Below a sketch will be given of the proofs of Theorems 1 and 2. Before this, however, I wish to note that the methods used here admit a generalization which makes it possible, to some extent, to study the spaces \(B\lambda(S^k, S^n)\), \(k \geq 1\), and also some other spaces of smooth and combinatorial embeddings.

Proof of Theorem 1. Fix on the circle \(S^1\) a point \(x_0\), and to every embedding \(f \in B\lambda(S^1, S^n)\) assign the differential \(df|_{x_0}\). As a result we obtain a mapping
\[ \Phi: B\lambda(S^1, S^n) \xrightarrow{B\lambda_0(S^1,S^n)} T^1(S^n), \]
where the set \(B\lambda_0(S^1, S^n)\) consists of all embeddings \(f \in B\lambda(S^1, S^n)\) with fixed differential at the point \(x_0\). Below it will be shown that \(\pi_i(B\lambda_0(S^1, S^n)) = 0\) if \(i \leq 2n - 7\). Since the mapping \(\Phi\) is a Serre fibration, Theorem 1 follows from this result.

Let \(I = [0,1]\) and \(\alpha = ({}^{1}/_{2}, \ldots, {}^{1}/_{2}, 0) \in \partial I^n\), \(\beta = ({}^{1}/_{2}, \ldots, {}^{1}/_{2}, 1) \in \partial I^n\). Denote by \(Q_{k,n}\) the space of all simple \(k\)-segment polygonal lines, lying

lying in the cube \(I^n\) and joining the points \(\alpha\) and \(\beta\). Dividing the first segment of a \(k\)-segment polygonal line in half, we obtain a mapping \(\varphi_k: Q_{k,n}\to Q_{k+1,n}\). Denote by \(Q_n\) the limit of the sequence

\[ Q_{1,n}\xrightarrow{\varphi_1} Q_{2,n}\xrightarrow{\varphi_2}\cdots \xrightarrow{\varphi_{k-1}} Q_{k,n}\xrightarrow{\varphi_k}\cdots \]

Then the following holds:

Lemma 1. The spaces \(B\Lambda_0(S^1,S^n)\) and \(Q_n\) are homotopy equivalent.

Therefore it is enough to show that \(\pi_i(Q_{k,n})=0\), if \(i\le 2n-7\).

Let \(P_{j,n}\) be the space of simple polygonal lines beginning at the point \(\alpha\). Then, assigning to each polygonal line \(g\in Q_{k,n}\) the polygonal line consisting of its first \(j\) segments, we obtain a mapping \(p_j:Q_{k,n}\to P_{j,n}\). Put \(Q^j_{k,n}=p_j(Q_{k,n})\). Then the equality \(p_jq_j=p_{j-1}\) uniquely defines a mapping \(q_j:Q^j_{k,n}\to Q^{j-1}_{k,n}\).

It is now not hard to show that the following is true.

Lemma 2. The mapping \(q_k\) is a homeomorphism; all fibers of the mapping \(q_{k-1}\) are \((n-4)\)-connected; all fibers of the mapping \(q_j\), \(j<k-1\), are homotopically trivial.

On the basis of what has been said one can establish that the space \(Q^{k-2}_{k,n}\) is homotopically trivial, and the space \(Q^{k-1}_{k,n}\cong Q_{k,n}\) is \((n-4)\)-connected.

Consider now the mapping \(q_{k-1}\) and its spectral sequence \(\{E^r_{p,q}\}\) of singular homologies. Then \(E^2_{p,q}=H_p(Q^{k-2}_{k,n};\mathcal H_q)\), where \(\mathcal H_q\) is a certain system of coefficients. From Lemma 2 it follows that \(\mathcal H_q=0\), if \(q\ne 0\), \(q\ne n-2\), \(q\ne n-3\), and since the space \(Q^{k-2}_{k,n}\) is homotopically trivial, \(E^2_{0,q}=0\), if \(q>0,\ n>4\).

The coefficient systems \(\mathcal H_{n-2}\), \(\mathcal H_{n-3}\) are already nontrivial. However, one can show that \(E^2_{n-2,q}=0\), if \(q\le n-5\); \(E^2_{n-3,q}=0\), if \(q\le n-4\). For example, the triviality of the group \(E^2_{n-2,q}\), \(q\le n-5\), follows from Lemma 3.

