A homotopy theoretical generalisation of the Bestvina-Brady construction
A simplicial complex here means an abstract simplicial complex and we assume that it always contains the empty set \(\emptyset\). For a finite simplicial complex \(K\) with the vertex set \([m]=\{1,2,\ldots ,m\}\) and the topological pair \((X,A)\), let \((X,A)^K\) denote the polyhedral product defined by \((X,A)^K=\bigcup_{\sigma\in K}(X,A)^{\sigma}\), where \((X,A)^{\sigma}=\{(x_1,\dots ,x_m)\in X^m:x_i\in\text{ if }i\notin\sigma\}\). When \((X,A)=(S^1,*)\), we write \(T_K=(S^1,*)^K\). For a finite flag complex \(L\), let \(G_L\) denote the right-angled Artin group associated to \(L\), and for an epimorphism \(\phi :G_L\to\mathbb{Z}\) let \(H_L\) denote the Bestivina-Brady group defined by the kernel of \(\phi\). When \(f:L\to K\) is a simplicial map and \(X\) is a commutative monoid with a submonoid \(A\subset X\), it induces the map \(\tilde{f}:X^n\to X^m\) given by \(\tilde{f}(x)_j=\prod_{f(i)=j}x_i\) and its restriction defines the map \(\tilde{f}:(X,A)^L\to (X,A)^K\), where \(L\) and \(K\) are simplicial complexes on the vertex sets \([n]\) and \([m]\), respectively. In particular, for any simplicial map \(f:L\to K\) and \((X,A)=(S^1,*)\), we denote by \(\tilde{T}_{(f,L,K)}\) the homotopy fibre of the map \(\tilde{f}:(S^1,*)^L=T_L\to (S^1,*)^K=T_K\). When \(L\) is a simplicial complex on \([m]\) and \(K\) is an \((m-1)\)-simplex \(\Delta^{m-1}\), we write \(\tilde{T}_{(f,L)}=\tilde{T}_{(f,L,K)}\). It is known that \(H_L\subset (S^1)^m\) is an \(H\)-space with \(\pi_1(T_L)=G_L\) and that the multiplication on \(S^1\) gives the map \(\mu :T_L\to S^1\) which induces the above epimorphism \(\phi\) on the fundamental groups. Thus, the pullback \(\tilde{T}_L\) of the universal cover of \(S^1\) along \(\mu\) is the space whose fundamental group is \(H_L\). In this paper, the authors study the homotopy theoretical generalization problem of the Bestvina-Brady group. More precisely, they investigate the homotopy type of the space \(\tilde{T}_{(f,L)}\) from the point of view of polyhedral products. In particular, they construct the cellular chain complex of \(\tilde{T}_{(f,L)}\) and its differentials explicitly. As an application they determine the homology of \(\tilde{T}_{(f,L)}\) explicitly and produce several interesting and useful examples.
scientific article; zbMATH DE number 6836769
scientific article