V. A. Rokhlin

NEW EXAMPLES OF FOUR-DIMENSIONAL MANIFOLDS

(Presented by Academician L. S. Pontryagin, 27 XI 1964)

1. Introduction.

In this note a manifold means an oriented smooth connected compact manifold, and the word submanifold has the analogous meaning. We study simply connected closed four-dimensional manifolds. If \(M\) is such a manifold, then the self-intersection index of an element of the integral group \(H_2(M)\) is a quadratic form on \(H_2(M)\) with discriminant \(\pm 1\), denoted by \(q(M)\). Considered up to isomorphism, the form \(q(M)\) is an invariant of the oriented homotopy type of the manifold \(M\) and, as was proved in 1949 by Whitehead \((^1)\) and L. S. Pontryagin \((^2)\), completely determines this type. Subsequently \((^{3-5})\), the invariants of the tangent bundle of the manifold \(M\) were also expressed in terms of \(q(M)\), and recently S. P. Novikov \((^6)\) and Wall \((^7)\) proved that \(q(M)\) determines \(M\) up to \(h\)-cobordism. Obviously,
\[ q(M_1 \# M_2) \simeq q(M_1) \oplus q(M_2). \]

The following facts from algebra make it possible to estimate the degree of effectiveness of the preceding classification theorems. We shall agree to call integral quadratic forms with discriminant \(\pm 1\) simply forms, and to assign a form to type II if all its values are even, and to type I otherwise. It is known that there exists only a finite number of nonisomorphic forms of a given rank; that an indefinite form is determined up to isomorphism by its rank, signature, and type; and that the signature of a form of type II is divisible by 8 (see, for example, \((^8,^9)\)). Further, a positive form decomposes in a unique way into an orthogonal sum of indecomposable forms \((^{10},^{11})\), and there exists an infinite number of indecomposable positive forms both of type I and of type II—for example, there is a construction assigning to each \(r \equiv 0 \pmod 4\) and \(>4\) an indecomposable positive form \(K_r\), belonging to type II if and only if \(r \equiv 0 \pmod 8\) \((^{12})\). There is no satisfactory classification of positive forms; only a list of indecomposable positive forms of ranks \(\leq 16\) has been compiled: there exist, up to isomorphism, 7 such forms, their ranks are \(1, 8, 12, 14, 15, 16, 16\), and only two of them, \(K_8\) and \(K_{16}\), belong to type II \((^{13})\).

The main unsolved problem in the homotopy theory of simply connected four-dimensional manifolds is which forms are realizable by manifolds, i.e., what must a form \(q\) be like for there to exist a (simply connected closed) manifold \(M\) with \(q(M) \simeq q\). Until now this problem had apparently been considered only in two papers \((^4,^8)\). In my note \((^4)\) it was proved that the signature of a realizable form of type II is divisible by 16 and that realizable forms of type II with signature 16 exist, while in Milnor’s survey \((^8)\) manifolds with a form of type II, signature 16 and rank 22 are indicated, and the existence of a manifold with a positive form of type II and signature 16 is put forward as a problem of first priority. (I note that the form of type II and signature 16, implicitly realized in \((^4)\), also had rank 22; the construction of the realizing manifold is reproduced below in § 6.) For completeness I shall also list the trivial facts: connected sums of manifolds diffeomorphic to \(P_2C\), \(\overline{P_2C}\), and \(S^2 \times S^2\) realize all forms
\[ I_p \oplus (-I_n), \]
where \(I_p\) is the orthogonal sum of \(p\) positive forms

rank 1 (in particular, all indefinite forms of type I) and all forms of type II with signature 0.

2. Formulation of results. The forms \(K_8 \oplus I_1\) and \(K_8 \oplus K_8\) are realizable. Every indefinite form of type II with signature divisible by 16 is realizable.

