On 2-Dimensional Homotopy Invariants of Complements of Knotted Surfaces

We prove that if $M$ is a CW-complex and $\ast$ is a 0-cell of $M$, then the crossed module ${\mathrm{\Pi }}_2(M,M^1,\ast )$ does not depend on the cellular decomposition of $M$ up to free products with ${\mathrm{\Pi }}_2(D^2,S^1,\ast )$, where $M^1$ 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 $\Pi_2(M,M^1,*) \to G$(whichisfinite)canbere-scaledtoahomotopyinvariant$I_G(M)$ (i. e. not dependent on the cellular decomposition of $M$). We describe an algorithm to calculate ${\pi }_2(M,M^{(1)},\ast )$ as a crossed module over ${\pi }_1(M^{(1)},\ast )$, in the case when $M$ is the complement of a knotted surface in $S^4$ 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.