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Let K be a CW complex
with an aspherical splitting, i.e., with subcomplexes K− and K+ such that (a)
K = K−∪K+ and (b) K−,K0 = K−∩K+,K+ are connected and aspherical. The
main theorem of this paper gives a practical procedure for computing the homology
H∗K of the universal cover K of K. It also provides a practical method for
computing the algebraic 2-type of K, i.e., the triple consisting of the fundamental
group π1K, the second homotopy group π2K as a π1K-module, and the first
k-invariant kK.
The effectiveness of this procedure is demonstrated by letting K denote
the complement of a smooth 2-knot (S4,kS2). Then the above mentioned
methods provide a way for computing the algebraic 2-type of 2-knots, thus
solving problem 36 of R. H. Fox in his 1962 paper, “Some problems in knot
theory.” These methods can also be used to compute the algebraic 2-type of
3-manifolds from their Heegaard splittings. This approach can be applied to many
other well known classes of spaces. Various examples of the computation are
given.
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