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Complements of discriminants of smooth maps: topology and applications. Transl. from the Russian by B. Goldfarb. Transl. ed. by S. Gelfand. (English) Zbl 0762.55001
Translations of Mathematical Monographs. 98. Providence, RI: American Mathematical Society (AMS). vi, 208 p. (1992).
The central point of the book is a proof that the space of smooth maps \(M\to\mathbb{R}^ n\) whose \(k\)-jets avoid a subset \(X\) of jet space \(J^ k(M,\mathbb{R}^ n)\) (satisfying some natural conditions) has the same homology as the space of sections of the jet bundle avoiding \(X\). The proof depends on an ingenious construction and a spectral sequence argument. The interest of the book comes from the fact that this (and related results) has a wide variety of applications to problems of independent interest. These are explored in separate chapters, after a lengthy introduction in which the main results are presented.
In Chapter 1, after recalling basic definitions, the author computes the cohomology of the braid group and related configuration spaces. Many of the results here were previously known, including the fact (see below) that the stable cohomology of the braid groups is isomorphic to that of the loop space \(\Omega^ 2 S^ 3\).
Chapter 2 discusses the complexity of algorithms, and obtains sharp estimates of topological complexity (defined in terms of the number of branding nodes in the process) for the problems of finding all (or at least one) solutions of a system of polynomial equations with accuracy within \(\varepsilon\), or finding “\(\varepsilon\)-roots”. The arguments ar very geometric; the homological genus of a covering (defined by considering that the set of solutions) plays a key role.
The main result is proved in Chapter 3, after the idea is expounded for a simpler special case: the space of functions on \(\mathbb{R}\) without zeros of multiplicity \(\geq 3\). The result contains a much simplified proof of an important theorem of K. Igusa [Ann. Math., II. Ser. 119, 1-58 (1984; Zbl 0548.58005)] and covers a much wider class of problems than an earlier result of Gromov (which obtained a homotopy equivalence between the spaces in question, under restrictive conditions). It is precise enough to be applied at once to obtain nontrivial calculations. Examples treated include the homology of iterated loop spaces of spheres and complements of arrangements of affine subspaces (of any dimension) in \(\mathbb{R}^ n\), both of which had been obtained earlier by special arguments.
The fourth chapter discusses the complements of discriminants of isolated singularities of functions on \(\mathbb{C}^ n\). Inclusions between such complements induce an inverse system of cohomology; each group \(H^ k\) is shown to stabilise, and the inverse limit is isomorphic to the cohomology of \(\Omega^{2n} S^{2n+1}\). A similar procedure leads to the calculation of limit cohomology for complements of caustics. The case \(n=1\) (complements of discriminants of polynomials in one variable) had been recently obtained by F. R. Cohen, R. L. Cohen, B. M. Mann and R. J. Milgram [Acta Math. 166, 163-221 (1991; Zbl 0741.55005)]. It seems that the general question on complements of discriminants was Vassiliev’s starting point. On the way he proves a conjecture of Arnold on stable irreducibility of strata in the discriminant.
The application which has attracted the most attention: to the cohomology of the space of knots: is reserved for the final chapter. Roughly speaking, the space of all maps \(S^ 1\to S^ 3\) is contractible, and is stratified, where strata of codimension \(k\) correspond to maps where the image involves \(k\) identifications of points. The ‘homology classes’ of these strata give cohomology classes of the complement. Deforming a knot to be trivial involves some crossing points, and a careful investigation of these leads Vassiliev to a new definition of knot invariants. These invariants are defined very carefully, and complete proofs are given. They have led to much recent work, some of which is briefly alluded to in the book.
The book is a work of stunning originality and an impressive unification of very diverse strands. Although it is written carefully, there are inevitably several rough edges. As far as the reviewer can tell, these are defects in the presentation and not in the mathematics. Likewise, although the translation is in general good, there are occasional lapses. None of these are such as to seriously trouble the interested reader. There is not much special notation: most is standard.
The book is carefully planned and well written: I much enjoyed reading it.

MSC:
55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology
55T99 Spectral sequences in algebraic topology
55S40 Sectioning fiber spaces and bundles in algebraic topology
57R45 Singularities of differentiable mappings in differential topology
68Q25 Analysis of algorithms and problem complexity
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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