MATHEMATICS

V. A. ROKHLIN and A. S. SHVARTS

ON THE COMBINATORIAL INVARIANCE OF PONTRYAGIN CLASSES

(Presented by Academician P. S. Aleksandrov on 2 II 1957)

1. Formulation of the main results. In this paper we prove the combinatorial invariance of the reduced Pontryagin classes (p_{4k}), i.e., of the characteristic Pontryagin classes considered as classes of weak (\nabla)-homology. The paper is closely connected with the work ((^{1})), to which we shall refer repeatedly. For the definition of a (C^1)-triangulation see ((^{2})). If (\varphi) is a continuous mapping, then (\varphi^{}) and (\varphi_{}) denote the corresponding homomorphisms of the groups of (\nabla)- and (\Delta)-homology.

Main theorem. Let (M_0^n, M_1^n) be smooth closed manifolds possessing isomorphic (C^1)-triangulations (K_0, K_1), and let (\varphi : M_1^n \to M_0^n) be a homeomorphic mapping determined by some isomorphism between (K_0) and (K_1). Then
[
\varphi^{*}\bigl(p_{4k}(M_1^n)\bigr)=p_{4k}(M_0^n)\quad (k=1,2,\ldots).
]

As a consequence we obtain:

If smooth closed orientable manifolds (M_0^{4l}, M_1^{4l}) possess isomorphic (C^1)-triangulations, then, with a suitable orientation, they have the same Pontryagin numbers.

As is known, two simplicial complexes are called combinatorially equivalent if they have isomorphic subdivisions. We shall call two triangulations of one and the same polyhedron strictly equivalent if there exist isomorphic subdivisions of them and such an isomorphism between these subdivisions that the corresponding homeomorphism of the polyhedron is homotopic to the identity. For example, as follows from Whitehead’s results ((^{2})), any two (C^1)-triangulations of a smooth closed manifold are strictly equivalent. According to the so-called “main hypothesis of combinatorial topology,” any two triangulations of one and the same polyhedron are combinatorially equivalent. This hypothesis becomes still stronger if combinatorial equivalence is replaced by strict equivalence. From our main theorem it immediately follows:

If the main hypothesis of combinatorial topology is true, then the Pontryagin numbers are topological invariants. If the strengthened main hypothesis is true, then the reduced Pontryagin classes are topological invariants.

Since the combinatorial invariance of the classes (p_{4k}) has been established, the question arises of defining them for combinatorial manifolds. We give such a definition for the classes (\sigma_{4k}) ((^{1})). The classes (p_{4k}) can be defined in terms of the classes (\sigma_{4k}).

The further results are obtained by comparing what was set out above with the recent work of Milnor ((^{3})). They concern the problem of introducing smoothness into a combinatorial manifold and are presented in § 5. We note that in Milnor’s work a hypothesis is refuted which in the theory of smooth manifolds served as an analogue of the main hypothesis of combinatorial topology: if two smooth manifolds are homeomorphic, then they are smoothly homeomorphic. It is not hard to show that homeomorphic but not smoothly homeomorphic mani-

Milnor manifolds (M_k^7) have isomorphic (C^1)-triangulations. Thus, under the conditions of the main theorem the manifolds (M_0^1) and (M_1^n) need not be smoothly homeomorphic.

2. Lemmas. In what follows, by (\Sigma^r) we mean a sphere triangulated by projecting onto it from its center the boundary of a regular ((r+1))-dimensional simplex inscribed in it, and by (c) the center of one of its (r)-dimensional simplices.

Lemma 1. Let (M^n) be a smooth closed manifold and (f:M^n\to\Sigma^r) a smooth map. If (c) is a regular point with respect to (f) (i.e., at all points of the set (f^{-1}(c)) the functional matrix of the map (f) has rank (r)), then there exists a (C^1)-triangulation of the manifold (M^n) and a map (\bar f:M^n\to\Sigma^r), simplicial with respect to it and homotopic to (f), such that (\bar f^{-1}(c)=f^{-1}(c)).

