MATHEMATICS

S. P. NOVIKOV

SOME PROPERTIES OF MANIFOLDS OF DIMENSION \(4k+2\)

(Presented by Academician L. S. Pontryagin, 4 VII 1963)

The present paper is adjacent to the author’s works \((^8)\) on the diffeomorphism of simply connected manifolds and to Kervaire’s work on the existence of nonsmoothable manifolds of dimension 10 \((^4)\). The central part of the paper, devoted to properties of 10-dimensional manifolds, also essentially uses ideas from works of Milnor and the author \((^5,\,^9)\) concerning generalized rings of internal homologies. Recall (see \((^8)\)) that in studying the homotopy group \(\pi_{N+n}(T_N)\) of the Thom space \(T_N\) of the normal bundle to a manifold \(M^n\) \((n=4k+2)\), we singled out a subset \(A \subset \pi_{N+n}(T_N)\), consisting of those elements \(a \in A\) for which \(H(a)=\varphi[M^n]\), where \(\varphi: H_k(M^n)\to H_{k+N}(T_N)\) is the Thom isomorphism and \(H:\pi_j(x)\to H_j(x)\) is the Hurewicz homomorphism. It was proved that a regular representative \(f_a:S^{N+n}\to T_N\) of an element \(a\in A\) can be chosen so that the manifold \(f_a^{-1}(M^n)=M_a^n\) has the following properties:

1. \(f_{a*}:H_i(M_a^n)\to H_i(M^n)\) is an isomorphism for \(i\ne 2k+1\).

2. \(\operatorname{Ker} f_{a*}=Z+Z\subset H_{2k+1}(M_a^n)\).

3. The Hurewicz homomorphism \(H:[\operatorname{Ker} f_{a*}\to \operatorname{Ker} f_{a*}\subset H_{2k+1}(M_a^n)]\) is an isomorphism.

4. If a cycle \(x\in \operatorname{Ker} f_{a*}\) is realized by an embedded sphere \(S^{2k+1}\subset M_a^n\) and \(n\ne 6,14\), then the normal bundle \(\nu(S^{2k+1},M_a^n)\) of the sphere \(S^{2k+1}\) in the manifold \(M_a^n\) depends only on the element \(x\), belongs to the group \(Z_2\), and determines a mapping \(\varphi:\operatorname{Ker} f_{a*}\to Z_2\) such that
   \[ \varphi(x+y)=\varphi(x)+X(y)+x\cdot y \mod 2. \]

If \(x,y\) is a basis of the group \(\operatorname{Ker} f_{a*}\), then we set \(\varphi(a)=\varphi(x)\varphi(y)\in Z_2\).

Theorem 1. The invariant \(\varphi(a)\) does not depend on the choice of the representative \(f_a\) having properties 1–4.

If \(n=6,14\), then the definition of the invariant must be changed: namely, in item 4 one should speak not of the normal bundle of the sphere \(S^{2k+1}\) in the manifold \(M_a^n\), but of the “framing” of the sphere. The exact definition of the invariant is given in \((^{10})\), and it is denoted by \(\psi(a)\in Z_2\). All algebraic properties of the invariant \(\varphi(a)\) carry over to the invariant \(\psi(a)\) for \(n=6,14\).

Theorem \(1'\). The invariant \(\psi(a)\) is uniquely and correctly defined. The proof of Theorem \(1'\) is identical to the proof of Theorem 1.

By \(\widetilde A\subset A\) we shall denote the subset of the set \(A\subset \pi_{N+n}(T_N)\) consisting of those elements \(a\in \widetilde A\) for which \(\varphi(a)=0\) (for \(n\ne 6,14\)) and \(\psi(a)=0\) (\(n=6,14\)). From the definition of the invariants \(\varphi\) and \(\psi\) and from Theorem 1, Corollaries 1 and 2 easily follow.

Corollary 1. For \(n=6,14\), the set \(\widetilde A\) contains exactly half of the elements of the set \(A\), independently of the manifold \(M^n\).