The last assertion exhausts the problem of realizability of indefinite forms. It is a consequence of the realizability of the form \(K_8 \oplus K_8\): if \(B\) is a manifold realizing this form, then an indefinite form of type II with signature \(16s \geqslant 0\) and rank \(r\) is realized by the manifold
\(sB \# \frac12(r-16s)(S^2 \times S^2)\), while the case \(s<0\) reduces to the case \(s>0\).

The forms \(K_8 \oplus I_1\) and \(K_8 \oplus K_8\) will be realized below. Since they do not decompose into realizable forms, the manifolds \(A\) and \(B\) realizing them do not decompose into connected sums not containing homotopy spheres; thus, to the three known manifolds indecomposable in this sense, \(P_2C\), \(\overline{P_2C}\), and \(S^2 \times S^2\), new ones are added. I also note that in the class of realizable positive forms there is no uniqueness of decomposition into indecomposable ones: for example,
\((K_8 \oplus I_1) \oplus (K_8 \oplus I_1) = (K_8 \oplus K_8) \oplus I_1 \oplus I_1\).
It is unknown whether the manifolds \(A \# A\) and \(B \# P_2C \# P_2C\) are diffeomorphic.

3. Membranes. Let \(M\) be a four-dimensional manifold and let \(F\) be its closed two-dimensional submanifold. A membrane on \(F\) is any submanifold \(P\) of the manifold \(M\), diffeomorphic to a disk, intersecting \(F\) exactly in its boundary \(\partial P\) and having with \(F\) (along \(\partial P\)) no common tangent planes. If one constructs on \(\partial P\) a vector field tangent to \(F\) and not tangent to \(P\), then the attempt to extend it without touching onto \(P\) leads to a two-dimensional obstruction with integer index (the orientations used of planes transversal to \(P\) must be compatible with the orientations of \(P\) and \(M\)), independent of the choice of the field on \(\partial P\). This index is called the index of the membrane \(P\) and is denoted by \(i(P)\). It does not depend on the orientation of \(P\) and changes sign when the orientation of \(M\) is changed.

If \(i(P)=0\), then the field extends onto \(P\) without touching. In this case one can perform a Morse surgery of the submanifold \(F\) along \(P\): replace the cylinder serving as a neighborhood of the circle \(\partial P\) in \(F\) by two nonintersecting smoothly attached disks close to \(P\). If \(\partial P\) does not separate \(F\), then the result will be a decrease of the genus of the surface \(F\) by one.

The following two methods make it possible to rebuild membranes. Let \(\Sigma\) be a submanifold of the manifold \(M\), diffeomorphic to \(S^2\), and not intersecting \(P\). The first method assumes that \(\Sigma\) does not intersect \(F\), and consists in forming the connected sum of \(P\) and \(\Sigma\) (along some path). The result is a new membrane \(P_1\) with \(\partial P_1=\partial P\) and \(i(P_1)=i(P)+\xi^2\), where \(\xi\) is the element of the group \(H_2(M)\) represented by the sphere \(\Sigma\). The second method assumes that \(\Sigma\) intersects \(F\) at one point, and moreover regularly and with index \(+1\), and is carried out in two steps. First step: the sphere \(\Sigma\) is deformed in a neighborhood of its intersection with \(F\) so that the intersection becomes a circle, and the closed complement of this circle in \(\Sigma\) becomes a membrane. Second step: this membrane is connected with \(P\) (along some path lying on \(F\)) into a new membrane \(P_1\). The first step can be carried out in two homologically different ways: in one way the degree \(v\) of the normal mapping of the sphere \(\Sigma\) onto \(F\) is \(+1\), in the other \(-1\). As is not difficult to show, \(i(P_1)=i(P)+\xi^2+v\), and it is clear that the circles \(\partial P\) and \(\partial P_1\) are homologous on \(F\).

4. Example. Put \(M=P_2C\) and take for \(F\) the torus defined in homogeneous coordinates \(x,y,z\) by the equation
\(y^2z=x^3-z^3\) and representing the tripled generator of the group \(H_2(P_2C)\). The purpose of this paragraph is to construct on \(F\) two membranes \(P_1,P_2\) of index 1, lying in the cell \(z\ne 0\) and intersecting in one point, which serves as a regular point of intersection of their boundaries on \(F\).