Proof. Introduce a Riemannian metric in (M^n). For sufficiently small (\varepsilon), the differential properties of the map (f) determine a natural decomposition of the (\varepsilon)-neighborhood of the submanifold (N^{n-r}=f^{-1}(c)) into the direct product (N^{n-r}\times V^r) of this submanifold by an (r)-dimensional ball, where the fiber (N^{n-r}\times b), corresponding to the center (b) of the ball, coincides with (N^{n-r}), and the fibers (a\times V^r) ((b\in N^{n-r})) are geodesic balls of radius (\varepsilon), normal to (N^{n-r}) ((^4)). Let (T^r) be a simplex, interior to (V^r), with center at the point (b) and with vertices (b_0,\ldots,b_r), and let (K) be some (C^1)-triangulation of the product (N^{n-r}\times T^r\subset N^{n-r}\times V^r), having the property that all its vertices lie in the layers (N^{n-r}\times b_j). Such a triangulation exists: it suffices to take any (C^1)-triangulation (K') of the manifold (N^{n-r}), enumerate its vertices in the sequence (a_0,\ldots,a_q), take as vertices in (K) the points (a_i\times b_j), and declare ((a_{i_0}\times b_{j_0}, a_{i_1}\times b_{j_1},\ldots,a_{i_\alpha}\times b_{j_\alpha})), where (i_0\leq i_1\leq\cdots\leq i_\alpha), to be a simplex in (K), if ((a_{i_0},a_{i_1},\ldots,a_{i_\alpha})) is a simplex in (K') and (j_0\leq j_1\leq\cdots\leq j_\alpha). We extend the triangulation (K) to all of (M^n); the existence of such an extension, as well as the existence of the original triangulation (K'), follows from the results of Whitehead ((^2)).

Let (c_0,\ldots,c_r,c_{r+1}) be the vertices in (\Sigma^r), with the vertex opposite (c) being (c_{r+1}). Construct the simplicial map (\bar f:M^n\to\Sigma^r) by setting (\bar f(a_i\times b_j)=c_j) ((i=0,\ldots,q;\ j=0,\ldots,r)) and (\bar f(x)=c_{r+1}) for every other vertex (x\in M^n). It is not difficult to verify that (\bar f^{-1}(c)=N^{n-r}) and that (\bar f) is homotopic to (f).

Lemma 2. Let (M) be a polyhedron; (L_0) and (L_1) its triangulations; (g_0) and (g_1) its maps into a triangulated polyhedron (\Sigma), simplicial with respect to (L_0) and (L_1) and homotopic to one another; and let (M\times I) be its product with the interval ([0,1]). If (L_0) and (L_1) have a common subdivision, then there exist a triangulation (L) of the prism (M\times I) and a map (g:M\times I\to\Sigma), simplicial with respect to (L), such that on (M\times 0) they coincide with (L_0\times0) and (g_0\times0), and on (M\times1) with (L_1\times1) and (g_1\times1).

The proof is elementary.

3. Proof of the main theorem. We may assume (M_0^n) and (M_1^n) to be orientable, since the nonorientable case reduces to the orientable one by passing to the orientable two-sheeted covering (cf. ((^1))). Next, it is possible to restrict ourselves to the case of odd (n), to which the case of even (n) reduces by multiplying the manifolds (M_0^n) and (M_1^n) by a circle. Finally, instead of the system of relations (\varphi^(p_{4k}(M_1^n))=p_{4k}(M_0^n)) ((k=1,2,\ldots)), it suffices to prove the equivalent system of relations (\varphi^(s_{4k}(M_1^n))=s_{4k}(M_0^n)) ((k=1,2,\ldots)) ((^1)).

