Bilinear and quadratic forms on torsion modules
Bilinear and quadratic forms on torsion modules
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扭转模块上的双线性和二次形式
DOI：
10.1016/0001-8708(77)90002-0
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发表时间：
1977
影响因子：
1.7
通讯作者：
Alan Durfee
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文献类型：
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作者：
Alan Durfee
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This paper investigates symmetric bilinear and quadratic forms on torsion modules over a Dedekind domain R with values in F/R, where F is the field of fractions of R; we show that these forms are related to the usual theory of bilinear forms on projective modules over R and its localizations. The algebraic results are of two types: classification results for complete discrete valuation rings (Sect. 3), and stable results for arbitrary Dedekind domains (Sects. 4, 5). Applications to the classical theory of bilinear and quadratic forms over R are also obtained. Although our interest is primarily in the integers Z and the p-adic integers Z,, these rings contain most of the difficulties of the general case. In Section 6 we specialize to the integers and apply the algebraic results to differential topology by classifying certain (n-2)-connected (2n-1)-manifolds. Links of isolated complex hypersurface singularities provide examples of such manifolds (Sect. 7).A symmetric bilinear form on a finite Abelian group first appeared as the linking form on the torsion subgroup of the first homology group of an oriented 3-manifold, and provided a convenient invariant to distinguish such manifolds as lens spaces. Partial algebraic classification of such forms was achieved by Seifert [27], Van Kampen [31], and Wall [33, 351. In higher dimensions, a linking form may be defined on the torsion subgroup of the (n-I)-dimensional homology of an oriented (2n-I)-manifold; this form is symmetric when 12 is even, and skew-symmetric when n is odd.(The classification of skew-symmetric forms is easy; see, for instance [33].) Quadratic forms on finite groups then appear as a refinement of the linking form when n is even, and figure in the classification of these manifolds [34].