MATHEMATICS

S. P. NOVIKOV

HOMOTOPIC PROPERTIES OF THE GROUP OF DIFFEOMORPHISMS OF THE SPHERE

(Presented by Academician L. S. Pontryagin on 30 VI 1962)

By \(\operatorname{diff} M^n\) and \(\operatorname{diff}^0 M^n\) we shall denote, respectively, the group of diffeomorphisms of a smooth manifold \(M^n\) preserving orientation, and its linearly connected component of the identity in the \(C^\infty\)-topology. Let \(W\) be a smooth manifold of class \(C^\infty\).

Definition 1. A mapping \(f : W \to \operatorname{diff} M^n\) will be called smooth of class \(C^r\) \((r \ge 0)\) if the mapping \(F(f) : W \times M^n \to W \times M^n\) such that
\[ F(f)(x,y)=(x,f_x(y)), \]
where \(x \in W\), \(y \in M^n\), is a diffeomorphism of class \(C^r\).

The following is easily proved.

Lemma 1. Every mapping \(f : W \to \operatorname{diff} M^n\) is approximated arbitrarily closely by a smooth mapping of class \(C^\infty\).

Our ultimate aim is to study the groups
\[ \operatorname{diff} S^{n-1}, \qquad \operatorname{diff}^0 S^{n-1}, \qquad \operatorname{diff} D^n, \qquad \operatorname{diff}^0 D^n, \qquad \overline K^n = K^n \cap \operatorname{diff}^0 D^n, \]
where \(K^n\) is the group of diffeomorphisms of the ball \(D^n\), fixed on the boundary \(S^{n-1}=\partial D^n\). A fibered space in the sense of Serre is defined:
\[ \operatorname{diff}^0 D^n \overset{\overline K^n}{\underset{p}{\longrightarrow}} \operatorname{diff}^0 S^{n-1}, \]
where \(p\) is the natural projection. The orthogonal group \(SO_n\) is naturally imbedded in the group \(\operatorname{diff}^0 S^{n-1}\). Milnor \((^{6-8})\) established, for a number of dimensions \(n\), the nontriviality of the groups \(\pi_0(\operatorname{diff} S^{n-1})\), and even of the group \(\pi_0(\operatorname{diff} S^{n-1})/p_*\pi_0(\operatorname{diff} D^n)\), which has been more or less completely computed for \(n \ge 6\). We shall study the groups \(\pi_i(\operatorname{diff}^0 S^{n-1})/p_*\pi_i(\operatorname{diff}^0 D^n)\) for a number of values of \(n\) and for \(i>0\). For this purpose we shall first study the relation between the Whitehead homomorphism \(J:\pi_i(SO_N)\to\pi_{N+i}(S^N)\) and the composition multiplication in the ring
\[ G=\sum_{i\ge 0}G_i \]
of stable homotopy groups of spheres \(G_i=\pi_{N+i}(S^N)\), \(N>i+1\). By \(J_i \subset G_i\) we shall denote the image \(J\pi_i(SO_N)\), and by \(\widetilde{\theta}_i \subset G_i\) we shall denote the subgroup of the group \(G_i\) each element of which is represented by a framed manifold combinatorially equivalent to the sphere \(S^i\). Obviously \(J_i \subset \widetilde{\theta}_i\). It is known that \(\widetilde{\theta}_i=G_i\) for \(i \not\equiv 2 \pmod 4\) or for \(i=10\), and that \(G_i/\widetilde{\theta}_i\) contains exactly 2 elements for \(i=2,6,14\) and no more than 2 elements in the remaining cases (these results are due to Kervaire, Milnor, and Smale \((^{5,6,10,11})\)). Using the technique of Morse modifications and surgeries of framed manifolds \((^{3,5})\), we prove comparatively simply the following theorem.

Theorem 1. Let \(\alpha \in G_i\), \(\beta \in G_j\), \(i>0\), \(j>0\). Then \(\alpha \circ \beta \in \widetilde{\theta}_{i+j}\) for all pairs \((i,j)\), except for the cases \(i=j=1,3,7\).

