Vol. 2 (1991), N0. 2

On the work of Vladimir Abramovich Rokhlin in Topology

by O. Ya. Viro and V. M. Kharlamov

As far back as his undergraduate stuidy at Moscow University, Vladimir Abramovich Rokhlin was interested in topology: he participated in the seminars of P. S. Aleksandrov and L. S. Pontryagin, published two papers in set-theoretic topology, and wrote a survey of the then new theory of homotopy groups (this survey was published after the war in 1946 in Uspekhi Matematicheskikh Nauk and for a long time served as one of the best introductions to the subject). Nevertheless, his basic area of interests in those years was measure theory and ergodic theory; in any case, that is what he chose when he became a graduate student.

Vladimir Abramovich turned seriously to topology at the beginning of the 1950's. Immediately before this he had helped Pontryagin prepare for publication a thorough exposition of results of the latter announced earlier on characteristic classes and homotopy classification of continuous mappings. The first topological work of Rokhlin in this period continued the work of Pontryagin.

Pontryagin found a way to reduce the problem of computing the homotopy groups of the spheres to geometric problems in the topology of smooth manifolds (Pontryagin's method), and computed the groups $\pi_{n+k}(S^n)$ with $k = l$ and $k = 2$. Starting out from Pontryagin's method, Rokhlin calculated the groups $\pi_{n+3}(S^n)$ (for $n\ge 5$ they are cyclic of order 24; recall that $\pi_{n+1}(S^n) = \mathbb Z_2$ for $n\ge3$ and $\pi_{n+2}(S^n) = \mathbb Z_2$ for $n\ge 2$). Although the computation of the group $\pi_{n+3}(S^n)$ was the main goal at first, an incomparably greater role in the development of topology was played by the investigation of three- and four-dimensional manifolds undertaken to achieve this goal: Rokhlin computed the cobordism groups of three-dimensional and four-dimensional manifolds (and it was in the works of Pontryagin and Rokhlin that the cobordism groups themselves had appeared), discovered the role of the signature in the topology of four-dimensional manifolds and its connection with the Pontryagin class, and, finally, proved his famous theorem on divisibility by 16 of the signature of a smooth closed spin-manifold. Moreover, and this was for Rokhlin himself, according to his testimony, the main difficulty overcome in computing the group $\pi_{n+3}(S^n)$, he found a new approach to the construction of topological invariants of closed manifolds and other analogous objects (Rokhlin and Pontryagin called this approach invariant-free''): in order to construct an invariant, one spans the manifold under consideration by a membrane in which the subsequent construction is realized, and then one proves that the characteristic obtained is independent of the membrane. This scheme for determining invariants turned out to be very effective also later: for example, the exotic smooth structures on spheres discovered by Milnor are recognized precisely according to this scheme.

The work of Rokhlin on four-dimensional manifolds, which was motivated by the computation of $\pi_{n+3}(S^n)$, exerted a significant influence on the subsequent development of the topology of manifolds. Rokhlin published it as a Doklady note (the last in the series [3]-[6] of four notes containing the computation of the groups $\pi_{n+3}(S^n)$). In addition to the results mentioned above, the note contains, for example, a discussion of the problem of realizability of an integral unimodular quadratic form as the intersection form of a smooth simply connected closed four-dimensional manifold (which, in view of the earlier result of Pontryagin and Whitehead, is equivalent to the problem of a homotopy classification of such manifolds). This problem is discussed in connection with the fact that, by Rokhlin's theorem on divisibility of the signature, certain forms, for example, $E_8$ (an even integer unimodular definite form of rank 8), are not realized in this way. As became clear later, this fact plays a crucial role in problems of smoothability and combinatorial triangulation of topological manifolds. The discovery at the end of the 1950's of nonsmoothable manifolds (of dimension $> 4$) is connected with the generalization of Rokhlin's theorem to higher dimensions discovered by Kervaire and Milnor in 1958, and the discovery at the end of the 1960's of nontriangulable manifolds (also of dimension $> 4$) is connected with Rokhlin's theorem itself. Finally, thirty years later Freedman showed that an arbitrary integer unimodular quadratic form, including $E_8$, is isomorphic to the intersection form of a simply connected closed four-dimensional manifold, and this directly relates the existence of nonsmoothable manifolds with Rokhlin's theorem.

