Part of this work was done when the second author was a member of the Hausdorff Center of Mathematics. At the time, he was supported by Wolfgang Lück's ERC Advanced Grant "KL2MG-interactions" (no. 662400) and the DFG Grant under Germany's Excellence Strategy - GZ 2047/1, Projekt-ID 390685813.
Part of this work was also done when both authors were visiting IMPAN at Warsaw during the Simons Semester "Geometric and Analytic Group Theory" which was partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund. We would also like to thank Kai-Uwe Bux for inviting us for a research visit at Bielefeld in May 2019 and many stimulating discussions. Special thanks go to Jonas Flechsig for his comments on preliminary versions of the paper. Furthermore, we want to thank Javier Aramayona and Stefan Witzel for discussions, Andrea Bianchi for comments and Matthew C. B. Zaremsky for some helpful communications and comments.
1. CONNECTIVITY TOOLS
In this section, we review some of the connectivity tools that we need for calculating the connectivity of our spaces. A good reference is [HV17, Section 2] although not all the tools we use can be found there.
1.1. Complete join.
The complete join is useful tool introduced by Hatcher and Wahl in [HW10, Section 3] for proving connectivity results. We review the basics here.
Definition 1.1. A surjective simplicial map $\pi : Y \to X$ is called a complete join if it satisfies the following properties:
(1) $\pi$ is injective on individual simplices.
(2) For each $p$-simplex $\sigma = \langle v_0, \dots, v_p \rangle$ of $X$, $\pi^{-1}(\sigma)$ is the join $\pi^{-1}(v_0) * \pi^{-1}(v_1) * \dots * \pi^{-1}(v_p)$.
Definition 1.2. A simplicial complex $X$ is called weakly Cohen-Macaulay of dimension $n$ if $X$ is $(n-1)$-connected and the link of each $p$-simplex of $X$ is $(n-p-2)$-connected. We sometimes shorten weakly Cohen-Macaulay to wCM.
The main result regarding complete join that we will use is the following.
Proposition 1.3. [HW10, Proposition 3.5] If $Y$ is a complete join complex over a wCM complex $X$ of dimension $n$, then $Y$ is also wCM of dimension $n$.
Remark 1.4. If $\pi: Y \to X$ is a complete join, then $X$ is a retract of $Y$. In fact, we can define a simplicial map $s: X \to Y$ such that $\pi \circ s = \text{id}_X$ by sending a vertex $v \in X$ to any vertex in $\pi^{-1}(v)$ and then extending it to simplices. The fact that $s$ can be extended to simplices is granted by the condition that $\pi$ is a complete join. In particular we can also conclude that if $Y$ is $n$-connected, so is $X$.
1.2. Bad simplices argument.
Let $(X, Y)$ be a pair of simplicial complexes. We want to relate the $n$-connectedness of $Y$ to the $n$-connectedness of $X$ via a so called bad simplices argument, see [HV17, Section 2.1] for more information. One identifies a set of simplices in $X \setminus Y$ as bad simplices, satisfying the following two conditions:
(1) Any simplex with no bad faces is in $Y$, where by a “face” of a simplex we mean a subcomplex spanned by any nonempty subset of its vertices, proper or not.
(2) If two faces of a simplex are both bad, then their join is also bad.
We call simplices with no bad faces good simplices. Bad simplices may have good faces or faces which are neither good nor bad. If $\sigma$ is a bad simplex, we say a simplex $\tau$ in Lk($\sigma$) is good for $\sigma$ if any bad face of $\tau * \sigma$ is contained in $\sigma$. The simplices which are good for $\sigma$ form a subcomplex of Lk($\sigma$) which we denote by $G_{\sigma}$ and call the good link of $\sigma$.
Proposition 1.5. [HV17, Proposition 2.1] Let $X, Y$ and $G_{\sigma}$ be as above. Suppose that for some integer $n \ge 0$ the subcomplex $G_{\sigma}$ of $X$ is $(n - \dim(\sigma) - 1)$-connected for all bad simplices $\sigma$. Then the pair $(X, Y)$ is $n$-connected, i.e. $\pi_i(X, Y) = 0$ for all $i \le n$.