Since we restrict ourselves to the Euclidean part \(z \ne 0\) of the manifold \(P_2C\), we may assume that \(F\) is defined by the equation \(y^2=x^3-1\) in complex Cartesian coordinates \(x,y\). Let \(K\) be a circle in the plane \(y=0\), containing two roots of the equation \(x^3=1\) as interior points and containing the third root neither in its interior nor on its boundary, and let \(L\) be one of the two loops on \(F\) covering the circle \(\partial K\). Construct on \(L\), as on a base, a cone with vertex \(x=y=0\), and smooth it near the vertex (which is not a point of local knotting). Then the cone turns into a membrane \(P_1\) of index \(1\), and, rotating it by \(120^\circ\) about the plane \(x=0\), we obtain a membrane \(P_2\). The boundaries of these membranes intersect at one point, regularly on \(F\), and the intersections not lying on the boundaries are removed by a small deformation.

1. Realization of the form \(K_8\oplus I_1\). Put \(X=\overline{P_2C}\#9P_2C\), and let \(\Sigma_0,\Sigma_1,\ldots,\Sigma_9\) be projective lines in the summands and \(\alpha_0,\alpha_1,\ldots,\alpha_9\) the generators of the group \(H_2(X)\) determined by them. Put, further,
   \(\beta=3\alpha_0+\alpha_1+\cdots+\alpha_8\), \(\gamma=\beta+\alpha_9\), and denote by \(G\) the annihilator of the class \(\beta\) in \(H_2(X)\) (with respect to the intersection index), and by \(G_1\) the annihilator of the pair \(\beta,\alpha_9\). Since \(\beta^2=-1\), \(\alpha_9^2=1\), \(\beta\alpha_9=0\), and the form \(q(X)\) has rank 10 and signature 8, it follows that
   \(H_2(X)=G\oplus Z\beta=G_1\oplus Z\alpha_9\oplus Z\beta\), and on \(G\) and \(G_1\) the form \(q(X)\) is positive and has discriminant 1 and ranks 9 and 8. Since, moreover, \(\xi^2\equiv \xi\gamma \mod 2\) for every \(\xi\in H_2(X)\), the form \(q(X)|_{G_1}\) belongs to type II. Consequently,
   \(q(X)|_{G_1}\simeq K_8\), and \(q(X)|_G\simeq K_8\oplus I_1\).

The connected sum \(E\) of the torus \(y^2z=x^3-z^3\), representing the class \(3\alpha_0\) in \(\overline{P_2C}\) (see § 4), and the spheres \(\Sigma_1,\ldots,\Sigma_8\) is a torus representing the class \(\beta\) in \(X\). The membrane \(P_1\), constructed in § 4, may be regarded as a membrane on \(E\); it has index \(-1\), and its boundary does not separate \(E\). Reconstructing it with the aid of the sphere \(\Sigma_9\) (the first method of § 3), we obtain a membrane of index \(-1+\alpha_9^2=0\) with the same boundary. Consequently, the class \(\beta\) is represented by a sphere. Since \(\beta^2=-1\), the boundary of a tubular neighborhood of this sphere is diffeomorphic to \(S^3\). Replacing this neighborhood by a four-dimensional ball, we obtain a simply connected manifold \(A\) with \(q(A)\simeq q(X)|_G\simeq K_8\oplus I_1\).

Remark. The manifold \(A\#\overline{P_2C}\) is diffeomorphic to \(X\).

