On 2-Dimensional Homotopy Invariants of Complements of Knotted Surfaces

2005/07/12 by Martins, João Faria
#57M05 #57M27 #57Q45 #FOS: Mathematics #Geometric Topology (math.GT) #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.math/0507239

Abstract

We prove that if M is a CW-complex and * is a 0-cell of M, then the crossed module Π₂(M,M¹,*) does not depend on the cellular decomposition of M up to free products with Π₂(D²,S¹,*), where M¹ is the 1-skeleton of M. 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,M¹,*) → G (which is finite) can be re-scaled to a homotopy invariant I_G(M) (i. e. not dependent on the cellular decomposition of M). We describe an algorithm to calculate π₂(M,M⁽¹⁾,*) as a crossed module over π₁(M⁽¹⁾,*), in the case when M is the complement of a knotted surface in S⁴ and M⁽¹⁾ is the 1-handlebody of a handle decomposition of M, which, in particular, gives a method to calculate the algebraic 2-type of M. In addition, we prove that the invariant I_G yields a non-trivial invariant of knotted surfaces.

Related

• Topological invariants of 3-dimensional manifold with boundary by using crossed module
• Invariants of knotted surfaces from link homology and bridge trisections
• Classifying the surface-knot modules
• Symmetric Products of Two Dimensional Complexes
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