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Cobordismes d’immersions lagrangiennes et legendriennes. (French) Zbl 0615.57001

Travaux en Cours, 20. Paris: Hermann. XVI, 198 p.; FF 180.00 (1987).
The study of cobordism groups of Lagrangian and Legendrian immersions was initiated by V. I. Arnol’d [Funkts. Anal. Prilozh. 14, No.3, 1-13 (1980; Zbl 0448.57017); ibid. 14, No.4, 8-17 (1980; Zbl 0472.55002)]. The aim of this book is to carry out a thorough study of these groups by the methods of cobordism theory. This is done by a judicious mixture of algebraic and geometric methods; the emphasis here is on the former, but much effort is given to explaining and illustrating the geometric results thereby obtained.
Readers wanting some preliminary orientation on Lagrangian and Legendrian immersions, and their connections with singularities, caustics and wave fronts, may want to consult Part III of the book ”Singularities of differentiable maps”, Volume I [V. I. Arnol’d, S. M. Gusejn- Zade and A. N. Varchenko (1985; Zbl 0554.58001)].
The present book begins with an introduction, in which the contents are summarized, and an initial chapter on Lagrangian immersions into cotangent bundles \(T^*X\), leading via the Gromov-Lees theorem to a homotopy-theoretic interpretation of the cobordism groups of exact Lagrangian immersions. By a Lagrangian immersion \(f: V\to T^*X\) one means an immersion of smooth manifolds, with dim V\(=\dim X\), for which the canonical symplectic 2-form \(\omega\) on \(T^*X\) satisfies \(f^*\omega =0\). \(T^*X\) also carries a canonical 1-form \(\alpha\) (of Liouville) with \(\omega =-d\alpha\), and one calls f an exact Lagrangian immersion if \(f^*\alpha\) is an exact form. In particular, when \(X={\mathbb{R}}^ n\) we obtain the notion of an exact Lagrangian immersion of \(V^ n\) into \(T^* {\mathbb{R}}^ n={\mathbb{C}}^ n\); the evident necessary condition that TV \(\otimes_{{\mathbb{R}}} {\mathbb{C}}\) be trivial is sufficient as well, in view of the Gromov-Lees theorem.
It takes a certain amount of effort to define the notion of cobordism of Lagrangian immersions. This leads to the cobordism groups \({\mathfrak N}L_ n\) and \(L_ n\) of exact Lagrangian immersions \(V^ n\to T^* {\mathbb{R}}^ n\), the unoriented and oriented cases, respectively. These are the coefficient groups for the more general theory based on immersions into \(T^*X\). The homotopy-theoretic interpretation is as follows. Let \(\Lambda_ n=U(n)/O(n)\) and \({\tilde \Lambda}_ n=U(n)/SO(n)\); these spaces carry universal bundles \(\lambda_ n\) and \({\tilde \lambda}_ n\) for real n-plane bundles with trivial complexifications. The resulting Thom spectra \(M\lambda\) and M\({\tilde \lambda}\) give rise to cobordism theories with coefficient rings \({\mathfrak N}L_*\) and \(L_*\), respectively. Geometrically, these theories are based on manifolds V for which TV \(\otimes_{{\mathbb{R}}} {\mathbb{C}}\) has a trivialization.
The first part of the book, comprising eight chapters, is a study of the rings \({\mathfrak N}L_*\) and \(L_*\). In fact, \({\mathfrak N}L_*\) was computed by L. Smith and R. E. Stong [J. Math. Mech. 17, 1087-1102 (1968; Zbl 0174.547)]; it embeds in the unoriented cobordism ring \({\mathfrak N}_*\) as a polynomial subalgebra on generators in all odd dimensions not of the form \(2^ k-1\). The oriented ring \(L_*\) poses a greater challenge and is examined from several angles. A related ring \(SL_*\) of ”special Lagrangian” immersions is introduced in order to construct a Rohlin-Wall exact sequence. As a final variation, a complex analogue \(L^ U_*\) is explored.
The second part of the book, the final six chapters, deals with the cobordism of Legendrian immersions into certain contact manifolds. By a contact structure on a manifold \(W^{2n+1}\) is meant a field of hyperplanes \({\mathcal H}\) on W, given locally as the kernel of a 1-form \(\alpha\) for which the restriction of \(d\alpha\) to \({\mathcal H}\) is nondegenerate. A Legendrian immersion is an immersion \(f: V^ n\to W^{2n+1}\) whose differential maps TV into \({\mathcal H}\); these arise in connection with the study of wave fronts. The contact manifolds considered are sphere bundles \(S(T^*X)\), projectivizations \(P(T^*X)\), and spaces of 1-jets \(J^ 1(X, {\mathbb{R}})\), with the greatest attention being given to the cobordism groups (oriented or not) of Legendrian immersions into \(S(T^* {\mathbb{R}}^{n+1})\) and \(P(T^* {\mathbb{R}}^{n+1})\). Algebraically, the results are less satisfying than in the Lagrangian case; on the other hand, a large selection of attractive examples is given in the Legendrian case.
Reviewer: P.Landweber

MSC:

57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
57R90 Other types of cobordism
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
57R42 Immersions in differential topology
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
57R45 Singularities of differentiable mappings in differential topology
57R20 Characteristic classes and numbers in differential topology