Geodesic
Truncated Quillen complexes of $p$-groups
Journal of Algebraic Combinatorics, Tome 40 (2014) no. 3, pp. 771-784.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Let $p$ be an odd prime and let $P$ be a $p$-group. We examine the order complex of the poset of elementary abelian subgroups of $P$ having order at least $p^2$. Bouc and Thévenaz showed that this complex has the homotopy type of a wedge of spheres. We show that, for each nonnegative integer $l$, the number of spheres of dimension $l$ in this wedge is controlled by the number of extraspecial subgroups $X$ of $P$ having order $p^{2l+3}$ and satisfying $\Omega_1(C_P(X))=Z(X)$. We go on to provide a negative answer to a question raised by Bouc and Thévenaz concerning restrictions on the homology groups of the given complex.
Classification : 55U10, 20D15, 20D30, 55P10
Keywords: Quillen complex, $p$-group, homology, order complex 
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     author = {Fumagalli, Francesco and Shareshian, John},
     title = {Truncated {Quillen} complexes of $p$-groups},
     journal = {Journal of Algebraic Combinatorics},
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     number = {3},
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     url = {http://geodesic.mathdoc.fr/item/JAC_2014__40_3_a5/}
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Fumagalli, Francesco; Shareshian, John. Truncated Quillen complexes of $p$-groups. Journal of Algebraic Combinatorics, Tome 40 (2014) no. 3, pp. 771-784. http://geodesic.mathdoc.fr/item/JAC_2014__40_3_a5/