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december 2017 Linear systems over $\mathbb Z[Q_{16}]$ and roots of maps of some 3-complexes into $M_{Q_{16}}$
Claudemir Aniz
Bull. Belg. Math. Soc. Simon Stevin 24(4): 501-522 (december 2017). DOI: 10.36045/bbms/1515035004

Abstract

Let $\mathbb Z[Q_{16}]$ be the group ring where $Q_{16}=\langle x,y|x^4=y^2,\,xyx=y\rangle$ is the quaternion group of order 16 and $\varepsilon $ the augmentation map. We show that, if $PX=K(x-1)$ and $PX=K(-xy+1)$ has solution over $\mathbb Z[Q_{16}]$ and all $m\times m$ minors of $\varepsilon (P)$ are relatively prime, then the linear system $PX=K$ has a solution over $\mathbb Z[Q_{16}]$, where $P=[p_{ij}]$ is an $m\times n$ matrix with $m\leq n$. As a consequence of such results, we show that there is no map $f:W\to M_{Q_{16}}$ that is strongly surjective, i.e., such that $MR[f,a]=\min\{\#(g^{-1}(a))|g\in [f]\}\neq 0$. Here, $M_{Q_{16}}$ is the orbit space of the 3-sphere $\mathbb S^3$ with respect to the action of $Q_{16}$ determined by the inclusion $Q_{16}\subseteq \mathbb S^3$ and $W$ is a $CW$-complex of dimension 3 with $H^3(W;\mathbb Z)=0$.

Citation

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Claudemir Aniz. "Linear systems over $\mathbb Z[Q_{16}]$ and roots of maps of some 3-complexes into $M_{Q_{16}}$." Bull. Belg. Math. Soc. Simon Stevin 24 (4) 501 - 522, december 2017. https://doi.org/10.36045/bbms/1515035004

Information

Published: december 2017
First available in Project Euclid: 4 January 2018

zbMATH: 06848698
MathSciNet: MR3743259
Digital Object Identifier: 10.36045/bbms/1515035004

Subjects:
Primary: ‎55M20, 55N25, 55S35

Rights: Copyright © 2017 The Belgian Mathematical Society

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Vol.24 • No. 4 • december 2017
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