Section One: Isometric Structures
The following definition is due to Kervaire [K2] and has its geo-
metrical origins in the work of J. Levine on the concordance group of
knots [Ll].
Let R be one of the ring of integers, % , the field of rational
numbers, Q , or any field, and e = _+l .
Definition 1.1. An e-symmetric isometric structure over R is a
triple (M, , ,t) where M is a finitely generated torsion free
R-module, , is an e-symmetric bilinear form on M with values in
R and t is an R-linear endomorphism satisfying:
i) , is nonsingular, that is, the adjoint homomorphism,
Ad , :M - HornR(M,R) given by Ad , (m) = c p where cp(y) = m,y ,
is an isomorphism.
ii) t(x),y = x,(l-t)(y) for all x,y in M .
Remark: One may also consider the isometry case by requiring the inver-
tibility of the endomorphism and replacing ii) by t(x),y = x,t (y) .
Two isometric structures are isomorphic if there is an R-module isomor-
phism preserving the bilinear forms and the associated endomorphism.
The rank of M will be the dimension of M % K , where K is the frac-
tion field of R . Since M is torsion free, the mapping i:M -• M ® K
given by i(m) = m & 1 is an injection and we will often identify M
with this image, which is an R-lattice (R-submodule of maximal rank) in
the vector space M ® K . Equivalently, we could view M ® K as the
localization of M with respect to the multiplicative set R\{0} .
Definition 1.2. As isometric structure is metabolic if there is an
R-submodule N satisfying:
i) N = N4" = (x:x,N = {x,n :n in N} = {0}} , the annihilator
of N .
ii) N is invariant under t .
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