As is known, the Morse surgery depends on the embedding of the sphere \(S^i\subset M^n\) with trivial normal bundle and on an element \(h\in\pi_i(SO_{n-i})\). As a result of the Morse surgery we arrive at the manifold \(M^n(S^i,h)\).

It is not hard to prove

Lemma 2. Let \(M^n=S^i\times S^j\). Then the manifold \(M^n(S^i,h)\) is diffeomorphic to \(S^n\).

The proof is based on the fact that every diffeomorphism \(\widetilde h:S^i\times D^j\to S^i\times D^j\) such that \(\widetilde h(x,y)=(x,\widetilde h_x(y))\) extends to a diffeomorphism \(\widetilde h:S^i\times S^j\to S^i\times S^j\) such that \(\widetilde h/S^i\times D^j=(\widetilde h_x\in SO_j)\).

From Lemma 2 it follows that

Lemma 3. Let \(\alpha\in J_i,\ \beta\in J_j\). Then \(\alpha\circ\beta\in J_{i+j}\), except in the cases \(i=j=1,3,7\).

We shall now denote by \(B(M^n)\subset G_n\), for a \(\pi\)-manifold \(M^n\), the set of elements \(\alpha\in B(M^n)\subset G_n\) having as a representative the manifold \(M^n\) with some framing. With the aid of Morse surgeries, Lemma 2 gives

Lemma 4. The formula holds
\[ B(S^i\times S^j)=J_{i+j}, \]

except in the cases \(i=j=1,3,7\).

From the results of Haefliger \((^{1,2})\), Smale \((^{11})\), and Kervaire \((^4)\) it is not difficult to derive

Lemma 5. Let \(\widetilde S^j\) be a smooth \(\pi\)-manifold homeomorphic to a sphere, and let \(i>j/2+1\). Then the manifold \(S^i\times \widetilde S^j\) is diffeomorphic to the direct product \(S^i\times S^j\).

The proof of Lemma 5 is obtained from Haefliger’s theorems on approximation of topological embeddings by smooth ones, Smale’s theorems on \(J\)-equivalence, and Kervaire’s theorem on normal bundles of homotopy spheres in Euclidean space.

From Lemmas 2–5 it follows that

Theorem 2. Let \(\alpha\in J_i,\ \beta\in \widetilde\theta_j\), with \(i>j/2+1\). Then \(\alpha\circ\beta\in J_{i+j}\), except in the cases \(i=j=1,3,7\).

In the case when the hypotheses of Theorem 2 are not fulfilled, i.e. if \(i\le j/2+1\), a situation is possible in which \(J_i\circ\widetilde\theta_j\not\subset J_{i+j}\) for suitably chosen dimensions \(i,j\). Let \(\beta\in\widetilde\theta_j\), and let the element \(\beta\) be represented by a framed homotopy sphere \(\widetilde S^j_\beta\) with some framing; moreover the sphere \(\widetilde S^j_\beta\) is determined uniquely modulo \(\theta^j(\partial\pi)\). As is known, the sphere \(\widetilde S^j_\beta\) decomposes into the union of two disks
\[ \widetilde S^j_\beta=D^j\cup_{q_\beta}D^j, \]
where \(q_\beta\in \operatorname{diff} S^{j-1}\), i.e. \(q_\beta\in\pi_0(\operatorname{diff} S^{j-1})\), and determines an element
\[ \widetilde q_\beta\in \pi_0(\operatorname{diff} S^{j-1})/p_*\pi_0(\operatorname{diff} D^j). \]
The following lemma is very important for our purposes.

