论文标题
高斯总和代数和拓扑
Gauss Sums in Algebra and Topology
论文作者
论文摘要
我们考虑与功能相关的高斯总和$ t \ to \ to \ mathbb r/\ mathbb z $，它满足了某种二次属性并研究其基本属性。这些属性和由于dirichlet所致的19世纪的高斯总和公式提供了Milgram Gauss Sum公式计算签名mod $ 8 $ 8 $的非单明双线性形式，而不是$ \ m athbb q $。布朗在非单明积分形式的签名mod 8上得出了一些结果。柯比（Kirby）和梅尔文（Melvin）给出了这种不变性形式的概括的公式，我们在这里进一步将其推广。 Milgram Gauss和公式和这些公式使我们能够在不诉诸于WITT组计算的情况下否定Brown的结果。假设有一些代数拓扑，我们将莫里塔（Morita）的定理计算签名mod $ 8 $从pontrjagin Square的定向的庞加莱双重性空间中，而无需使用Bockstein光谱序列。由于我们使用可能是单数的形式，因此我们还获得了莫里塔（Morita）定理的带有边界的庞加莱空间定理。最后，我们将结果应用于$ h^1（m; \ Mathbb z/2 \ Mathbb z）$ sq^1x \ cup y $的双线性形式。
We consider Gauss sums associated to functions $T\to \mathbb R/\mathbb Z$ which satisfy some sort of quadratic property and investigate their elementary properties. These properties and a Gauss sum formula from the nineteenth century due to Dirichlet give the Milgram Gauss sum formula computing the signature mod $8$ of a non-singular bilinear form over $\mathbb Q$. Brown derived some results on the signature mod 8 of non-singular integral forms. Kirby and Melvin gave a formula for a generalization of this invariant to possibly non-singular forms and we further generalize it here. The Milgram Gauss sum formula and these formulas allow us to reprove Brown's result without resort to Witt group calculations. Assuming a bit of algebraic topology, we reprove a theorem of Morita's computing the signature mod $8$ of an oriented Poincaré duality space from the Pontrjagin square without using Bockstein spectral sequences. Since we work with forms which may be singular, we also obtain a version of Morita's theorem for Poincaré spaces with boundary. Finally we apply our results to the bilinear form $Sq^1x\cup y$ on $H^1(M;\mathbb Z/2\mathbb Z)$ of an orientable 3-manifold.
