##
**A la recherche de la topologie perdue. 1: Du côté de chez Rohlin. 2: Le côté de Casson.**
*(French)*
Zbl 0597.57001

Progress in Mathematics, Vol. 62. Boston/Basel/Stuttgart: Birkhäuser. XXIII, 244 p. DM 94.00 (1986).

The book named after the world-wide known Marcel Proust novel ”A la recherche du temps perdu” is aimed at collecting and analyzing several poorly available or unpublished papers which constitute an essential part of the foundation of 4-dimensional topology.

The book is dedicated to the memory of V. A. Rohlin (1919-1984) whose famous theorem on the divisibility by 16 of the signature of any smooth, closed, oriented, spin 4-manifold plays a crucial role in the topology of 4-manifolds. The book starts off with the biography of Rohlin which briefly describes his life and mathematical interests (the latter included measure theory, ergodic theory, topology, and real algebraic geometry). One may add to this biography that during his life Rohlin collaborated with quite a number of outstanding mathematicians: L. S. Pontryagin, A. N. Kolmogorov, S. P. Novikov, M. L. Gromov and others. Rohlin succeeded in creating a strong topological team in Leningrad which nowadays includes his former pupils O. Ya. Viro, V. M. Kharlamov, Ya. M. Eliasberg, N. V. Ivanov, N. Yu. Netzvetaev, S. M. Finashin, the reviewer and other mathematicians.

The biography is followed by a list of Rohlin’s publications (58 items) and a general introduction to the book. The rest of the book is divided into two parts. The first part ”Du côté de chez Rohlin” is concerned with the Rohlin signature theorem, its various proofs and generalizations. This part includes the following six sections written by different authors.

(1) ”Four articles of Rohlin”. These 4 articles published in the book in French translation first appeared in Russian in 1951-1952. The articles contain among other things the signature theorem with a rather sketchy proof.

(2) L. Guillou, A. Marin, ”Comments on the 4 preceding articles”. These detailed comments provide the necessary background for reading Rohlin’s articles and clarify his concise arguments. It is characteristic that the four articles of Rohlin occupy 23 pages of the book and the comments occupy 71 pages.

(3) L. Guillou, A. Marin, ”An extension of Rohlin’s signature theorem”. The paper extends Rohlin’s theorem to the case of non-spin 4- manifolds. This extension asserts that \(\sigma\) (M)-F\(\cdot F\equiv 2\alpha (M,F)\) mod 16 where M is any closed, smooth, oriented 4-manifold, \(\sigma\) (M) is its signature, F is any closed surface in M, homologically dual to \(w_ 2(M)\), \(\alpha\) (M,F) is the Brown invariant of a certain quadratic form \(H_ 1(F; {\mathbb{Z}}/2{\mathbb{Z}})\to {\mathbb{Z}}/4{\mathbb{Z}}\) associated with the imbedding \(F\hookrightarrow M\). (In the case of orientable F this extension was obtained by Rohlin in the 1960s.)

(4) Y. Matsumoto, ”An elementary proof of Rohlin’s signature theorem and its extension by Gillou and Marin”. The paper presents a quick and clear proof of the theorems based on the Arf invariants of links.

(5) A. Marin, \(''{\mathbb{C}}P^ 2/\sigma\) or Kuiper and Massey in the country of conics”. In 1973/74 W. Massey and N. Kuiper independently constructed homeomorphisms of the quotient of \({\mathbb{C}}P^ 2\) by the involution of complex conjugation onto \(S^ 4\). A. Marin shows that the approaches of these two authors are essentially equivalent and that their homeomorphisms are actually diffeomorphisms. The subject of this paper is related to the preceding part of the book via the theory of plane real algebraic curves.

(6) The last section of Part 1 describes the historical evolution of Rohlin’s signature theorem.

Part 2 of the book is entitled ”Le côté de Casson”. It contains 3 papers.

(1) C. McA. Gordon, ”On the G-signature theorem in dimension four”. The paper presents an elementary proof of the 4-dimensional Atiyah-Singer signature theorem. ”The proof uses no analysis and only a little bordism”.

(2) A. J. Casson, C. McA. Gordon, ”Cobordism of classical knots”. This most important paper has circulated for about ten years as a preprint. The authors introduce new signature-type invariants of classical knots and use these invariants (and the Atiyah-Singer signature theorem) to exhibit non-slice knots with algebraically slice Seifert forms. An appendix to this section written by P. M. Gilmer describes subsequent applications and extensions of the Casson-Gordon invariants.

(3) A. J. Casson, ”Three lectures on new infinite constructions in 4-dimensional manifolds” (notes prepared by L. Guillou). In this paper Casson introduces and studies his famous flexible handles (Casson handles). This study gives a strong impetus to further development of 4- dimensional topology. Note that M. Freedman’s topological classification of simply connected closed 4-manifolds is based on his thorough analysis of the Casson handles crowned with the theorem that topologically the Casson handles are ordinary handles.

