Hyperplanes in abelian groups and twisted signatures
| Autor: | Eismeier, Mike Miller, Sagerman, Aiden |
|---|---|
| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.topol.2023.108692 |
| Popis: | 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. Comment: Accepted version, to appear in Topology & its Applications |
| Databáze: | arXiv |
| Externí odkaz: |
|