1. Realization of the form \(K_8\oplus K_8\). Return to the manifold \(X\). The connected sum \(T\) of the torus \(E\) and the sphere \(\Sigma_9\) is a torus representing the class \(\gamma\). Since \(\gamma^2=0\), the boundary of a tubular neighborhood of the torus \(T\) has the form \(T\times S^1\). Cutting it out, we obtain a simply connected manifold \(W\) with boundary \(\partial W=T\times S^1\) and with a form isomorphic to \(K_8\) (the form of a four-dimensional manifold \(M\) with boundary is defined as the self-intersection index, considered on \(H_2(M)/i_*H_2(\partial M)\), where \(i:\partial M\to M\) is the inclusion; if \(M\) is simply connected, then \(i_*\) is a monomorphism). The membranes \(P_1,P_2\), constructed in § 4, may be regarded as membranes on \(T\) in \(X\), and in \(W\) from them there remain disks \(D_1,D_2\), while from the sphere \(\Sigma_9\) there remains a disk \(D_3\). The generators of the group \(H_2(\partial W)\) are represented by the tori
   \(T_1=\partial P_2\times S^1\), \(T_2=\partial P_1\times S^1\), \(T_3=T\times c\), where \(c\in S^1\). The intersections
   \(D_1\cap T_1\), \(D_2\cap T_2\), \(D_3\cap T_3\) are one-point and regular, and the disks \(D_1,D_2,D_3\) become, after a suitable displacement, membranes: \(D_1\) a membrane of index \(-1\) on \(T_2\) and \(T_3\); \(D_2\) a membrane of index \(-1\) on \(T_3\) and \(T_1\); \(D_3\) a membrane of index \(1\) on \(T_1\) and \(T_2\). The boundaries of these membranes do not separate \(T_1,T_2,T_3\).

Glue the manifold \(W\) to its second copy by means of the automorphism of the boundary \(\partial W=T\times S^1\) composed of the identity transformation of the torus \(T\) and the reflection of the circle \(S^1\) in its diameter. We obtain a closed simply connected manifold \(Y\) with a form \(q(Y)\) of type II, of rank 22 and signature 16, in which the disks \(D_1,D_2,D_3\) and their doubles \(D'_1,D'_2,D'_3\) are smoothly glued into spheres \(S_1,S_2,S_3\). The nonzero intersection indices of the elements \(\sigma_1,\sigma_2,\sigma_3,\tau_1,\tau_2,\tau_3\) of the group \(H_2(Y)\), represented by the spheres \(S_1,S_2,S_3\) and the tori \(T_1,T_2,T_3\), are given by the table
\(\sigma_1^2=\sigma_2^2=-2\), \(\sigma_3^2=2\), \(\sigma_1\tau_1=\sigma_2\tau_2=\sigma_3\tau_3=1\), and on the annihilator \(G_2\) of this sextuple of classes the form \(q(Y)\) is isomorphic to \(K_8\oplus K_8\).

We reconstruct on the torus \(T_1\) the membrane \(D_3\) by means of the sphere \(S_1\), on the torus \(T_2\) the membrane \(D_3\) by means of the sphere \(S_2\), and on the torus \(T_3\) the membrane \(D_1'\) by means of the sphere \(S_3\), putting \(\nu=1,1,-1\) (the second method of § 3). We obtain membranes with indices

\[ i(D_3)+\sigma_1^2+1=0,\qquad i(D_3)+\sigma_2^2+1=0, \]

\[ i(D_1')+\sigma_3^2-1=0 \]

and boundaries that do not split \(T_1,T_2,T_3\). Consequently, the classes \(\tau_1,\tau_2,\tau_3\) are represented by spheres. These spheres are easily made disjoint by moving apart, before the reconstructions, the tori \(T_1,T_2,T_3\), the spheres \(S_1,S_2,S_3\), and the membranes; and since \(\tau_1^2=\tau_2^2=\tau_3^2=0\), one can perform Morse reconstructions along them, replacing their nonintersecting tubular neighborhoods by products of a three-dimensional ball with a circle. As a result, \(Y\) is transformed into a simply connected manifold \(B\) with form

\[ q(B)\simeq q(Y)|_{G_2}\simeq K_8\oplus K_8 . \]

Remark. The manifold \(B\#3(S^2\times S^2)\) is diffeomorphic to \(Y\).

Leningrad State University
named after A. A. Zhdanov

Received
16 XI 1964

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