The fundamental crossed module of the complement of a knotted surface

Abstract

Abstract. We prove that if M is a CW-complex and M 1 is its 1-skeleton, then the crossed module Π 2 (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 Π 2 (D 2 , S 1 ). From this it follows that if G is a finite crossed module and M is finite, then the number of crossed module morphisms Π 2 (M, M 1 ) → G can be re-scaled to a homotopy invariant I G (M ), depending only on the algebraic 2-type of M . We describe an algorithm for calculating π 2 (M, M (1) ) as a crossed module over π 1 (M (1) ), in the case when M is the complement of a knotted surface Σ in S 4 and M (1) is the handlebody of a handle decomposition of M made from its 0-and 1-handles. 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 I G yields a non-trivial invariant of knotted surfaces in S 4 with good properties with regard to explicit calculations.

Higher lattices, discrete two-dimensional holonomy and topological phases in (3 + 1)D with higher gauge symmetry

“…Crossed modules of groups are discussed in [4,11,28]. Crossed modules of groupoids, discussed extensively in this paper, appear in [18, §6.2] and [11,33,19].…”

“…Let x be an interior point of open cell c 2 P corresponding to P . We have a commutative diagram (28), where all morphisms are induced by inclusion. The vertical line p x corresponds to the identity map Z → Z, in the sense that it sends the positive generator K ∈ H 2 (Σ) ∼ = Z to the positive generator K x of H 2 (Σ, Σ \ {x}) ∼ = Z.…”

“…Proof. As mentioned above, this follows from the long homotopy exact sequence of the triple (M 3 , M 2 , M 1 ), applied to each choice of base point x ∈ M 0 ; details can be found in [28], for the case of CW-complexes with a single base-point. Alternatively we can also use the higher-dimensional van Kampen theorem of Brown and Higgins; see [13,14,15,11] and [18, §6], stating that (under mild conditions) the fundamental crossed module functor preserves colimits.…”

Link invariants from finite categorical groups

“…This invariant I G (K) depends only on the homotopy type of the complement C K [15,16], thus it is a function of the knot group alone. Our main insight is that imposing a suitable restriction on the type of such colourings, and then counting the possibilities, gives a finer invariant.…”

“…We observe that the quotient π 1 (Y D )/im ∂ is isomorphic to the fundamental group of the link complement C K = π 1 (S 3 \ n(K)) for any diagram D, since quotienting π 1 (Y D ) by im ∂ corresponds to imposing the Wirtinger relations [8], which produces the Wirtinger presentation of π 1 (C K ), coming from the particular choice of diagram. Thus Π 2 (X D , Y D ), whilst not being itself a link invariant (unless [15,16] considered up to crossed module homotopy), contains an important link invariant, namely π 1 (C K ), by taking the above quotient. The guiding principle in the construction to follow is to extract additional Reidemeister invariant information from the crossed module Π 2 (X D , Y D ).…”