  We give explicit examples of pairs of one-ended, open 4-manifolds whose
end-sums yield uncountably many manifolds with distinct proper homotopy types.
This answers strongly in the affirmative a conjecture of Siebenmann regarding
the nonuniqueness of end-sums. In addition to the construction of these
examples, we provide a detailed discussion of the tools used to distinguish
them; most importantly, the end-cohomology algebra. Key to our Main Theorem is
an understanding of this algebra for an end-sum in terms of the algebras of the
summands together with ray-fundamental classes determined by the rays used to
perform the end-sum. Differing ray-fundamental classes allow us to distinguish
the various examples, but only through the subtle theory of infinitely
generated abelian groups. An appendix is included which contains the necessary
background from that area.