Orient (\Sigma^r) and (M_0^n), and denote by (H_{4k}=H_{4k}(M_0^n)) the integral (4k)-dimensional (\Delta)-homology group of the manifold (M_0^n). By the theorem of Serre ((^5))—Thom ((^4)), there exists a natural number (m) (depending on (n) and (k)) such that every class of the form (m u'), (u'\in H_{4k}), contains an oriented submanifold (N_0^{4k}) of the smooth manifold (M_0^n), serving as the oriented inverse image of a point (c\in\Sigma^r), (r=n-4k), under some smooth map (f_0:M_0^n\to\Sigma^r), for which (c) is a regular point: (N_0^{4k}=f_0^{-1}(c)). Transfer the orientation of (M_0^n) by means of (\varphi) to (M_1^n), construct for the map (f_0\varphi^{-1}:M_1^n\to\Sigma^r) a smooth map (f_1:M_1^n\to\Sigma^r) homotopic to it, for which (c) is also a regular point, and put (N_1^{4k}=f^{-1}(c)). The submanifolds (N_0^{4k}) and (N_1^{4k}) are normalizable (i.e., possess a system of (n-4k) independent exterior vector fields) and belong to the classes (u=m u') and (\varphi_u), respectively, whence ((s_{4k}(M_0^n),u)=\alpha_k\sigma(N_0^{4k})) and ((s_{4k}(M_1^n),\varphi_u)=\alpha_k\sigma(N_1^{4k})) ((^1)). Therefore, to prove the relations (\varphi^*(s_{4k}(M_1^n))=s_{4k}(M_0^n)) it is enough to show that (\sigma(N_1^{4k})=\sigma(N_0^{4k})).

Apply Lemma 1 to (f_i:M_i^n\to\Sigma^r) ((i=0,1)): let (\overline K_i) be the triangulation and (\overline f_i) the map whose existence it asserts. On (M_i^n) we have two (C^1)-triangulations (K_i) and (\overline K_i), and, according to Whitehead’s results (see § 1), there exist isomorphic subdivisions (K_i'), (\overline K_i') of the complexes (K_i), (\overline K_i) and an isomorphism (K_i'\to\overline K_i') between these subdivisions such that the corresponding homeomorphism (\varphi_i:M_i^n\to M_i^n) is homotopic to the identity. Consider the homeomorphism (\psi:M_0^n\to M_1^n) defined by the formula (\psi=\varphi_1\varphi\varphi_0^{-1}), and, bearing in mind the notation of Lemma 2, set (M=M_1^n), (L_0=\psi(\overline K_0)), (L_1=\overline K_1), (\Sigma=\Sigma^r), (g_0=\overline f_0\psi^{-1}), (g_1=\overline f_1). It is not difficult to verify that (g_0) and (g_1) are homotopic to one another. Further, (\varphi(K_0')) and (K_1'), as subdivisions of one and the same complex (K_1), have a common subdivision (K), and it is clear that (\varphi_1(K)) is a common subdivision of the complexes (L_0) and (L_1). Thus Lemma 2 is applicable. Orient the prism (M\times I) so that (\Delta(M\times I)=M\times 1-M\times 0), and set (in the notation of Lemma 2) (N=g^{-1}(c)). This is an oriented ((4k+1))-dimensional (h)-manifold with boundary (its orientation is determined by the orientations of (\Sigma^r) and (M\times I) and by the map (g)), and (\Delta N=N_1-N_0), where (N_0=\psi(N_0^{4k})\times0), (N_1=N_1^{4k}\times1). Consequently, (\sigma(N_1)=\sigma(N_0)) ((^{6,7})) and (\sigma(N_1^{4k})=\sigma(N_0^{4k})).

Remark. The main theorem is also true for open manifolds and manifolds with boundary. The proof becomes somewhat more complicated.

4. Definition of the classes (\sigma_{4k}) for combinatorial manifolds. This definition is suitable for any combinatorial (h)-manifold, i.e., a complex (K^n) whose body (|K^n|) is an (h)-manifold (the presence of a boundary is also allowed). Suppose first that (n) is odd and (|K^n|) is oriented and closed, and orient (|K^n|) and (\Sigma^r), (r=n-4k). From the theorem of Serre ((^5)) it is not difficult to derive that there exists a natural number (m) (depending only on (n) and (k)) such that in every class (m u), (u\in H_{4k}(K^n)), there is an oriented (h)-manifold of the form (f^{-1}(c)), where (f) is a map of the polyhedron (|K^n|) into (\Sigma^r), simplicial with respect to some subdivision (K_1) of the complex (K^n). It turns out that the number (\tau/m), where (\tau) is the signature of this (h)-manifold, is uniquely determined by the class (u) (i.e., does not depend on the choice of (m), (K_1), and (f)) and is a linear function of (u); thus the formula ((\sigma_{4k},u)=\tau/m) defines a certain class (\sigma_{4k}=\sigma_{4k}(K^n)) of rational (\nabla)-homology of the polyhedron (|K^n|). If (n) is even, then (\sigma_{4k}(K^n)) is defined as (i^*(\sigma_{4k}(K^n\times S^1))), where (S^1) is a triangulation of the circle; (i) is the natural embedding (K^n\to K^n\times S^1). If

