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(*) Partially supported by the L-O-O-P Fund.
ABSTRACT.
Let K
be a CW complex with an aspherical splitting, i.e., with subcomplexes K-
and K+
such that (a) K = K- UNION K+
and (b) K-,
K0 = K- INTERSECT 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 of computing the algebraic 3-type
of K, i.e., the
triple consisting of the fundamental group p1(K),
the second homotopy group
p2(K)
as a p1(K)-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
3-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 3-type
of 3-manifolds
from their Heegard splittings. This approach can be applied to many well
known classes of spaces. Various examples of the computation are given.
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(*) Partially supported by the L-O-O-P Fund.