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The complex of free factors of a free group. (English) Zbl 0935.20015
The natural inclusion \(\mathbb{Z}^n\to\mathbb{Q}^n\) of the free Abelian group \(\mathbb{Z}^n\) into the vector space \(\mathbb{Q}^n\) gives a one-to-one correspondence between the proper direct summands of \(\mathbb{Z}^n\) and proper subspaces of \(\mathbb{Q}^n\). So the poset of partially ordered sets of direct summands of \(\mathbb{Z}^n\) is isomorphic to the spherical building \(X_n\) for \(\text{GL}(n,\mathbb{Q})\). The Solomon-Tits theorem says that this geometric realization of the poset of proper subspaces of an \(n\)-dimensional vector space has the homotopy type of a bouquet of spheres of dimension \(n-2\) [L. Solomon, Theory of finite groups, Symp. Harvard Univ. 1968, 213-221 (1969; Zbl 0216.08001)]. If \(FC_n\) is the geometric realization of the poset of the free factors of the free group \(F_n\), then this is a natural analogue of \(X_n\). The authors give an analogue of the above-mentioned theorem of Solomon-Tits by proving that the geometric realization of the poset of free factors of \(F_n\) has the homotopy type of a bouquet of spheres of dimension \(n-2\).
For the proof they use a surjective map of the poset \(B_n\) of simplices of a subcomplex of the “sphere complex” defined by the first author onto the poset \((FC_n)^{op}\). Then they give some properties of \(B_n\) and \(FC_n\) and compute the homology of \(B_n\) by applying Quillen’s spectral theory.

MSC:
20E05 Free nonabelian groups
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
57M07 Topological methods in group theory
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