MATHEMATICS

V. A. ROKHLIN

INTERNAL HOMOLOGIES

(Presented by Academician P. S. Aleksandrov on 23 XI 1957)

1. Statement of the main theorem. In my note (¹) the natural homomorphism \(h^k\) of the group \(\mathfrak{S}^k\) of internal homologies into the group \(\mathfrak{N}^k\) of internal homologies mod 2 was studied; namely, the kernel of the homomorphism \(h^k\) (it is equal to \(2\mathfrak{S}^k\)) and the image \(h^k(\mathfrak{S}^k)\) were found. An exhaustive characterization of this image was, however, impossible until the groups \(\mathfrak{N}^k\) themselves had been sufficiently studied. At present their structure is known. Let \([M^k]\) denote the element of the group \(\mathfrak{N}^k\) defined by the manifold \(M^k\), and let \(\mathfrak{N}\) denote the algebra that is the direct sum of all groups \(\mathfrak{N}^k\), with multiplication defined by the usual multiplication of manifolds. As Thom showed (²), for each natural number \(r\) not of the form \(2^s-1\), there exists a manifold \(P(r)\) of dimension \(r\) such that the elements \([P(r)]\) form an independent mod 2 system of generators of the algebra \(\mathfrak{N}\); in other words, all possible products
\[ [P(r_1)] \times [P(r_2)] \times \cdots \times [P(r_l)] \]
with \(r_1+r_2+\cdots+r_l=k\) form an independent mod 2 system of generators of the group \(\mathfrak{N}^k\). Concrete manifolds \(P(r)\) for even \(r\) and for \(r=5\) were indicated by Thom, and for the remaining \(r\) by Dold (³). It is precisely these manifolds that we shall denote by \(P(r)\). Moreover, for \(P(r)\) with odd \(r\) we need know only that they are orientable. \(P(r)\) with even \(r\) is the real projective space \(PR(r)\). According to the Pontryagin–Thom theorem, \([M_1^k]=[M_2^k]\) if and only if \(M_1^k\) and \(M_2^k\) have the same characteristic numbers; a characteristic number of a manifold \(M^k\) is a scalar product of the form \((\rho,m^k)\), where \(\rho\) is a product of weight \(k\) of Stiefel–Whitney classes
\[ \left(\rho=\prod_{\alpha=1}^{l} w_{r_\alpha},\quad r_1+\cdots+r_l=k\right), \]
and \(m^k\) is the fundamental \(\Delta\)-class mod 2 of the manifold \(M^k\). Relying on these facts, I give in this note an effective description of \(h^k(\mathfrak{S}^k)\).

In § 2 it will be shown that
\[ PR(n)\times PR(n)\sim PC(n)\pmod 2\;(\mathrm{in}) \tag{1} \]
(\(PC(n)\) is complex projective space; formula (1) with \(n=2\) is found in (¹,²)). Consequently, the generators
\[ [P(r_1)]\times[P(r_2)]\times\cdots\times[P(r_l)], \]
in which each factor of even dimension occurs an even number of times, belong to \(h^k(\mathfrak{S}^k)\). They are called generators of the first kind; the remaining generators are generators of the second kind; and the generators for which all \(r_\alpha\) are even and pairwise distinct are special. Every generator of the second kind is a product of a generator of the first kind by a special generator. A characteristic number \((\rho,m^k)\) in which \(\rho\) contains the class \(w_1\) as a factor is called a \(w_1\)-characteristic number. The set of manifolds \(M^k\) for which \([M^k]\in <!-- source-page: 002 --> \(\in h^k(\mathfrak{S}^k)\), is denoted by \(K_1\); the class of manifolds \(M^k\) for which all \(w_1\)-numbers are equal to zero, by \(K_2\); the class of manifolds \(M^k\) for which \([M^k]\) belongs to the subgroup of the group \(\mathfrak{N}^k\) generated by generators of the first kind, by \(K_3\); the class of manifolds \(M^k\) for which \(A^{k-1} \sim 0 \pmod 2\) (intr.), \(B^{k-2} \sim 0 \pmod 2\) (intr.), by \(K_4\) (for the definition of \(A^{k-1}\) and \(B^{k-2}\) see in (1)); the class of manifolds \(M^k\) for which \(A^{k-1} \sim 0\) (intr.), \(B^{k-2} \sim 0 \pmod 2\) (intr.), by \(K_5\).