Corollary 2. If \(M^n=M_1^n\# M_2^n\), then the set \(\widetilde A\) coincides with the set \(A\) for the manifold \(M^n\) if and only if \(\widetilde A\) coincides with \(A\) for each of the manifolds \(M_1^n\) and \(M_2^n\).

We now study the distribution of the invariant \(\varphi\) on the set \(A\). Suppose, for simplicity, that the manifold \(M^n\) is such that \(H^{2k+1}(M^n,Z)\otimes Z_2=0\) and \(n=4k+2,\ k\ne 0,1,3\).

Let us note that \(\pi_{N+n}(\widetilde T_N)=Z+\widetilde\pi\), where the group \(\widetilde\pi\) is finite. The set \(\widetilde A\) is the collection of all elements of the form \(1+\gamma\), where \(1\in Z\) and \(\gamma\in\widetilde\pi\). It is assumed that the decomposition of the group \(\pi_{N+n}(T_N)\) into a direct sum has been chosen and fixed.

Theorem 2. The formula
\[ \varphi(1+\gamma+\delta)=\varphi(1+\gamma)+\varphi(1+\delta)+\varphi(1+0), \qquad \text{where } \gamma,\delta\in\widetilde\pi, \]
holds.

The proof of Theorem 2 is not difficult and follows from the representation
\[ 1+\gamma+\delta=(1+\gamma)+(1+\delta)-(1+0), \]
on the basis of which the element \(1+\gamma+\delta\) can be realized by a “good” representative \(f:S^{N+n}\to T_N\), and one can trace the surgeries of the full inverse image \(f^{-1}(M^n)\).

In our case the direct decomposition \(\pi_{N+n}(T_N)=Z+\widetilde\pi\) can be chosen so that \(\varphi(1+0)=0\); therefore, putting \(\varphi(\gamma)=\varphi(1+\gamma)\), we obtain a homomorphism \(\varphi:\widetilde\pi\to Z_2\). Hence follows

Corollary 3. If \(H^{2k+1}(M^n,Z)\otimes Z_2=0\), then the set \(\widetilde A\) contains either half of the set \(A\), or coincides with the set \(A\).

We have carried out a study of the relation between the sets \(\widetilde A\) and \(A\), sufficiently complete for dimension \(n=4k+2\) with \(k=1,3\), and giving some information about higher dimensions. The first nontrivial case is \(k=2\) (\(n=10\)), on which we shall dwell further. The nontriviality of this case is due to the fact that for the sphere the sets \(\widetilde A\) and \(A\) coincide, and it is quite unclear what the situation will be for other manifolds. Our goal is a generalization of the invariant \(\varphi(M^{10})\in Z_2\), defined by Kervaire \((^4)\) for 4-connected 10-dimensional manifolds, and the application of the Kervaire invariant to the solution of certain problems.

Since the cohomology operation
\[ \operatorname{Sq}^2\operatorname{Sq}^4:H^5(X,Z)\to H^{11}(X,Z_2) \]
is identically zero by virtue of the relation
\[ \operatorname{Sq}^2\operatorname{Sq}^4=\operatorname{Sq}^6+\operatorname{Sq}^5\operatorname{Sq}^1, \]
there is defined a “secondary” cohomology operation
\[ \Phi:\operatorname{Ker}\operatorname{Sq}^4\to\operatorname{Coker}\operatorname{Sq}^2, \qquad \text{where } \operatorname{Ker}\operatorname{Sq}^4\subset H^5(X,Z). \]

Lemma 1. The operation \(\Phi\) has the following property:
\[ \Phi(x+y)=\Phi(x)+\Phi(y)+xy. \]

The proof of Lemma 1 is very simple.

Lemma 2. If \(\pi_1(M^{10})=0\) and \(w_2(M^{10})=0\) for a topological manifold \(M^{10}\), then the operation
\[ \Phi:H^5(M^{10},Z)\to H^{10}(M^{10},Z_2)=Z_2 \]
is always defined and unambiguous.

