Book Review: Complements of discriminants of smooth maps\/}: {\it Topology and applications

Abstract

The book under review is the translation of the author's second(doctoral) thesis. Its joint theme is discriminants and theirmost famous application, invariants of knots (now calledVassiliev invariants).\par From the introduction: ``Adiscriminant is a subset of a function space consisting offunctions (or maps) with singularities of some fixed type. Manyobjects in mathematics can be described as complements of(properly defined) discriminants. Among them are: spaces of Morseand generalized Morse functions, spaces of polynomials (orsystems of polynomial equations) without multiple roots, sets ofnonsingular deformations of complex hypersurfaces, iterated loopspaces of certain spaces, and spaces of knots (i.e. nonsingularimbeddings S\sp 1 \to S\sp 3 or {\bf R}\sp 1\to {\bf R}\sp 3).''\par The book contains five chapters andfive appendices. (1) Cohomology of braid groups and configurationspaces. This is an introductory chapter and collects technicalresults on the cohomology of configuration spaces used in laterchapters. In particular, it presents results of V. I. Arnold, D.B. Fuchs, G. Segal, F. Cohen, P. May and F. V. Vainshtein on thecohomology of braid groups with constant coefficients, withcoefficients in the sheaf \pm{\bf Z} and in the Coxeterrepresentation. (2) Applications: complexity of algorithms andsuperpositions of algebraic functions. In this chapter theprevious results are applied to the theory of computationalcomplexity and to the theory of algebraic functions. Theconfiguration space, R\sp 2(d), can be considered as the spaceof all complex polynomials of the formz\sp d+{λ}\sb 1z\sp {d 1}+\cdots + {λ}\sb {d1}z+{λ}\sb d without multiple roots. This space has anatural d! sheeted covering. S. Smale used topologicalproperties of this covering to estimate the topologicalcomplexity of an algorithm for calculating the roots ofpolynomials (i.e. the minimal number of branching nodes overalgorithms calculating roots). The author dramatically improvesSmale's estimates. His main tools are the Schwarz genus of afibration (i.e. the smallest cardinality of a cover of the baseby open domains such that there is a continuous section of thefibration over each of these domains) and use of the cohomologyof braid groups to approximate Schwarz genera. Using similarideas the author estimates the topological complexity of solvingpolynomial systems in several complex variables. Furthermore, theuse of the genus of a covering related to the algebraic functionsgives new obstructions to the representation of an algebraicfunction as a superposition of functions in fewer variables. (3)Topology of spaces of real functions without complicatedsingularities. The topological structure of the space of smoothmaps of a manifold to {\bf R}\sp n without complicatedsingularities is an important characteristic of the manifold.Their study was initiated by the work of Smale and Cerf. Chapter3 contains computations of the cohomology rings of such spaces.The author proves, in particular, the following reductiontheorem: If the codimension of the discriminant in the functionspace is at least 2, then instead of spaces of admissiblefunctions one can consider spaces of admissible sections of jetbundles. In particular, the space {\scr F} {Σ}\sb k offunctions {\bf R}\sp 1\to {\bf R}\sp 1 without zeros ofmultiplicity k (k\geq 3) and identically equal to 1 outsidesome compact set is homotopy equivalent to the loop spaceΩS\sp {k 1}. The author also computes the cohomologyrings of the spaces of real polynomials x\sp d+a\sb 1x\sp {d1}+\cdots +a\sb {d 1}x+a\sb d without roots of multiplicity k,for any k and d. With d increasing, these cohomology rings(and homotopy groups) stabilize to those of {\scr F}{Σ}\sb k. (4) Stable cohomology of complements ofdiscriminants and caustics of isolated singularities ofholomorphic functions. The author proves, in particular, that thestable cohomology ring of the complement of the discriminant forsingularities of holomorphic functions on {\bf C}\sp n isisomorphic to the cohomology ring of the space of 2n loops ofthe (2n+1) dimensional sphere,H\sp {*}({Ω}\sp {2n}S\sp {2n+1}). (5) Cohomology of thespace of knots. The author constructs a new series of numericalinvariants of knots. This construction is based on the study ofthe discriminant (which in this case is defined as the set ofmaps S\sp 1 \to S\sp 3 with singularities or selfintersections). A numerical invariant of knots is tautologicallya class in the 0 dimensional cohomology group of the complementof the discriminant in the space of all maps S\sp 1 \to S\sp 3.Each of these invariants can be given by a discriminant: to eachnonsingular piece of the discriminant (i.e. a connected componentof the set of its nonsingular points) we have to assign an indexwhich is the difference of the values of the invariant forneighboring knots separated by this piece. On the other hand,suppose that to each nonsingular component of the discriminant wehave assigned a numerical index. In order for this collection todefine an invariant of the isotopy type of knots it shouldsatisfy the following condition on homology: a linear combinationof these components with the appropriate coefficients should nothave a boundary in the space of all maps S\sp 1 \to S\sp 3.Enumerating such admissible collections is a problem in homologytheory and is achieved by standard methods of this theory. Theauthor constructs a spectral sequence that produces such acollection of indices. This spectral sequence E\sp {p,q}\sb ris generated by a natural stratification of the discriminant bythe types of degeneration of the corresponding maps. The knotinvariants correspond to the elements of the groups E\sp { i,i}\sb {\infty}, i\geq 1.\par The author computes his invariantsup to the term E\sp { 4,4} and gets five nontrivial knotinvariants. He notices that the first of them coincides with thesecond coefficient of the Alexander Conway polynomial. The mainquestion asked is whether the set of invariants is complete(distinguishes nonequivalent knots). In particular, the authorasks whether a given knot is equivalent to its mirror image iffits invariants and the invariants of the mirror image are thesame.\par {Reviewer's remarks: The invariants introduced by theauthor (now called Vassiliev or finite type invariants) becamethe major topic of knot theory (including graphs and linksembedded in any 3 manifold). From my point of view their biggestattraction is that they can be computed in polynomial time,unlike the Jones type polynomials (up to the famous{\rm P}eq {\rm NP} conjecture). Furthermore, the completionof the space of invariants forms a well structured object (e.g. aHopf algebra).}\par Finite type invariants were alsoindependently introduced in 1988 by M. N. Gusarov [Zap. Nauchn.Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 193 (1991),Geom. i Topol. 1, 4 9, 161; MR\Cite{Gusarov91:new:i}[93b:57007]]. The Gusarov approach ispurely combinatorial. Gusarov, Bar Natan, Birman and Lin provedthat all Jones type invariants can be obtained from Vassilievinvariants. For a survey on Vassiliev invariants one can refer tothe paper by J. S. Birman [Bull. Amer. Math. Soc. (N.S.) 28(1993), no. 2, 253 287; MR 94b:57007]