Main theorem. \(K_1 = K_2 = K_3 = K_4 = K_5\).

It is obvious that \(K_1 \subset K_2\), \(K_5 \subset K_4\). From formula (1) it follows that \(K_3 \subset K_1\). Therefore it is enough to prove formula (1) and the relations \(K_2 = K_4\), \(K_2 \subset K_3\), \(K_1 \subset K_5\). This will be done in §§ 2—5. Some consequences of the main theorem are given in § 6.

2. Proof of formula (1). Let \(a\) and \(b\) be generators of the one-dimensional groups of \(V\)-homology mod 2 of the manifolds \(P R(n)\) standing on the left, and let \(c\) be a generator of the two-dimensional group of \(V\)-homology mod 2 of the manifold \(P C(n)\). For the left-hand side

\[ w_r=\sum_{i=0}^{r} \binom{n+1}{i}\binom{n+1}{r-i} a^{r-i}\otimes b^i . \tag{2} \]

If \(w_r\) enters into the product \(\prod_{\alpha=1}^{l} w_{r_\alpha}\) of weight \(r_1+\cdots+r_l=2n\), then the terms of the sum (2) equidistant from the beginning and the end give, upon multiplication by the remaining classes \(w_{r_\alpha}\), identical results. Consequently, when computing the corresponding number they may be discarded, and the class \(w_r\) may be replaced by zero if \(r\) is odd, and by the middle term

\[ \binom{n+1}{s}\binom{n+1}{s} a^s\otimes b^s = \binom{n+1}{s}a^s\otimes b^s, \]

if \(r=2s\). For the right-hand side, \(w_r=0\) if \(r\) is odd, and \(w_{2s}=\binom{n+1}{s}c^s\). Consequently, the left- and right-hand sides have the same numbers, which proves formula (1).

3. Proof of the relation \(K_2 = K_4\). Let \(m^k, a^{k-1}, b^{k-2}\) be the fundamental \(\Delta\)-classes mod 2 of the manifolds \(M^k, A^{k-1}, B^{k-2}\), and let \(w_r, u_r, v_r\) be the Stiefel—Whitney classes of these manifolds; let \(\bar u_r, \bar v_r\) be the normal classes determined by the embeddings \(i: A^{k-1}\to M^k\), \(j: B^{k-2}\to M^k\). The complete skew products with bases \(A^{k-1}, B^{k-2}\) determined by these embeddings are sums (in the sense of Whitney) of tangent and normal skew products, and their characteristic classes are \(i^*w_r\) and \(j^*w_r\). Hence,

\[ i^*w_r=u_r+u_{r-1}\bar u_1,\quad j^*w_r=v_r+v_{r-1}\bar v_1+v_{r-2}\bar v_2 \quad (r=1,2,\ldots;\ v_{-1}=0). \]

From the orientability of \(A^{k-1}\) and from the fact that the classes \(i_*a^{n-1}\), \(j_*b^{n-2}\) are dual in \(M^k\) to the classes \(w_1, w_1^2\), it follows that \(u_1=0\), \(\bar v_1=0\), \(\bar v_2=v_1^2\). Thus, finally:

\[ i^*w_r=u_r+u_{r-1}\bar u_1\quad (u_1=0);\qquad j^*w_r=v_r+v_{r-2}v_1^2 . \tag{3} \]

These equalities can be solved inductively with respect to \(u_r\) and \(v_r\):

\[ u_r=i^*\varphi_r(w_1,\ldots,w_r);\qquad v_r=j^*\psi_r(w_1,\ldots,w_r), \tag{4} \]

where \(\varphi_r,\psi_r\) are homogeneous polynomials of weight \(r\). From (4) it follows that for every product \(\xi=\prod_{\alpha=1}^{l} u_{r_\alpha}\) of weight \(k-1\) there exists such a homogeneous polynomial \(\varphi(w_1,\ldots,w_{k-1})\) of weight \(k-1\), and for every product \(\eta=\prod_{\beta=1}^{m} v_{s_\beta}\) of weight \(k-2\) such a homogeneous polynomial \(\psi(w_1,\ldots,w_{k-2})\) of weight

