#
The Homotopy Groups of Knots I. How to Compute the Algebraic 3-type

**by**
###
Samuel J. Lomonaco, Jr.^{(*)}

**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_{-},
K_{0} = 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 p_{1}(K),
the second homotopy group

**p**_{2}(K)
as a p_{1}(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
(S**^{4}, kS^{2}).
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.

___________________________________

(*) Partially supported by the L-O-O-P Fund.