Truncated Quillen complexes of \(p\)-groups (Q472631)

For a partially ordered set \(\mathcal{P}\), the order complex \(\Delta \mathcal{P}\) means the abstract simplicial complex whose \(k\)-dimensional faces are all chains \(x_0<x_1<\cdots <x_k\) of length \(k\) with \(x_i\in\mathcal{P}\). For a prime \(p\) and a group \(G\), let \(\mathcal{A}_p(G)\) denote the poset of \(p\)-subgroups of \(G\) whose order is induced by the inclusion, and the order complex \(\Delta \mathcal{A}_p(G)\) is called the Quillen complex of \(G\). For a finite \(p\)-group \(P\), let \(\mathcal{A}_{\geq 2}(P)\) denote the partially ordered set of elementary abelian subgroups of \(P\) having order at least \(p^2\). It is known that the subcomplex \(\Delta \mathcal{A}_{\geq 2}(P)\) has the homotopy type of a wedge of spheres. In this paper, the authors try to investigate how many spheres of each dimension appear in this wedge decomposition. In particular, they show that for each non-negative integer \(l\) the number of spheres of dimension \(l\) is controlled by the number of extraspecial subgroups \(X\) of \(P\) having order \(p^{2l+3}\) which satisfy the condition \(\Omega_1(C_P(X))=Z(X)\). As an application, they can also give a negative answer to some question raised by Bouc and Thévenaz concerning restrictions on the homology groups of the given complex.