By V. A. Vassiliev

This publication stories a wide category of topological areas, lots of which play a tremendous position in differential and homotopy topology, algebraic geometry, and disaster conception. those contain areas of Morse and generalized Morse services, iterated loop areas of spheres, areas of braid teams, and areas of knots and hyperlinks. Vassiliev develops a basic strategy for the topological research of such areas. one of many significant effects here's a method of knot invariants extra strong than all recognized polynomial knot invariants. additionally, a deep relation among topology and complexity idea is used to procure the easiest identified estimate for the numbers of branchings of algorithms for fixing polynomial equations. during this revision, Vassiliev has additional a bit at the fundamentals of the speculation and class of embellishes, details on purposes 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 idea and Feynman diagrams, will locate this e-book an updated reference in this interesting region of mathematics.

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

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**Sample text**

AK that consists of all collections of points in Fix a flag R'0 c R" c 1RTs-' R Td t r which the first 21 KI I points form a collection realizing a point of the set AK, the next 2IK2I points are obtained from a collection ' E AK2 by translation along the vector N. a /a x1 (where the number N is sufficiently large), the points are obtained from a collection c E AK3 by translation +21K, I along 2N . a /ax1 , etc. Finally, for each number m > 21K' + next 21K31 I define a cycle {K1, ... , Kt ; ml C 1R (m) diffeomorphic to A(K1 , ...

Me). 2 follows immediately from this formula applied to the case t = 2. In fact, the group H`n-1(Br(m), ±Z) is generated by a single cell e(m). By (14), this group is nontrivial if and only if all possible binomial coefficients ('7) (0 < 1 < m) do not generate the group of integers. The latter happens if and only if m is a power of a prime number. For the proof of the theorem we make the sections of the local system ±7L agree over different cells. For each cell (m1, ... ) mt) consider a path in R2 (m) that lies entirely in this cell, except for the end belonging to a cell of maximal codimension (m) , in such a way that, if a point z, from the collection {z1, ...

Oo these homomorphisms define a map O: H*(Br, Z2) -> H*(Br, 7G2) 0 H*(Br, Z2) , which, together with the usual cohomology product, defines the Hopf algebra structure on H*(Br, 7G2) . THEOREM ([FUCI152]). The map 0 sends the generator (sl , ... , sq) of the ring H*(Br, Z2) to the element 10 (sl ) 0 0 a ) sq) + (SI 5 ... ) Sq) 01+ E (si S ir (sj, sjq-1 §2. COHOMOLOGY OF BRAID GROUPS WITH TRIVIAL COEFFICIENTS 25 where the sum is taken over all subdivisions of the collection sl , ... , sq into two nonempty collections.