Lemma 3. Under the conditions of Lemma 2, the operation \(\Phi\) defines a unique homomorphism
\[ \Phi:\operatorname{Tor} H^5(M^{10},Z)\to H^{10}(M^{10},Z_2). \]

Lemma 3 follows easily from Lemmas 1 and 2.

In what follows we shall study only manifolds having the following properties:

1. \(\pi_1(M^{10})=0\).
2. \(w_2(M^{10})=0\).
3. The homomorphism \(\Phi:\operatorname{Tor} H^5(M^{10},Z)\to H^{10}(M^{10},Z_2)\) is trivial.

For topological manifolds possessing properties 1–3, we define the “generalized Kervaire invariant” as follows: a) choose in the group \(H^5(M^{10},Z)/\mathrm{Torsion}\) a basis \(x_1,\ldots,x_{2l}\) such that \(x_{2i-1}x_{2i}\ne0\) for \(1\le i\le l\) and \(x_kx_s=0\) otherwise; b) by virtue of condition 3, the operation \(\Phi\) is uniquely defined on the group \(H^5(M^{10},Z)/\mathrm{Torsion}\) and takes its values in \(Z_2\); the sum
\[ \Phi(M^{10})=\sum_{i=1}^{l}\Phi(x_{2i-1})\Phi(x_{2i}) \]
does not depend on the basis and is called the “generalized Kervaire invariant”; c) the invariant \(\Phi(M^{10})\) is a homotopy invariant of the manifold.

The following important result holds.

Lemma 4. If the manifold \(M^{10}\) is smooth and is the boundary of a smooth oriented manifold \(W^{11}\) such that \(w_2(W^{11})=0\), then \(\Phi(M^{10})=0\).

By analogy with the papers \((^5,^9)\) we shall consider the ring of spinor internal homologies (“cobordisms”)
\[ V_{\mathrm{Spin}}=\sum V^i_{\mathrm{Spin}},\qquad V^i_{\mathrm{Spin}}=\pi_{N+n}(M\mathrm{ Spin}\, N), \]
where \(M\mathrm{ Spin}\, N\) is the Thom complex of the spinor group. As is known, the tangent (or stable normal) bundle of a manifold reduces to the group \(\mathrm{Spin}\) if \(w_1(M^n)=w_2(M^n)=0\). Thus, from Lemma 4 the following follows.

Lemma 5. The Kervaire invariant defines a single-valued homomorphism
\[ \Phi:\widetilde V^{10}_{\mathrm{Spin}}\to Z_2,\qquad \text{where } \widetilde V^{10}_{\mathrm{Spin}}\subset V^{10}_{\mathrm{Spin}}. \]

Proof. The additivity of the invariant \(\Phi\) is obvious. If a manifold defines the zero element of the group \(V^{10}_{\mathrm{Spin}}\), then there is stretched over it a film \(W^{11}\) such that \(w_2(W^{11})=0\), and therefore \(\Phi=0\). However, by virtue of restriction 3, the invariant \(\Phi\), perhaps, is not defined for all elements of the group \(V^{10}_{\mathrm{Spin}}\). The lemma is proved.

We give, without proof, a number of results on the ring
\[ V_{\mathrm{Spin}}=\sum V^i_{\mathrm{Spin}}. \]

I. The groups \(V^i_{\mathrm{Spin}}\) for \(i\leq 10\) are given by the following table:

| \(i\) | \(=\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(V^i_{\mathrm{Spin}}\) | \(=\) | \(Z\) | \(Z_2\) | \(Z_2\) | 0 | \(Z\) | 0 | 0 | 0 | \(Z+Z\) | \(Z_2+Z_2\) | \(Z_2+Z_2+Z_2\) |

II. The generators of the groups \(V^k_{\mathrm{Spin}}\) for \(k\leq 10\) may be chosen as follows:
\[ 1\in V^0_{\mathrm{Spin}},\qquad x_1\in V^1_{\mathrm{Spin}},\qquad x_1^2\in V^2_{\mathrm{Spin}},\qquad x_4\in V^4_{\mathrm{Spin}}, \]
\[ x_8\in V^8_{\mathrm{Spin}},\qquad y_8\in V^8_{\mathrm{Spin}},\qquad 4y_8=x_4^2,\qquad x_1x_8\in V^9_{\mathrm{Spin}}, \]
\[ x_1y_8\in V^9_{\mathrm{Spin}},\qquad x_1^2x_8\in V^{10}_{\mathrm{Spin}},\qquad x_1^2y_8\in V^{10}_{\mathrm{Spin}},\qquad Z_{10}\in V^{10}_{\mathrm{Spin}}. \]

