论文标题
阿贝尔群体和扭曲的签名
Hyperplanes in abelian groups and twisted signatures
论文作者
论文摘要
我们调查以下问题：如果$ a $ a $ a'$是有限循环群的产品，什么时候存在同构$ f：a \ to a'$，它保留了坐标超平面的结合（等效地，以便$ f（x）$在$ x $的情况下只有某些坐标为零）？ 我们表明，如果存在这样的同构，则$ a $ a'$具有相同的循环因素；如果所有循环因子的订单都大于$ 2 $，则地图$ f $是对角线的，因此将坐标超平面发送到协调超平面。因此，人们可以从了解结合的知识中恢复坐标超平面。 该结果适应为具有一定多重性属性的不变性应用。 As a model application, we show using twisted signatures that there exists a family of compact 4-manifolds $X(n)$ with $H_1 X(n) = \mathbb Z/n$ with the property that $\prod X(n_i) \cong \prod X(n'_j)$ if and only if the factors may be identified (up to permutation), and that the induced map on first homology is (up to permutation) represented by对角矩阵。
We investigate the following question: if $A$ and $A'$ are products of finite cyclic groups, when does there exist an isomorphism $f: A \to A'$ which preserves the union of coordinate hyperplanes (equivalently, so that $f(x)$ has some coordinate zero if and only if $x$ has some coordinate zero)? We show that if such an isomorphism exists, then $A$ and $A'$ have the same cyclic factors; if all cyclic factors have order larger than $2$, the map $f$ is diagonal up to permutation, hence sends coordinate hyperplanes to coordinate hyperplanes. Thus one can recover the coordinate hyperplanes from knowledge of their union. This result is well-adapted for application to invariants with a certain multiplicativity property. As a model application, we show using twisted signatures that there exists a family of compact 4-manifolds $X(n)$ with $H_1 X(n) = \mathbb Z/n$ with the property that $\prod X(n_i) \cong \prod X(n'_j)$ if and only if the factors may be identified (up to permutation), and that the induced map on first homology is (up to permutation) represented by a diagonal matrix.
