4
1. INTRODUCTION
of C\ which fits into an exact sequence
0 sCi —Ci Q/Z 0.
(For details, see Section 2.1.) In [12, 7] it was shown that Cn contains a
subgroup isomorphic to Z°° by investigating a signature invariant of Qn and
pulling it back via tn.
In spite of the importance of Qn and $
n
in the study of Cn, several inter-
esting questions on their structures have not been answered. For example,
it has not been known whether Qn and Cn have torsion elements. Also there
has been no geometric answer to the question how much structure of Cn can
be revealed via &n.
1.2. Main results
1.2.1. The structure of Qn. As an answer to the above questions, we
give a full calculation of the structure of the limit Qn.
THEOREM
1.1. The group Gn is isomorphic to Z°° 0 (Z/2)°° 0 (Z/4)°°.
Although it is abstractly isomorphic to the integral (algebraic) knot
concordance group of Levine, we do not have any natural identification which
is topologically meaningful. In fact it turns out that, from a topological
viewpoint, their structures are drastically different. It will be discussed in a
later subsection.
In Chapter 3, we construct a complete set of invariants of Qn, and by
realizing and computing them, we prove Theorem 1.1. Briefly, our invariants
of Qn can be described as follows. We need to start with known invariants
of the integral algebraic concordance group Gn,c- An algebraic number z
is called reciprocal if z and
z~x
are conjugate, i.e., if they share the same
irreducible polynomial over Q. It is known that the concordance group of
Seifert matrices maps into the direct sum of Witt groups of nonsingular
hermitian forms on finite dimensional vector spaces over number fields Q(z)
equipped with the involution z =
z~x,
where z runs over reciprocal numbers.
This associates to a Seifert matrix A a Witt class of a hermitian form Az
over Q(z), which is called the ^-primary part of A. The signature of Az
(defined for \z\ 1 only), the modulo 2 residue class of the dimension r of
Az, and the discriminant
disA, = (-l)*^detA,€ ^
z + z
' ^
{uu I u G
Q(^)x}
give rise to invariants of the integral algebraic concordance group.
To construct invariants of Qn, we take "limits" of the above invariants.
Let P be the set of all sequences a = (..., c*2, ai) of reciprocal numbers ai
such that
(ari)r
oti for all i and r. (P can be viewed as the limit of an
inverse system consisting of the sets of reciprocal numbers and morphisms
z
zr.)
Let Po be its subset consisting of a = (a^) with \ai\ = 1. For
an element A in Gn represented by a Seifert matrix A of complexity c, we
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