The fate of Rokhlin's proofs of his theorems mentioned above on four-dimensional manifolds, and first and foremost the theorem on divisibility of the signature by 16, was unusual. While the results themselves were acknowledged at once and became widely used, apparently the original proofs of Rokhlin were not understood for a long time. At first glance this happened because the published text of the proofs was very concise, and their understanding was complicated by an error made in the first of the four notes and corrected in the fourth. But the main reason is deeper. At roughly the same time, i.e., at the beginning of the 1950's, new and powerful methods were discovered in the homotopy theory, and the connection between homotopy problems and geometric problems discovered by Pontryagin and exploited by Pontryagin and Rokhlin for solving homotopy problems started working in the reverse direction. The computation of homotopy groups became easier by the new more formal and more algebraic methods, and it became natural to derive Rokhlin's theorem, which had originally played the role of a by-product (if not a lemma) in the computation of $\pi_{n+3}(S^n)$, from the results of the computation of these groups. This scheme was followed by all the papers that were published after Rokhlin's notes in the 1950's and 1960's and expounded his results on four-dimensional manifolds in detail (including a proof published by Rokhlin himself in 1958 [15]). In the 1970's an understanding of the role of Rokhlin's theorem and the intrinsic demands of four-dimensional topology led first to a reinvention of the original geometric proof, and then to a true reading of Rokhlin's papers (see the comments by L. Guillou and A. Marin on Rokhlin's four notes in their book, A la recherche de la topologie perdue (Birkhauser, Boston, 1986, pp. 25-95)).

But let us return to the 1950's. After computing the cobordism groups of dimensions 3 and 4, Rokhlin turned to study of the cobordism groups in higher dimensions and the theory of Pontryagin characteristic classes. The geometric techniques developed by Rokhlin in these years enabled him ([10], [14], [16]) to compute, in particular, the 2-primary component of the periodic part of the cobordism group for oriented manifolds (the famous exact sequence of Rokhlin).

In his work on Pontryagin classes, we take the liberty of singling out one theme that is central, in our view. It was conceived as far back as the 1953 note on four-dimensional manifolds. This is the connection between Pontryagin classes and the signature. To a great extent, the special role of the signature in the topology of manifolds was understood thanks to Rokhlin. He and Thom (independently) discovered the invariance of the signature with respect to cobordisms, and the formula $p_1 = 3\sigma$ connecting the signature $\sigma$ and the first Pontryagin number of a smooth closed 4-manifold. This formula demonstrates, in particular, the topological and homotopy invariance of the first Pontryagin class of a smooth closed four-dimensional manifold. Using as a basis the multidimensional generalization of this formula (the Hirzebruch-Thom-Rokhlin formula) and special properties of the signature, Rokhlin and Schwarz [l3] (and at about the same time, Thom) proved the invariance of the rational Pontryagin classes with respect to piecewise linear homeomorphisms, and gave a combinatorial (not local) definiton of the Pontryagin classes of a piecewise linear manifold. Rokhlin had several other results in this direction. To a great degree, his work prepared the way for Novikov's complete solution, obtained in the mid-1960's, of the problem of topological invariance of rational Pontryagin classes. It should be noted also that in 1966 Rokhlin and Novikov proved the topolgical invariance of certain integer classes that are multiples of the Pontryagin-Hirzebruch classes. It was this work of theirs that contained the first statement of the additivity of the signature upon gluing together two manifolds along a whole component of the boundary, a fundamental property of the signature later used in diverse investigations of the topology of manifolds; the work was never published in its entirety, but the main results are presented in S. P. Novikov, Essays on topology and related topics, Springer-Verlag, 1970. Now this theorem is known as Novikov additivity''.