Lemma 6. Let \(\alpha\in J_i,\ \alpha\circ\beta\notin J_{i+j}\). Then there exists a smooth map \(h:S^i\to SO_j\) such that the diffeomorphism \(F_\beta(h)(x,y)=(x,[q_\beta h_x q_\beta^{-1}](y))\) does not extend to a diffeomorphism \(S^i\times D^j\to S^i\times D^j\), where
\[ F_\beta(h):S^i\times S^{j-1}\to S^i\times S^{j-1}. \]

Proof. Consider the direct product \(S^i\times\widetilde S^j_\beta\) and specify on it two different framings: the trivial one and the framing corresponding to the element \(\alpha\circ\beta\notin J_{i+j}\). Perform two Morse surgeries and pass to the manifolds \(M^n(S^i,h_1)\) and \(M^n(S^i,h_2)\), where \(h_i:S^i\to SO_j\) and \(n=i+j\). Choose the map \(h_1\) so that it would be possible to drag ...

on \(M^n(S^i,h_1)\) a trivial framing, and \(h_2\) so that on \(M^n(S^i,h_2)\) one could drag the framing defining the element \(\alpha\circ\beta\). Both manifolds \(M^n(S^i,h_1)\) and \(M^n(S^i,h_2)\) are homotopy spheres, but \(M^n(S^i,h_1)\in \theta^n(\partial\pi)\), while \(M^n(S^i,h_2)\notin \theta^n(\partial\pi)\). Therefore the map \(h=h_1h_2^{-1}\) is such that the diffeomorphism
\[ F(h):\quad S^i\times D^j\to S^i\times D^j, \]
where \(F(h)(x,y)=(x,h_x(y))\), does not extend to a diffeomorphism
\[ S^i\times \widetilde S^j_\beta\to S^i\times \widetilde S^j_\beta . \]
This is equivalent to saying that the diffeomorphism
\[ F_\beta(h):\quad S^i\times S^{j-1}\to S^i\times S^{j-1}, \]
where \(F_\beta(h)(x,y)=(x,[q_\beta h_x q_\beta^{-1}](y))\), does not extend to \(S^i\times D^j\). Thus the lemma is proved.

From Lemmas 5 and 1 we obtain the following

Corollary 1. Let \(\alpha\in J_i,\ \beta\in \hat\theta_j,\ \alpha\circ\beta\notin J_{i+j}\). Then there exists a diffeomorphism \(q_\beta:S^{j-1}\to S^{j-1}\), \(q_\beta\notin \operatorname{diff}^0 S^{j-1}\), and there exists an element \(h\in \pi_i(SO_j)\) such that the element
\[ q_\beta h q_\beta^{-1}\in \pi_i(\operatorname{diff}^0 S^{j-1}) \]
does not belong to
\[ \rho_*\pi_i(\operatorname{diff}^0 D^j) \]
(one may assume that \(q_\beta\in \pi^0(\operatorname{diff} S^{j-1})\)).

For the application of the last results it is necessary to know the structure of the groups \(G_i\), the products \(G_i\circ G_j\), the image \(\operatorname{Im}J\), and the subgroups \(\hat\theta_i\). We give a table of these groups for \(i\le 14\) and a multiplication table for \(G_i\circ G_j\).

One can choose generators
\[ x_i^{(p)},\ y_i^{(p)},\ z_i^{(p)}\in G_i \]
(\(p\) is a prime number)
\[ x_1^{(2)},\ x_2^{(2)},\ x_3^{(2)},\ x_3^{(3)},\ x_6^{(2)},\ x_7^{(2)},\ x_7^{(3)},\ x_7^{(5)},\ x_8^{(2)},\ y_8^{(2)},\ x_9^{(2)},\ y_9^{(2)},\ z_9^{(2)},\ x_{10}^{(2)},\ x_{10}^{(3)},\ x_{11}^{(2)},\ x_{11}^{(3)},\ x_{11}^{(7)},\ x_{13}^{(3)},\ x_{14}^{(2)}, \]
such that:

1. \(2x_1^{(2)}=0,\ 2x_2^{(2)}=0,\ 8x_3^{(2)}=0,\ 3x_3^{(3)}=0,\ 2x_6^{(2)}=0,\ 16x_7^{(2)}=0,\ 3x_7^{(3)}=0,\)
   \[ 5x_7^{(5)}=0,\quad 2x_8^{(2)}=0,\quad 2y_8^{(2)}=0,\quad 2x_9^{(2)}=0,\quad 2y_9^{(2)}=0,\quad 2z_9^{(2)}=0,\quad 2x_{10}^{(2)}=0, \]
   \[ 3x_{10}^{(3)}=0,\quad 8x_{11}^{(2)}=0,\quad 9x_{11}^{(3)}=0,\quad 7x_{11}^{(7)}=0,\quad 3x_{13}^{(3)}=0,\quad 2x_{14}^{(2)}=0. \]