III. The element \(x_1\in V^1_{\mathrm{Spin}}\) is represented by the circle \(S^1\subset R^{N+1}\) with nontrivial framing.

IV. The group \(V^8_{\mathrm{Spin}}\) is generated by the following manifolds: a) the quaternionic projective plane \(P^2(Q)\); b) the 8-dimensional 3-connected almost parallelizable Milnor manifold \(M^8_0\) with index \(I(M^8_0)=8\cdot 28\).

V. The generator \(Z_{10}\in V^{10}_{\mathrm{Spin}}\) is represented by a manifold \(M^{10}\) such that
\[ w_4w_6(M^{10})\ne 0. \]

The subgroup
\[ V^1_{\mathrm{Spin}}V^1_{\mathrm{Spin}}V^8_{\mathrm{Spin}}\subset V^{10}_{\mathrm{Spin}} \]
is singled out by the condition \(w_4w_6=0\). The results given in items I–V may be obtained by analogy with the papers of Milnor \((^5)\) and the author \((^9)\) on rings of generalized internal homologies by means of Adams’ spectral method \((^1)\) and the \(A\)-genus \((^2)\).

Lemma 6. The homomorphism
\[ \Phi:\widetilde V^{10}_{\mathrm{Spin}}\to Z_2 \]
is trivial on the subgroup
\[ V^1_{\mathrm{Spin}}V^1_{\mathrm{Spin}}V^8_{\mathrm{Spin}}\subset V^{10}_{\mathrm{Spin}}. \]

The proof of Lemma 6 is nontrivial and substantially uses item IV on the geometric generators of the group
\[ V^8_{\mathrm{Spin}}=Z+Z. \]
The most difficult part is the analysis of the element represented by the manifold \(P^2(Q)\times S^1\times S^1\). In essence, one must explicitly trace the Morse surgeries in the manifolds
\[ M^8_0\times S^1\times S^1 \quad\text{and}\quad P^2(Q)\times S^1\times S^1 \]
over one-dimensional cycles.

From the lemmas it easily follows:

Theorem 3. The invariant \(\Phi(M^{10})\) is a single-valued function of the product \(w_4w_6(M^{10})\), and \(\Phi(M^{10})=0\) if \(w_4w_6(M^{10})=0\), for a smooth manifold \(M^{10}\).

Thus,
\[ \Phi=\Phi(w_4w_6) \]
for smooth manifolds.

Remark. The author believes that \(\Phi(w_4,w_6)\equiv 0\) for smooth manifolds; for this it is enough to construct such a smooth manifold \(M^{10}\) that
\[ w_4w_6(M^{10})\ne 0 \quad\text{and}\quad \Phi(M^{10})=0. \]

Theorem 4. If, for a smooth manifold \(M^{10}\), the invariant \(\Phi(M^{10})\) is defined, then for it \(\tilde A=A\), i.e. the invariant \(\varphi(\alpha)\equiv 0\) for all \(\alpha\in A\).

Proof. Let \(\alpha\) be such that \(\varphi(\alpha)=1\). Take a representative
\(M^{10}_{\alpha}=f_{\alpha}^{-1}(M^{10})\), possessing properties 1–4 indicated at the beginning of the paper. Obviously,
\(\varphi(M^{10}_{\alpha})=\varphi(M^{10})+1\) and
\(w_4w_6(M^{10}_{\alpha})=w_4w_6(M^{10})\); since \(M^{10}_{\alpha}\) is smooth, we arrive at a contradiction with Theorem 3.