In the 1960's Rokhlin again turned to the topology of 4-manifolds. His attention was attracted to this area by a note of Kervaire and Milnor in which it was proved that certain two-dimensional homology classes of a four-dimensional smooth manifold are not realized by a smoothly imbedded sphere. The Kervaire Milnor theorem generalized an example in Rokhlin's note [6], its statement generalized the Rokhlin theorem, and its proof was obtained by an easy reduction to the latter. Rokhlin attacked from different directions the problem of realizability of the two-dimensional homology classes of a 4-manifold by an imbedded smooth surface of specified genus. As a result he discovered a generalization of his theorem in which the manifold is not assumed to be a spinmanifold, and which involves the Arf-invariant of the quadratic form associated with the characteristic surface of the manifold (notice that as far back as [6], in the proof of the theorem on divisibility of the signature, Rokhlin singled out an auxiliary assertion that is an important complement to the theorem and turned out to be a special case of the generalization of it to non-spin 4-manifolds). This generalization was presented by Rokhlin in his report at the 1966 International Congress in Moscow, but for a long time did not appear in print. Rokhlin put it in his 1972 note [27], where this generalized theorem was used for proving the conjecture of Gudkov (see below). Another approach to the problem of realizability of two-dimensional homology classes that are not necessarily characteristic classes was proposed by Rokhlin in [26]. This approach was based on the use of branched coverings and yielded low estimates of the genus of a surface realizing a divisible homology class (closely related estimates were obtained independently by Wu-Chung Hsiang and R. H. Szczarba).

In 1972 Rokhlin turned to an investigation of the topological properties of real algebraic varieties. Despite significant achievements, this area had remained isolated from the general progress of topology before the 1970's. A breakthrough came due to the work of Arnol'd and Rokhlin. The starting point was the conjecture of Gudkov about the disposition of the ovals for a plane real algebraic curve. Arnol'd connected the problem of the mutual position of ovals to the topology of 4-manifolds and, using methods of four-dimensional topology, proved new prohibitions on mutual position of the ovals. In particular, he proved a weakened variant of Gudkov's conjecture. Then Rokhlin proved the full conjecture. At first he presented a proof [27] based on the use of a subtle tool in the topology of 4-manifolds - a generalization of the theorem on divisibility of the signature to non-spin 4-manifolds. (As mentioned above, it was here that the generalization first appeared in print. The proof of the Gudkov conjecture given in [27] contains an error, noted by Marin in 1977. Marin suggested a correction for the error that required a further generalization of Rokhlin's theorem. Such a generalization was then found by Guillou and Marin. See their book, cited above, for details.) A little later in the same year 1972 Rokhlin [28] proved by other means a far-reaching generalization of Gudkov's conjecture, with curves replaced by varieties of arbitrary dimension.

Subsequent work of Rokhlin in this area not only yielded a number of new results, but also led to a new way of looking at the topology of real algebraic varieties. The new point of view turned out to be exceptionally fruitful and has become common since then. Its finite formulation was presented by Rokhlin in his a report, Two aspects of the topology of real algebraic curves'' at the 1982 Leningrad International Topology Conference. Its essence, roughly speaking, is that the subject of the topology of real algbraic varieties is not only the topological properties of the set of real points of the variety, but also the relations between these properties and the position of this set in the set of complex points of the variety. And while in the first of his papers on the topology of real algebraic varieties Rokhlin solved purely real problems, and passage to the complex domain played only the role of a technical device in them, in his later papers the study of the complex characteristics of a real algebraic variety acquired an independent value.

