Displaying similar documents to “Mapping class group and the Casson invariant”

Homomorphic extensions of Johnson homomorphisms via Fox calculus

Bernard Perron (2004)

Annales de l’institut Fourier

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Using Fox differential calculus, for any positive integer k , we construct a map on the mapping class group ℳ g , 1 of a surface of genus g with one boundary component, such that, when restricted to an appropriate subgroup, it coincides with the k + 1 t h Johnson-Morita homomorphism. This allows us to construct very easily a homomorphic extension to ℳ g , 1 of the second and third Johnson-Morita homomorphisms.

Virtual strings

Vladimir Turaev (2004)

Annales de l'Institut Fourier

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A virtual string is a scheme of self-intersections of a closed curve on a surface. We study algebraic invariants of strings as well as two equivalence relations on the set of strings: homotopy and cobordism. We show that the homotopy invariants of strings form an infinite dimensional Lie group. We also discuss connections between virtual strings and virtual knots.