Lemma 3. Let \(X=\{x\in Q^{k-2}_{k,n}\mid q^{-1}_{k-1}(x)\text{ is homotopically trivial}\}\), and let \(L\) be a finite polyhedron of dimension \(\le n-5\). Then for an arbitrary mapping \(f:L\to Q^{k-2}_{k,n}\) there exists a mapping \(f':L\to Q^{k-2}_{k,n}\), arbitrarily close to \(f\), such that \(f'(L)\subset X\).

Finally, since \(E^2_{p,q}=E^\infty_{p,q}\), \(n\ge 4\), it follows from the triviality of the groups \(E^2_{p,q}\), \(0<p+q<2n-7\), Lemma 2 and Hurewicz’s theorem that Theorem 1 follows.

Remark. One can show that, generally speaking, \(H_{2n-6}(Q_{k,n})\ne 0\).

Proof of Theorem 2. Theorem 2 is a consequence of the more general Theorem 3.

Put \(\widetilde N^n=N^n\setminus x_0\), \(x_0\in \operatorname{int}N^n\). Then the following holds.

Theorem 3. Let \(N^n=S^n\). Then

\[ \pi_i(B\Lambda_0(S^k,\widetilde N^n)) = \pi_i(B\Lambda(S^k,N^n))+\pi_i(S^{\,n-k-1}). \]

We shall first show that Theorem 2 follows from Theorem 3. To this end consider the fibrations

\[ \Phi:\ B\Lambda(S^k,R^n)\xrightarrow{B\Lambda(S^k,R^n)} R^n\times V_{n,k}, \qquad p:\ V_{n,k+1}\xrightarrow{S^{\,n-k-1}} V_{n,k} \]

and note that every frame \(\xi\in V_{n,k-1}\) defines in a natural way an embedding \(q(\xi)\in B\Lambda(S^k,R^n)\), which realizes the sphere \(S^k\) in the “standard” way in the \((k+1)\)-plane spanned by the frame \(\xi\).

The mapping \(q\) induces the following commutative diagram

\[ \begin{array}{cccccc} \cdots \to (B\Lambda_0(S^k,R^n)) & \to & \pi_i(B\Lambda(S^k,R^n)) & \xrightarrow{\Phi_*} & \pi_i(R^n\times V_{n,k}) & \to \cdots \\ \ \ \uparrow s_* & & \uparrow q_* & & \uparrow t_* & \\ \cdots \to \pi_i(S^{n-k-1}) & \to & \pi_i(V_{n,k+1}) & \xrightarrow{p_*} & \pi_i(V_{n,k}) & \to \cdots \end{array} \]

in which \(t_*\) is an isomorphism, \(s_*\) is a monomorphism, defined by the decomposition

\[ \pi_i\bigl(B\Lambda_0(S^k,R^n)\bigr)=\pi_i\bigl(B\Lambda_0(S^k,S^n)\bigr)+\pi_i(S^{n-k-1}). \]

It follows easily from this that \(q_*\) is a monomorphism. Taking \(k=1\) and applying Theorem 1, we obtain Theorem 2.

Proof of Theorem 3. Let one of the components \(\widetilde S^{\,n-1}\) of the boundary \(\partial L^n\) of the manifold \(L^n\) be diffeomorphic to the sphere \(S^{n-1}\), and let \(\widetilde S^{\,k-1}\) be the image in \(\widetilde S^{\,n-1}\), under this diffeomorphism, of the sphere \(S^{k-1}\) standardly embedded in \(S^{n-1}\). Consider the space \(X_k(\widetilde S^{\,n-1},L^n)\) of all embeddings of the ball \(D^k\) in \(L^n\) satisfying the conditions:

a) the embedding \(f|_{\partial D^k}=g\) does not depend on \(f\in X_k(\widetilde S^{\,n-1},L^n)\), \(g(\partial D^k)=\widetilde S^{\,k-1}\);

\(\beta)\) the differential \(df|_{\partial D^k}\) does not depend on \(f\in X_k(\widetilde S^{\,n-1},L^n)\) and is transversal to \(\partial L^n\).

Let \(N^n\) be some manifold, \(x\in \operatorname{int} N^n\). Cutting out from \(N^n\) the interior of a small ball \(D_x^n\) with center at \(x\), we obtain the manifold \(L^n=L(N^n)\). As the component \(\widetilde S^{\,n-1}\) of the boundary \(\partial L^n\) of this manifold we choose \(\partial D_x^n\), and put \(X_k(N^n)=X_k(\widetilde S^{\,n-1},L(N^n))\). The following lemma holds:

Lemma 4. The spaces \(B\Lambda_0(S^k,N^n)\) and \(X_k(N^n)\) are homotopy equivalent.