This change of paradigm can be seen in two Rokhlin's papers. In [31] Rokhlin observed that every $M$-curve has natural orientations that are determined by the position of the real curve in its complexification (following Rokhlin, one calls them complex orientations), and in the case of a curve of even degree he established a relation between them and the position of the curve in the real plane. This fundamental relation subsequently came to be called Rokhlin's formula, and almost no modern investigation of the topology of real algebraic curves has avoided using this formula (it was later extended by Mishachev to $M$-curves of odd degree and by Rokhlin ([33], [34]) to arbitrary curves of type I, i.e., to curves separating their complexifications). The first systematic investigation of the connections between the real topology of a curve and its position in the complexification was undertaken by Rokhlin in [34]. In the same article he turned to a study of rigid isotopies (following Rokhlin, this is how we call isotopies formed of nonsingular curves of the same degree). This paper was significant not only in the new results it contained, but also in the stimulating influence it had on subsequent investigations in the topology of real algebraic varieties. In his last published work [35] Rokhlin proved new inequalities connecting the numerical topological characteristics of real curves of type I (see above). This paper indicated, in particular, the opportunities for new applications of the theory of branched coverings to the study of real algebraic varieties.

Rokhlin's contribution to the topology of real algebraic varieties has determined to a great extent the modern look of this area. His work enlarged the arsenal of technical means, shaped a new, more comprehensive way of looking at the basic objects and problems, and served as a basis and model for subsequent investigations. Under his leadership, his students (Cheponkus, Chislenko, Fiedler, Finashin, Kharlamov, Makeev, Mishachev, Slepyan, Viro and Zvonilov) carried out broad systematic investigations of the topological properties of real algebraic manifolds and obtained many outstanding results.

Significant events took place in the topology of 4-manifolds at the beginning of the 1980's. A number of problems in four-dimensional topology were solved due chiefly to the work of Freedman and Donaldson. A whole complex of problems in the topology of 4-manifolds was affected directly or indirectly. Under the influence of these events Rokhlin spent much time thinking actively about these problems during the last years of his life, and gave survey talks on the latest achievements in this area. These investigations of his remained uncompleted.

List of V. A. Rokhlin's work on topology

1. On a property of metric spaces, Sb. Nauchn. Student Rab. Moskov. Gos. Univ. 5 (1938), 19-22. (Russian)

2. 0n the theory of compressions, Sb. Nauchn. Student Rab. Moskov. Gos. Univ. 18 (1940), 37-40. (Russian)

3. Homotopy groups, Uspekhi Mat. Nauk 1 (1946), nos. 5-6, 175-223. (Russian)

4. A survey of results in resutls in the homotopy theory of continuous mappings of a sphere into a sphere, Uspekhi Mat. Nauk 15 (1950), no. 6, 88-101. (Russian)

5. On a mapping of the $(n + 3)$-dimensi0nal sphere into the $n$-dimensional sphere, Dokl. Akad. Nauk SSR 80 (1951), 541-544. (Russian)

6. The classification of mappings of the $(n + 3)$-sphere into the $n$-sphere, Dokl. Akad. Nauk SSSR 81 (1951), 19-22. (Russian)

7. A three-dimensional manifold is the boundary of a four-dimensional manifold, Dokl. Akad. Nauk SSSR 81 (1951), 355-357. (Russian)

8. New results in the theory of four-dimensional manifolds, Dokl. Akad. Nauk SSSR 84 (1952), 221-224. (Russian)

9. An intrinsic definition of the Pontryagin characteristic classes, Dokl. Akad. Nauk SSSR 84 (1952), 449-452. (Russian)

10. Intrinsic homology, Dokl. Akad. Nauk SSSR 89 (1953), 789-792. (Russian)

11. Characteristic cycles of smooth manifolds, Proc. Third All-Union Math. Congress, Vol. 2, 1956. (Russian)

12. On the Pontryagin characteristic classes, Dokl. Akad. Nauk SSSR 113 (1957), 276-279. (Russian)

13. With A.S. Schwvarz, 0n combinatorial invariance of the Pontryagin classes, Dokl. Akad. Nauk SSSR 114 (1957), 490-493. (Russian)

l4. Intrinsic homology. II, Dokl. Akad. Nauk SSSR 119 (1958), 876-879. (Russian)

15. The relation between characteristic classes of a four-dimensional manifold, Uchen. Zap. Kolomenskogo Ped. Inst. 2:2 (1958) 3-17 (Russian)

16. The theory of intrinsic homology, Uspekhi Mat. Nauk 14 (1959), no. 4 (88). 3-20; English transl. in Amer. Math. Soc. Transl. (2) 30 (I963).

