## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 24, Number 4 (2017), 501-522.

### Linear systems over $\mathbb Z[Q_{16}]$ and roots of maps of some 3-complexes into $M_{Q_{16}}$

#### 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 $H^3(W;\mathbb Z)=0$.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 24, Number 4 (2017), 501-522.

**Dates**

First available in Project Euclid: 4 January 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1515035004

**Digital Object Identifier**

doi:10.36045/bbms/1515035004

**Mathematical Reviews number (MathSciNet)**

MR3743259

**Zentralblatt MATH identifier**

06848698

**Subjects**

Primary: 55M20: Fixed points and coincidences [See also 54H25] 55S35: Obstruction theory 55N25: Homology with local coefficients, equivariant cohomology

**Keywords**

quaternion group quaternionic determinant strongly surjective map

#### Citation

Aniz, Claudemir. 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 (2017), no. 4, 501--522. doi:10.36045/bbms/1515035004. https://projecteuclid.org/euclid.bbms/1515035004