We shall call an embedding
\[ \varphi:D^k\times D^{n-k}\to L(N^n) \]
admissible if \(\varphi_{D^k\times 0}\in X_k(N^n)\) and \(\varphi(\partial D^k\times D^{n-k})\subset \widetilde S^{\,n-1}\). Two admissible embeddings \(\varphi,\varphi'\) will be called equivalent if there exists a mapping \(\psi:D^k\to SO(n-k)\) such that \(\varphi(x,y)=\varphi'(x,\psi(x)\cdot y)\), \(x\in D^k\), \(y\in D^{n-k}\). The class of admissible equivalent embeddings \(\varphi\) for which \(\varphi|_{D^k\times 0}=f\) will be called a tubular neighborhood of the embedding \(f\in X_k(N^n)\).

Let \(B\subset X_k(N^n)\) be compact and \(N^n=\widetilde N^n\cup x_0\).

Lemma 5. There exists a family of tubular neighborhoods \(u_f\), continuous relative to \(f\), \(f\in B\), containing within themselves the point \(x_0\), if \(N^n=S^n\).

Every embedding
\[ s:D^k\to L(N^n) \]
having the form \(\varphi\cdot s'\), where \(s'\) is a sectioning surface of the fibration
\[ D^k\times D^{n-k}\to D^k, \]
and the mapping
\[ \varphi:D^k\times D^{n-k}\to L(N^n) \]
is admissible, will be called a sectioning surface of the tubular neighborhood determined by the embedding \(\varphi\). We may regard the space \(L(\widetilde N^n)\) as a subspace of the space \(L(N^n)\). Under this condition the following is true:

Lemma 6. Let \(B\subset L(\widetilde N^n)\) be some compact set, and let the family \(u_f\), \(f\in B\), be the same as in Lemma 5. Then there exists a continuous family \(s_f\) of nonzero sectioning surfaces of the family \(u_f\), passing through the point \(x_0\).

Let now \(N^n=S^n\), \(\mu:S^i\to X_k(N^n)\) be some mapping, \(B=\mu(S^i)\), and \(a\in\partial D^k\). Then, putting

\[ H(y)=s_{\mu(y)}(a)\in \widetilde S^{\,n-1}\setminus \widetilde S^{\,k-1},\quad y\in S^i, \]

we obtain a mapping
\[ H:S^i\to \widetilde S^{\,n-1}\setminus \widetilde S^{\,k-1}. \]

Lemma 7. The homotopy class of the mapping \(H\) depends only on the homotopy class of the mapping \(\mu\) and determines an epimorphism

\[ h:\pi_i\bigl(X_k(\widetilde N^n)\bigr)\to \pi_i\bigl(\widetilde S^{\,n-1}\setminus \widetilde S^{\,k-1}\bigr)\approx \pi_i(S^{n-k-1}). \]

Let, further, the embedding \(\chi: I \to L(N^n)\) be such that \(\chi(0)=x_0\), \(\chi(1)\in \widetilde S^{\,n-1}\setminus \widetilde S^{\,k-1}\), and \(\chi((0,1))\cap \partial L^k=\varnothing\). Let \(\overline X_k(N^n)\subset X_k(S^n)\) consist of those embeddings \(f\in X_k(N^n)\) for which \(f(D^k)\cap \chi(I)=\varnothing\). Then the following holds.

Lemma 8. The space \(X_k(N^n)\) contracts onto \(\overline X_k(N^n)\subset X_k(N^n)\).

Since \(\overline X_k(N^n)\subset X_k(\widetilde N^n)\), it follows from Lemma 8 that \(\pi_i(X_k(N^n))\) is a direct summand of the group \(\pi_i(X_k(\widetilde N^n))\). Moreover, it is not hard to show that \(\pi_i(\overline X_k(N^n))=\operatorname{Ker} h\). Therefore, by Lemma 7,

\[ \pi_i\bigl(X_k(\widetilde N^n)\bigr) = \pi_i\bigl(\overline X_k(N^n)\bigr) + \pi_i(S^{n-k-1}), \]

which proves Theorem 3.

Moscow Institute
of Radio Electronics and Mining Electromechanics

Received
17 January 1964