2. \((x_1^{(2)})^2=x_2^{(2)},\ (x_1^{(2)})^3=4x_3^{(2)},\ (x_3^{(2)})^2=x_6^{(2)},\ x_1^{(2)}x_7^{(2)}=x_8^{(2)},\ (x_1^{(2)})^2x_7^{(2)}=(x_3^{(2)})^3=x_9^{(2)},\)
   \[ x_1^{(2)}y_8^{(2)}=y_9^{(2)},\quad (x_1^{(2)})^2y_8^{(2)}=0,\quad (x_1^{(2)})^3x_7^{(2)}=0,\quad x_1^{(2)}z_9^{(2)}=x_{10}^{(2)},\quad (x_1^{(2)})^2z_9^{(2)}=4x_{11}^{(2)},\quad (x_7^{(2)})^2=x_{14}^{(2)}, \]
   \[ x_{11}^{(2)}x_3^{(2)}=0,\quad x_3^{(3)}x_{10}^{(3)}=x_{13}^{(3)}. \]

3. \(x_1^{(2)},\ x_3^{(2)},\ x_3^{(3)},\ x_7^{(2)},\ x_7^{(3)},\ x_7^{(5)},\ x_8^{(2)},\ x_9^{(2)},\ x_{11}^{(2)},\ x_{11}^{(3)},\ x_{11}^{(7)}\in \operatorname{Im}J\); the remaining generating elements do not belong to \(\operatorname{Im}J\).

4. All generators, except
   \[ (x_1^{(2)})^2=x_2^{(2)},\quad (x_3^{(2)})^2=x_6^{(2)},\quad (x_7^{(2)})^2=x_{14}^{(2)}, \]
   belong to the subgroups \(\tilde\theta_i\).

Further, concerning the \(p\)-components of the groups \(G_i^{(p)}\), it is known that:

1) \(G_{2p-3}^{(p)}=Z_p=J_{2p-3}^{(p)}\) (generator \(x_{2p-3}^{(p)}\));

2) \(G_{2p(p-1)-2}^{(p)}=\hat\theta_{2p(p-1)-2}^{(p)}=Z_p\) (generator \(x_{2p(p-1)-2}^{(p)}\)), and for \(p>2\) the group
\[ I_{2p(p-1)-2}^{(p)}=0; \]

3) the elements
\[ x_{2p-3}^{(p)}\circ \left(x_{2p(p-1)-2}^{(p)}\right)^k\notin \operatorname{Im}J \]
for \(k\le p-2\), and
\[ x_{2p-3}^{(p)}\circ \left(x_{2p(p-1)-2}^{(p)}\right)^{p-1}\ne 0,\qquad \left(x_{2p(p-1)-2}^{(p)}\right)^p\ne 0. \]
(For results concerning multiplication in the homotopy groups of spheres, see (9).)*

* We note that earlier it was believed that
\[ G_{14}=\pi_{N+14}(S^N)=Z_2+Z_2. \]
However, this result is incorrect. The author showed that \(G_{14}=Z_2\), using the fact that
\[ J_8\circ J_{11}\subset J_{14}=0 \]
and that
\[ G_{14}=J_7\circ J_7\cup J_3\circ J_{11}. \]

Now it remains for us, using the preceding results and the information on the groups \(G_n,\theta_n,J_n\) and the multiplication \(G_i\circ \widetilde{G}_j\), to indicate cases of nontriviality of the groups
\(A_{i,j}=\pi_i(\operatorname{diff}^0 S^{j-1})/p_*\pi_i(\operatorname{diff}^0 D^j)\).