\(k-2\), that \(\xi=i^*\varphi\), \(\eta=j^*\psi\). If \(M^k \in \mathrm K_2\), then

\[ (\xi,a^{k-1})=(i^*\varphi,a^{k-1})=(\varphi,i_*a^{k-1})=(w_1\varphi,m^k)=0, \]

\[ (\eta,b^{k-2})=(j^*\psi,b^{k-2})=(\psi,j_*b^{k-2})=(w_1^2\psi,m^k)=0. \]

This means that all the residues of \(A^{k-1}\) and \(B^{k-2}\) are zero, i.e. that \([A^{k-1}]=0\) and \([B^{k-2}]=0\), i.e. that \(M^k \in \mathrm K_4\). Thus, \(\mathrm K_2 \subset \mathrm K_4\).

Now let \(M^k \in \mathrm K_4\). From (3) it follows that \(\varphi_r\) in formula (4) can be represented in the form \(w_r+w_1\omega_{r-1}\), where \(\omega_{r-1}\) is a polynomial in \(w_1,\ldots,w_{r-1}\). Consequently, \(\varphi\) in the formula \(\xi=i^*\varphi\) can be represented in the form \(\rho+w_1\chi\), where

\[ \rho=\prod_{\alpha=1}^l w_{r_\alpha}, \]

and \(\chi\) is a polynomial in \(w_1,\ldots,w_{k-2}\). Formula (3) allows \(j\chi\) to be represented as some polynomial \(\zeta(v_1,\ldots,v_{k-2})\). Since \(\rho=\varphi+w_1\chi\), we have

\[ (w_1\rho,m^k)=(w_1\varphi,m^k)+(w_1^2\chi,m^k)=(\xi,a^{k-1})+(\eta,b^{k-2})=0. \]

This proves that all the \(w_1\)-residues of \(M^k\) are zero, i.e. that \(M^k \in \mathrm K_2\). Thus, \(\mathrm K_4 \subset \mathrm K_2\).

1. Proof of the relation \(\mathrm K_2 \subset \mathrm K_3\). It is enough to prove that if \(M^k \in \mathrm K_2\) and

\[ [M^k]=\sum_{i=1}^p [N_i^k], \]

where the \([N_i^k]\) are pairwise distinct generators of the second kind, then \([M^k]=0\), i.e. \(p=0\). Suppose first that all \([N_i^k]\) are special generators, and let \([N_1^k]\) be the largest among them: if
\[ N_1^k=PR(r_1)\times PR(r_2)\times\cdots\times PR(r_l),\quad N_i^k=PR(s_1)\times PR(s_2)\times\cdots\times PR(s_m), \]
with \(i>1\) and \(r_1>r_2>\cdots>r_l,\ s_1>s_2>\cdots>s_m\), \(r_\alpha=s_\alpha\) for \(\alpha<\beta\), \(r_\beta\ne s_\beta\), then \(r_\beta>s_\beta\). Define a homomorphism of the ring of symmetric polynomials in \(t_1,\ldots,t_k\) over the prime field of characteristic 2 into the ring of \(\nabla\)-homologies mod 2 of a \(k\)-dimensional manifold, which assigns to the \(r\)-th elementary symmetric function the class \(w_r\); denote by \(\sigma(r)\) the class corresponding to the polynomial \(t_1^r+\cdots+t_k^r\), and put
\[ \tau=\sigma(1)\sigma(r_1-1)\times\sigma(r_2)\cdots\sigma(r_l). \]
This is a homogeneous polynomial of weight \(k\) in \(w_1,\ldots,w_k\), and, since \(\sigma(1)=w_1\), the corresponding residue is a \(w_1\)-residue. Consequently, for \(M^k\) it is zero. As the computation shows, for \(N_1^k\) and \(N_i^k\) with \(i>1\) it is equal, respectively, to 1 and 0, and the relation
\[ [M^k]=\sum [N_i^k] \]
gives \(0=1\).