The book is dedicated to the memory of V. A. Rohlin (1919-1984) whose famous theorem on the divisibility by 16 of the signature of any smooth, closed, oriented, spin 4-manifold plays a crucial role in the topology of 4-manifolds. The book starts off with the biography of Rohlin which briefly describes his life and mathematical interests (the latter included measure theory, ergodic theory, topology, and real algebraic geometry). One may add to this biography that during his life Rohlin collaborated with quite a number of outstanding mathematicians: L. S. Pontryagin, A. N. Kolmogorov, S. P. Novikov, M. L. Gromov and others. Rohlin succeeded in creating a strong topological team in Leningrad which nowadays includes his former pupils O. Ya. Viro, V. M. Kharlamov, Ya. M. Eliasberg, N. V. Ivanov, N. Yu. Netzvetaev, S. M. Finashin, the reviewer and other mathematicians.

The biography is followed by a list of Rohlin’s publications (58 items) and a general introduction to the book. The rest of the book is divided into two parts. The first part ”Du côté de chez Rohlin” is concerned with the Rohlin signature theorem, its various proofs and generalizations. This part includes the following six sections written by different authors.

(1) ”Four articles of Rohlin”. These 4 articles published in the book in French translation first appeared in Russian in 1951-1952. The articles contain among other things the signature theorem with a rather sketchy proof.

(2) L. Guillou, A. Marin, ”Comments on the 4 preceding articles”. These detailed comments provide the necessary background for reading Rohlin’s articles and clarify his concise arguments. It is characteristic that the four articles of Rohlin occupy 23 pages of the book and the comments occupy 71 pages.

(3) L. Guillou, A. Marin, ”An extension of Rohlin’s signature theorem”. The paper extends Rohlin’s theorem to the case of non-spin 4- manifolds. This extension asserts that \(\sigma\) (M)-F\(\cdot F\equiv 2\alpha (M,F)\) mod 16 where M is any closed, smooth, oriented 4-manifold, \(\sigma\) (M) is its signature, F is any closed surface in M, homologically dual to \(w_ 2(M)\), \(\alpha\) (M,F) is the Brown invariant of a certain quadratic form \(H_ 1(F; {\mathbb{Z}}/2{\mathbb{Z}})\to {\mathbb{Z}}/4{\mathbb{Z}}\) associated with the imbedding \(F\hookrightarrow M\). (In the case of orientable F this extension was obtained by Rohlin in the 1960s.)

(4) Y. Matsumoto, ”An elementary proof of Rohlin’s signature theorem and its extension by Gillou and Marin”. The paper presents a quick and clear proof of the theorems based on the Arf invariants of links.

(5) A. Marin, \(''{\mathbb{C}}P^ 2/\sigma\) or Kuiper and Massey in the country of conics”. In 1973/74 W. Massey and N. Kuiper independently constructed homeomorphisms of the quotient of \({\mathbb{C}}P^ 2\) by the involution of complex conjugation onto \(S^ 4\). A. Marin shows that the approaches of these two authors are essentially equivalent and that their homeomorphisms are actually diffeomorphisms. The subject of this paper is related to the preceding part of the book via the theory of plane real algebraic curves.

(6) The last section of Part 1 describes the historical evolution of Rohlin’s signature theorem.

Part 2 of the book is entitled ”Le côté de Casson”. It contains 3 papers.

(1) C. McA. Gordon, ”On the G-signature theorem in dimension four”. The paper presents an elementary proof of the 4-dimensional Atiyah-Singer signature theorem. ”The proof uses no analysis and only a little bordism”.

(2) A. J. Casson, C. McA. Gordon, ”Cobordism of classical knots”. This most important paper has circulated for about ten years as a preprint. The authors introduce new signature-type invariants of classical knots and use these invariants (and the Atiyah-Singer signature theorem) to exhibit non-slice knots with algebraically slice Seifert forms. An appendix to this section written by P. M. Gilmer describes subsequent applications and extensions of the Casson-Gordon invariants.

(3) A. J. Casson, ”Three lectures on new infinite constructions in 4-dimensional manifolds” (notes prepared by L. Guillou). In this paper Casson introduces and studies his famous flexible handles (Casson handles). This study gives a strong impetus to further development of 4- dimensional topology. Note that M. Freedman’s topological classification of simply connected closed 4-manifolds is based on his thorough analysis of the Casson handles crowned with the theorem that topologically the Casson handles are ordinary handles.

Reviewer: V.Turaev

### MSC:

57-06 | Proceedings, conferences, collections, etc. pertaining to manifolds and cell complexes |

57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

58J40 | Pseudodifferential and Fourier integral operators on manifolds |

57R20 | Characteristic classes and numbers in differential topology |

57R19 | Algebraic topology on manifolds and differential topology |