The Fundamental Crossed Module of the Complement of a Knotted Surface
نویسنده
چکیده
We prove that if M is a CW-complex and M1 is its 1-skeleton then the crossed module Π2(M,M 1) depends only on the homotopy type of M as a space, up to free products, in the category of crossed modules, with Π2(D 2, S1). From this it follows that, if G is a finite crossed module and M is finite, then the number of crossed module morphisms Π2(M,M 1) → G can be re-scaled to a homotopy invariant IG(M), depending only on the homotopy 2-type of M . We describe an algorithm for calculating π2(M,M (1)) as a crossed module over π1(M (1)), in the case when M is the complement of a knotted surface Σ in S4 and M (1) is the handlebody made from the 0and 1-handles of a handle decomposition of M . Here Σ is presented by a knot with bands. This in particular gives us a geometric method for calculating the algebraic 2-type of the complement of a knotted surface from a hyperbolic splitting of it. We prove in addition that the invariant IG yields a non-trivial invariant of knotted surfaces in S 4 with good properties with regards to explicit calculations. ∗Also at: Universidade Lusófona de Humanidades e Tecnologia, Av do Campo Grande, 376, 1749-024, Lisboa, Portugal.
منابع مشابه
On 2-Dimensional Homotopy Invariants of Complements of Knotted Surfaces
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