Now let the \([N_i^k]\) be arbitrary generators of the second kind. Each of them has the form \([O]\times[N]\), where \(O\) is a generator of the first kind and \(N\) is a special generator. Collecting terms with the same \(O\), we obtain
\[ [M^k]=\sum [O_j]\times[M_j], \]
where the \([O_j]\) are pairwise distinct, and the \([M_j]\) are sums of special generators. Construct for \(M_j\) the submanifolds \(A_j\) and \(B_j\). Analogous submanifolds for \(\sum O_j\times M_j\) may be taken to be \(\sum O_j\times A_j\) and \(\sum O_j\times B_j\), and, since \(\sum O_j\times M_j\), together with \(M^k\), belongs to \(\mathrm K_2=\mathrm K_4\), it follows that
\[ \left[\sum O_j\times A_j\right]=0,\qquad \left[\sum O_j\times B_j\right]=0. \]
But this is possible only under the condition that \([A_j]=0,\ [B_j]=0\), i.e. only under the condition that \(M_j\in \mathrm K_4=\mathrm K_2\). According to what was proved in the preceding paragraph, it follows that \([M_j]=0\); thus \([M^k]=0\).

1. Proof of the relation \(\mathrm K_1 \subset \mathrm K_5\). Let \(M^k \subset \mathrm K_1\), and let \(L^{k+1}\) be a manifold with boundary \(M^k+M_1^k\), where \(M_1^k\) is an orientable manifold. Construct in \(L^{k+1}\) an orientable submanifold \(A^k\), serving as эҭ

which is a \(k\)-dimensional (integer) Stiefel \(\Delta\)-cycle. Next, in \(A^k\) we construct a submanifold \(B^{k-1}\) serving as a \(\Delta\)-cycle \((\bmod 2)\) of the singularities of the external vector field on \(A^k\) in \(L^{k+1}\). Owing to orientability, the submanifolds \(A^k\) and \(B^{k-1}\) of \(M_1^k\) can be chosen so that the oriented boundary for \(A^k\) is \(A^{k-1}\), and the boundary for \(B^{k-1}\) is \(B^{k-2}\). Therefore \(A^{k-1} \sim 0\) (int.) and \(B^{k-2} \sim 0 \bmod 2\) (int.), i.e. \(M^k \in K_5\).

6. Consequences of the main theorem.

A. In the algebra \(\mathfrak{N}\), the product of a nonorientable element by a nonzero orientable element is a nonorientable element.

This follows, for example, from the equality \(K_1 = K_3\).

B. If \(M^k \sim 0 \bmod 2\) (int.) and \(2M^k \sim 0\) (int.), then \(M^k \sim 0\) (int.).

Proof. Let \(L^{k+1}\) be an oriented manifold with boundary \(2M^k\), and let \(M^{k+1}\) be a nonorientable closed manifold into which the manifold \(L^{k+1}\) is transformed under the natural identification of two copies of the manifold \(M^k\) constituting its boundary. \(M^k\) serves as a \(k\)-dimensional Stiefel \(\Delta\)-cycle of the manifold \(M^{k+1}\) and possesses in \(M^{k+1}\) an external vector field without singularities. Consequently, \(M^{k+1} \in K_4\), and, since \(K_4 = K_5\), it follows that \(M^k \sim 0\) (int.).

C. In \(\mathfrak{N}^k\) there are no elements of finite order divisible by 4.

Proof. Suppose \(4mM^k \sim 0\) (int.). Put \(M_1^k = 2mM^k\). Then \(M_1^k \sim 0 \bmod 2\) (int.), \(2M_1^k \sim 0\) (int.). Consequently, \(M_1^k \sim 0\) (int.), and \(4m\) is not the order of the manifold \(M^k\).

Kolomna Pedagogical Institute

Received
20 XI 1957

References

\({}^{1}\) V. A. Rokhlin, DAN, 89, 789 (1953). \({}^{2}\) R. Thom, Comm. Math. Helv., 28, 17 (1954). \({}^{3}\) A. Dold, Math. Zs., 65, 25 (1956).