17. Differential topology, Proc. Fourth All-Union Math. Congress, Vol. I. l963. pp. 218-219. (Russian)

l8. Imbedding of nonorientable three-dimensional manifolds in five-dimensional Euclidean space, Dokl. Akad. Nauk SSSR 160 (1965). 549-551; English transl. in Soviet Malh. Dokl. 6 (1965).

I9. Dijfeomorphisms of the manifold $S^3\times S^2$. Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 1386-1387. (Russian)

20. New examples of four-dimensional manifolds, Dokl. Akad. Nauk SSSR 162 (1965), 273-276; English transl. in Soviet Math. Dokl. 6 (I965).

21. The Pontryagin-Hirzebruch class of codimension 2, Izv. Akad. Nauk SSSR Ser. Mat. 30:3 (1966) 705-718; English transl. in Amer. Math. Soc. Transl. (2) 71 (1968)

22. Imbeddings and immersions in Riemannian Geometry. Uspekhi Mat. Nauk, 23:4 (1968) (142), 245. (Russian)

23. On the normal Euler numbers of the projective plane and the Klein bottle in four-dimensional Euclidean space, Dokl. Akad. Nauk SSSR 191 (1970), 27-29; English transl. in Soviet Math. Dokl. 11 (1970).

24. Two-dirnensional homology and two-dimensional submanifolds of 4-manifolds, Uspekhi Mal. Nauk 25 (1970), no. 3 (153), 258. (Russian)

25. joint with M. L. Gromov, Embeddings and immersions in Riemannian geometry, Uspekhi Mat. Nauk 25 (1970), no. 5 (l55). 3-62; English transl. in Russian Math. Surveys 25 (1970).

26. Two-dimensional submanifolds of four-dimensional manifolds, Funktsional. Anal. i Prilozhen. 5 (I972), no. 4, 48-60; English transl. in Functional Anal. Appl. 5 (I972).

27. Proof of Gudkov's conejcture. Funktsional. Anal. i Prilozhen. 6 (1972), no. 2, 62-64; English transl. in Functional Anal. Appl. 6 (I972).

28. Congruences modulo 16 in Hilbert's sixteenth problem, Funktsional. Anal. i Prilozhen. 6 (1972). no. 4, 58-64; English transl. in Functional Anal. Appl. 6 (1972).

29. Congruences modulo 16 in Hilbert's sixteenth problem. 1. Funktsional. Anal. i Prilozhen. 7 (1973), no. 2. 9l-92: English transl. in Functional Anal. App]. 7 (1973).

30. Work in recent years on the topology of real algebraic varieties, Uspekhi Mat. Nauk 29 (1974), no. 3 (177), I8O. (Russian)

31. Complex orientations of real algebraic curves, Funktsional. Anal. i Prilozhen. 8 (1974), no. 4, 7l-75; English transl. in Functional Anal. Appl. 8 (1974).

32. with D. B. Fuks. An elementary course in topology. Geometric chapters, Nauka, Moscow, 1977. (Russian)

33. The type and signature of a real plane algebraic curve, Uspekhi Mat. Nauk 1978, 33:3, 145. (Russian)

34. Complex topological characteristics of real algebraic curves, Usepkhi Mat. Nauk 33 (1978), no. 5 (203), 77-89; English transl. in Russian Math. Surveys 33 (1978).

35. New inequalities in the topology of real planar algebraic curves, Funktsional Anal. i Prilozhen. 14 (1980), no. I, 37-43; English transl. in Functional Anal. Appl. 14 (1980).