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Let $G$ be a group. Its subgroup lattice, denoted $\Sigma G$, consists of all subgroups of $G$ partially ordered by inclusion. The topology of this poset is quite trivial, since it always has a maximal $(G)$ and minimal $(1)$ element. On the other hand, we can remove these offending maximal and minimal elements to get something interesting: define the reduced lattice $\Sigma'G$ which contains all the non-trivial proper subgroups of $G$ ordered by inclusion.

Define an operator $\mathcal{L}$ as follows $$\mathcal{L}G = \pi_1(\Sigma'G)$$ i.e., $\mathcal{L}$ maps each group $G$ to the fundamental group of $\Sigma'G$, choosing arbitrary basepoints in each connected component if necessary. Note that $\mathcal{L}$ is not an endo-functor on the category of groups since reduced lattices need not map to reduced lattices under a group homomorphism (a non-trivial kernel might map to $1$ for instance).

Which groups are fixed (up to isomorphism) by $\mathcal{L}$, i.e., satisfy $\mathcal{L}G \simeq G$?

There is tons of literature on subgroup lattices (including a paper which seemed relevant but turned out to only consider intervals in the lattice) so I'm not sure where to look for helpful results. I don't expect a complete solution, it will be nice to even just have some idea of what types of groups can arise as fundamental groups of reduced lattices.

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    $\begingroup$ Are there any non-trivial examples where ${\mathcal L}G$ is computed? $\endgroup$
    – user6976
    Jul 29, 2013 at 22:44
  • $\begingroup$ In the paper the OP linked it points out that it is known that for a finite group you get something shellable (which implies homotopy equivalent to a wedge of spheres) iff the group is solvable. $\endgroup$
    – Benjamin Steinberg
    Jul 30, 2013 at 0:33
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    $\begingroup$ arxiv.org/pdf/0911.4136.pdf computes the homotopy type of PSl(2,7) $\endgroup$
    – Benjamin Steinberg
    Jul 30, 2013 at 0:38
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    $\begingroup$ math.cornell.edu/~kbrown/ramras.pdf discusses examples of simple groups which give wedges of circles, so free groups. It is conjectured that finite groups give only wedges of spheres of possibly different dimensions. $\endgroup$
    – Benjamin Steinberg
    Jul 30, 2013 at 0:46
  • $\begingroup$ The published paper has more info than what was in the thesis, I think (or at least better proofs). The arXiv version is probably good enough: arXiv:math/0210001. Probably anything I know (knew?) about this is in there... $\endgroup$
    – Dan Ramras
    Aug 1, 2013 at 6:04

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