If (|K^n|) is nonorientable, then (\sigma_{4k}(K^n)) is defined by passing to the orientable double covering. If (K^n) has a boundary, then (\sigma_{4k}(K^n)) is defined by doubling (K^n).

This definition is combinatorially invariant. (\sigma_{4k}) is a rational (V)-homology class that becomes integral after multiplication by a number depending only on (n) and (k). If (K^n) is a (C^1)-triangulation of a smooth manifold, then such a number is (\alpha_k), namely (\alpha_k\sigma_{4k}=s_{4k}). In the general case this formula may be taken as the definition of the classes (s_{4k}), and hence of the classes (p_{4k}).

The proof of the assertions made essentially repeats the arguments of § 3. However, it also relies on Lemma 3, in which, as above, (|K^n|) is closed, (K^n) and (\Sigma^r) are oriented, (n) is odd, and (r=n-4k).

Lemma 3. Let (f_0:|K^n|\to\Sigma^r) and (f_1:|K^n|\to\Sigma^r) be maps that are simplicial with respect to certain subdivisions of the complex (K^n). If the oriented (h)-manifolds (f_0^{-1}(c)) and (f_1^{-1}(c)) belong to the same class of integral (\Delta)-homologies, then there exists a map (g:\Sigma^r\to\Sigma^r) such that (gf_0) and (gf_1) are homotopic.

The proof is analogous to the proof of Serre’s theorem ((^5)).

1. Combinatorial manifolds and smoothness. We shall call a formal manifold ((^2)) admitting smoothness if it is combinatorially equivalent to a (C^1)-triangulation of some smooth manifold. We shall call a closed oriented combinatorial (h)-manifold (K^n) internally homologous to a smooth manifold if there exists a smooth oriented manifold with a (C^1)-triangulation (K_1^n) such that the difference (K_1^n-K_0^n) is isomorphic to the boundary of some oriented combinatorial (h)-manifold (K^{n+1}). From Cairns’s results ((^8)) it follows that every formal manifold (K^n) with (n\le 4) admits smoothness.

Consider the manifolds (B_k^8) and (X_k^8), constructed by Milnor ((^3)). Let (L_k^8) be a triangulation of the second, obtained from some (C^1)-triangulation of the first. (L_k^8) is a formal manifold. We shall show that:

For (k^2\not\equiv 1\pmod 7), the closed oriented formal manifold (L_k^8) admits no smoothness and, moreover, is not internally homologous to a smooth manifold.

Proof. According to § 4, for (L_k^8) the class (\frac17(9\sigma_4^2+45\sigma_8)) is defined, and from Milnor’s results it is not difficult to derive that this class cannot be integral. Meanwhile, for a smooth manifold it, being equal to (p_8), is integral; and for a combinatorial (h)-manifold internally homologous to a smooth one, it therefore must also be integral.

As a consequence we obtain:

If the fundamental hypothesis of combinatorial topology is true, then no smoothness can be introduced on the topological manifold (X_k^8).

Ivanovo State Pedagogical Institute

Received
30 I 1957

CITED LITERATURE

(^1) V. A. Rokhlin, DAN, 113, No. 2 (1957).
(^2) J. H. C. Whitehead, Ann. Math., 41, 809 (1940).
(^3) J. Milnor, Ann. Math., 64, 399 (1956).
(^4) R. Thom, Comm. Math. Helv., 28, 17 (1954).
(^5) J. P. Serre, Ann. Math., 58, 258 (1953).
(^6) V. A. Rokhlin, DAN, 84, 221 (1952).
(^7) R. Thom, Ann. Ecole Norm. Sup., (3), 69, 109 (1952).
(^8) S. S. Cairns, Ann. Math., 45, 218 (1944).