Let \(M^{10}\) be a topological manifold (or, more generally, a polyhedron possessing Poincaré duality). Following the schemes of the papers of the author \((^8)\) and Browder \((^3)\), one can prove the following assertion:

Theorem 5. If, for a polyhedron \(M^{10}\), the invariant \(\Phi\) is defined, then the following two conditions are necessary and sufficient for \(M^{10}\) to have the homotopy type of a smooth manifold: a) \(\Phi(M^{10})=\Phi(w_4w_6)\); b) there exists an \(SO_N\)-bundle \(\nu\) over \(M^{10}\) such that the cycle \(\varphi[M^{10}]\in H_{N+10}(T_N)\) is spherical, where \(T_N\) is the Thom complex of the bundle \(\nu\) (\(\nu\) will be the normal bundle of the smooth manifold sought).

We note that Kervaire in \((^4)\) constructed a 4-connected manifold satisfying condition b) and not satisfying condition a). Thus, both conditions are essential. Moreover, for \(5\le n\le 17\) dimension \(n=10\) alone presented difficulty (the cases \(n=6,14\) are simple). For \(n=4k+2\), where \(k\ge 4\), new difficulties arise. The author believes that for even \(k\) a generalization of the invariant \(\Phi\) is possible on the basis of the relation
\[ Sq^2Sq^{2k}=Sq^{2k+2}+Sq^{2k+1}Sq^1 \]
of the Steenrod algebra and the study of the ring \(V_{\mathrm{spin}}\).

We indicate some results on spin cobordisms.

Lemma 7. If \(\pi_1(M^n)=0\) and \(w_2(M^n)=0\), then the stable normal (tangent) \(SO_N\)-bundle reduces to the group \(\mathrm{Spin}\,N\), and uniquely. For \(n\ge 3\) every element of the group \(V^n_{\mathrm{spin}}\) is represented by a simply connected manifold.

Consider the natural homomorphism “forgetting the framing”
\[ \rho:G_i\to V^i_{\mathrm{spin}}, \]
where \(G_i=\pi_{N+i}(S^N)\). The following very substantial fact holds.

Lemma 8. For \(3\le i\le 8\) the image of the homomorphism \(\rho:G_i\to V^i_{\mathrm{spin}}\) is trivial. For \(i=9,10\) the image of the homomorphism \(\rho\) is isomorphic to \(Z_2\).

For the proof one must consider the Milnor manifold \(M^8_0\) indicated above and perform surgeries in the manifold \(M^8_0\times S^1\) for \(i=9\) and \(M^8_0\times S^1\times S^1\) for \(i=10\), dragging along the nontrivial “spin framings.” As a result, the corresponding elements of the groups \(V^9_{\mathrm{spin}}\), \(V^{10}_{\mathrm{spin}}\) are realized by homotopy spheres.

Theorem 6. There exist smooth manifolds of the homotopy type of spheres of dimensions 9 and 10 which are not boundaries of any smooth manifolds with trivial Stiefel class \(w_2=0\).

Corollary. For \(n=9,10\), membership of a smooth simply connected manifold in the class of spinor inner homologies is not a combinatorial invariant (in contrast to the usual inner homologies, corresponding to the groups \(O\) and \(SO\)) and is not determined by the homotopy type and the tangent bundle.

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
13 VI 1963

REFERENCES

1. J. F. Adams, Comm. Math. Helv., 32, No. 3, 180 (1958).
2. A. Borel, F. Hirzebruch, Am. Math. J., 80, 459 (1958).
3. W. Browder, Colloquium on Algebraic Topology, Aarchus Univ., 1962.
4. M. Kervaire, Comm. Math. Helv., 34, No. 4, 257 (1960).
5. J. Milnor, Am. Math. J., 82, No. 3, 505 (1960).
6. J. Milnor, M. Kervaire, Proc. Intern. Congress Math. 1958, 1960, p. 454.
7. J. Milnor, Proc. of Symp. in Pure Mathematics, 1961, p. 39.
8. S. P. Novikov, DAN, 143 No. 5, 1046 (1962).
9. S. P. Novikov, Matem. sborn., 57, 407 (1962).