Knots, Links, Braids and 3-Manifolds: An Introduction to the by V. V. Prasolov, A. B. Sossinsky

By V. V. Prasolov, A. B. Sossinsky

This e-book is an advent to the notable paintings of Vaughan Jones and Victor Vassiliev on knot and hyperlink invariants and its contemporary changes and generalizations, together with a mathematical remedy of Jones-Witten invariants. It emphasizes the geometric facets of the speculation and treats subject matters equivalent to braids, homeomorphisms of surfaces, surgical procedure of 3-manifolds (Kirby calculus), and branched coverings. This beautiful geometric fabric, fascinating in itself but no longer formerly amassed in e-book shape, constitutes the foundation of the final chapters, the place the Jones-Witten invariants are developed through the rigorous skein algebra process (mainly as a result of the Saint Petersburg school).

Unlike numerous fresh monographs, the place all of those invariants are brought through the use of the delicate summary algebra of quantum teams and illustration concept, the mathematical must haves are minimum during this booklet. a variety of figures and difficulties make it appropriate as a direction textual content and for self-study.

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Additional resources for Knots, Links, Braids and 3-Manifolds: An Introduction to the New Invariants in Low-Dimensional Topology (Translations of Mathematical Monographs)

Example text

The controlled equivariant boundary theorem is closely related to:the equivariant product structure theorem. Namely, let a G-manifold M be provided with a locally smooth PL structure ~0 consider locally smooth PL structures ~ near aM . We on M x R which agree with ~0 x R near aM x R (so-called structures rel 8). We say that a structure 5] on M x R tel a admits a product structure if 5] is G-isotopic rel a to a structure of the form e x R , where e is a locally smooth PL structure on M which agrees with ~'0 near a M .

Then M is G-dlffeomorphic to OM x <:0,1). In conclusion one has to mention that there is an alternative approach to the equivariant Siebenmann's theorem given by Steinberger and West [49], [51] following Chapman's ideas [10]. They develop the equivariant version of the boundary theorem in controlled setting thus obtaining an equivariant analogue of Chapman's result ( [51] thin. 4 ) but their end theorem is only formulated in [51] and it relies heavily on the fundamental paper [50] which is still (as far as author knows) in preparation.

O(WH(g))*]) ~ I~o(Z[,o(WH(M))*~]) is an isomorphism. 3) dimMHa > 6. Then M is G-dlffeomorphic to OM x <:0,1). In conclusion one has to mention that there is an alternative approach to the equivariant Siebenmann's theorem given by Steinberger and West [49], [51] following Chapman's ideas [10]. They develop the equivariant version of the boundary theorem in controlled setting thus obtaining an equivariant analogue of Chapman's result ( [51] thin. 4 ) but their end theorem is only formulated in [51] and it relies heavily on the fundamental paper [50] which is still (as far as author knows) in preparation.

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