By V. A. Vassiliev

This e-book stories a wide type of topological areas, a lot of which play a massive position in differential and homotopy topology, algebraic geometry, and disaster concept. those comprise areas of Morse and generalized Morse capabilities, iterated loop areas of spheres, areas of braid teams, and areas of knots and hyperlinks. Vassiliev develops a common strategy for the topological research of such areas. one of many vital effects here's a procedure of knot invariants extra strong than all recognized polynomial knot invariants. furthermore, a deep relation among topology and complexity thought is used to procure the easiest identified estimate for the numbers of branchings of algorithms for fixing polynomial equations. during this revision, Vassiliev has further a bit at the fundamentals of the speculation and class of adorns, info on functions of the topology of configuration areas to interpolation idea, and a precis of modern effects approximately finite-order knot invariants. experts in differential and homotopy topology and in complexity conception, in addition to physicists who paintings with string concept and Feynman diagrams, will locate this booklet an updated reference in this interesting quarter of mathematics.

Readership: Physicists who paintings with string idea and Feynman diagrams, and experts in differential and homotopy topology and in complexity conception.

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Orderings of these roots. -+ PROPOSITION. The space %in -I is a space of type K(Br(m), 1) : its imbedding into the space of all nondiscriminant polynomials (4) is a homotopy equiv- alence. The space Min is a space of type K (I (m) , 1), where I (m) is the colored braid group. o SMALE'S THEOREM (see [Smale4 1). For any m there is a number go > 0 such that for all E E (0, E01 the topological complexity of the problem P(m, g) and the genus of the covering fm are related by the inequality (5) T(m, c) ?

The genus of a map X -> Y is a generalization of the category of the space Y Y. DEFINITION (see [LSh]). The category of a topological space Y is the min- imal cardinality of an open cover of Y consisting of sets contractible in Y. Suppose Y is path connected. Consider the Serre fibration S -> Y with the space consisting of all paths [0, 1] -> Y sending 0 to the base point and projection that associates to each path its endpoint. PROPOSITION ([Schwarz]). The category of a path connected space Y coincides with the genus of its Serre fibration.

If YY! is a power of a prime number then for sufficiently small e > 0 the topological complexity of the problem Pl(m, e) of finding one of the roots of a polynomial (4) with accuracy e equals m - I. Denote by prim(m) the greatest divisor of m which is a power of a prime number. §3. 4. THEOREM. For any natural m and for sufficiently small e > 0 the topological complexity of the problem P, (m , e) is at least prim(m) - I. 5. PROPOSITION. e. for any 8 > 0 there is m(8) such that prim(m) > (1 - 8) In m for m > m(6)).

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