Theorem 3. The groups
\[ A_{i,j}=\pi_i(\operatorname{diff}^0 S^{j-1})/p_*\pi_i(\operatorname{diff}^0 D^j) \]
have the following form:

1) \(A_{1,8}\supset Z_2\);

2) \(A_{1,9}\supset Z_2\);

3) \(A_{2p-3,\,2kp(p-1)-2k}\otimes Z_p \supset Z_p+\cdots+Z_p\) \((p-1\) summands\(),\ p\geqslant 3\) for \(k\leqslant p-2\);
\[ A_{3,10}\otimes Z_3 \supset Z_3+Z_3 \quad \text{for } p=3. \]

The proof of Theorem 3 follows at once from the lemmas and from the structure of the ring \(G=\sum G_i\). Since \(\pi_1(SO_n)=Z_2\) \((n>2)\), it follows that \(h\in Z_2\) (see Lemma 6), and the element \(\beta\) has order 2, \(\beta\in G_8(G_9)\). Therefore
\(q_\beta hq_\beta^{-1}\in\pi_1(\operatorname{diff}^0 S^7)\) also has order 2 and
\(q_\beta^2\in\operatorname{diff}^0 S^7\). Hence the group
\(\pi_1(\operatorname{diff}^0 S^7)\supset Z_2+Z_2\) with generators \(h\) and
\(q_\beta hq_\beta^{-1}\notin \operatorname{Im}p_*\). Thus \(A_{1,8}\supset Z_2\). Similarly for \(A_{1,9}\). Assertions 1) and 2) are proved. Let us prove assertion 3). Note that
\(\pi_{2p-3}(SO_j)=Z\) for \(j>2p-2\). Let \(h\) be a generator of the group
\(\pi_{2p-3}(SO_j)\), and
\(\beta=\chi(p)_{2p(p-1)-2}^{\,k}\), \(k\leqslant p-2\). Then
\(h, q_\beta hq_\beta^{-1},\ldots,q_\beta^{p-1}hq_\beta^{1-p}\) are distinct elements of the group
\(\pi_{2p-3}(\operatorname{diff}^0 S^{j-1})\), \(j=2kp(p-1)-2k\), and all these elements are of infinite order. However, relations of the form
\[ \lambda_1ph=\lambda_2p(q_\beta hq_\beta^{-1})=\cdots=\lambda_p\circ p(q_\beta^{p-1}hq_\beta^{1-p}) \]
are possible, whence the desired result follows. The theorem is proved.

As usual, by \(B_G\) we shall denote the classifying space of the group \(G\).

Corollary 2. The classifying space \(B_{\operatorname{diff}S^{n-1}}\) is not homotopically simple for \(n=8,9,2kp(p-1)-2k,\ k\leqslant p-2\); namely: the group \(\pi_1\) acts nontrivially on the groups respectively
\[ \pi_2(B_{\operatorname{diff}S^7}),\quad \pi_2(B_{\operatorname{diff}S^8}),\quad \pi_{2p-2}(B_{\operatorname{diff}S^{2kp(p-1)-2k-1}}),\quad k\leqslant p-2. \]

From the fibration, in the sense of Serre,
\[ \operatorname{diff}^0D^n \xrightarrow{\ \overline{K}^n\ } \operatorname{diff}^0S^{n-1}, \]
where
\[ \overline{K}^n=K^n\cap \operatorname{diff}^0D^n, \]
we obtain

Corollary 3. a) There exists a diffeomorphism \(F:D^n\to D^n\) such that
\(F\in \operatorname{diff}^0D^n\), \(F/\partial D^n=1\), and \(F\) is not isotopic to the identity in the group \(K^n\) for \(n=8,9\); b) the groups
\[ \pi_{2p-4}\bigl(K^{2k(p-1)p-2k}\bigr)\neq 0 \]
\((p>2)\) for \(k\leqslant p-2\) (\(p\) a prime number).

Corollary 4. There exist sphere bundles over spheres with structural group \(\operatorname{diff}^0S^{n-1}\), not equivalent to orthogonal bundles in the group \(\operatorname{diff}^0S^{n-1}\), but equivalent to orthogonal bundles in the group \(\operatorname{diff}S^{n-1}\).

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
23 VI 1962

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