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Given a group, there is another way to define its "(co-)homology" using a classifying space. Specifically, one takes the partially ordered set of its proper non-trivial subgroups (if they exist), and defines the (co-)homology of the group to be the singular (co-)homology of the classifying space of the partially ordered set viewed as a small category. There are many different way to apply this method, for example, one can take the $p$-subgroups instead; or one could take the poset of subgroups which containing some particular subgroup. And there has been some known interesting results, see for example

Brown, K. Euler characteristics of groups: the $p$-fractional part. (1975)

Quillen, D. Homotopy Properties of the Poset of Nontrivial $p$-Subgroups of a Group (1978)

Group (co-)homology obtained this way, in general is quite different from the usual ones (for discrete groups or topological groups). The classifying spaces here tend to be contractible frequently. Typically, people use this for finite groups, which sometimes provides links to homological theory of rings (see, e.g. Stanley–Reisner rings).

This method could be easily generalized to well-powered categories (i.e. categories whose objects have small collections of sub-objects). Yet, I always thought this is not the "right" kind of (co-)homology theory.

Is my feeling heuristically correct or wrong, or is it just some random feeling? I would also like to know if there is any interesting/useful application of this approach (other than groups). Thanks!

Addendum Dan and Ralph's comments made me realize that it was my own wishful thinking to name this "group (co-)homology". Quillen certainly defined these for group and used these in his above-cited article, but he never explicitly called them group (co-)homology. Of course, everything is useful in some sense; I guess by "not right" I meant that they can not provide much information about the group itself. But then you'll ask me what I mean by "much information"....

So let me only ask this: What are the other interesting applications of this kind of (co-)homology theory using posets of sub-objects in other areas, excluding applications to general posets and similar applications to groups? (Apologies for this messy post.)

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What does it mean to be the "right" kind of (co)homology? There are different kinds of cohomology suited to studying different kinds of phenomena. –  Qiaochu Yuan Aug 14 '11 at 21:12
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Considered as a functor of the underlying group, I don't think this will satisfy any of the properties one would expect of a "cohomology theory." That doesn't mean it's useless, it's just a different sort of invariant. –  Dan Ramras Aug 14 '11 at 21:17
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Can you give a reference where group cohomology is defined in the way you described it ? Brown surely doesn't use this definition (while applying posets of subgroups is folklore in group cohomology - see for instance Brown's book). –  Ralph Aug 14 '11 at 22:18
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up vote 4 down vote accepted

So the general idea is to take a particular lattice $L$ of subobjects, remove the top and bottom element to get $L - \{0, 1\}$, and then study the homology of the nerve of the poset $P$.

There is an old theorem of Jon Folkman: if the lattice $L$ is a geometric lattice, then the nerve of $P$ turns out to be a bouquet of spheres of dimension given by the length of any maximal ascending path in $P$, and the number of spheres is given by the value $\mu(0, 1)$ of the Möbius function of the lattice $L$ (see Euler characteristic of a poset).

  • J. Folkman, The homology groups of a lattice, J. Math. Mech. 15 (1966), 631-636.

This would apply to some of the examples you mention, which are finite modular lattices and thus a fortiori geometric lattices. The typical examples of non-modular geometric lattices, on the other hand, tend to come from matroid theory.

Another application is to free Lie algebras: as shown in some beautiful work of Joyal, in

  • A. Joyal, Foncteurs analytiques et espèces de structures. Combinatoire Énumérative (Lecture Notes in Mathematics 1234), Springer-Verlag, 1985

the $S_n$ representation on the space of Lie polynomials in $n$ distinct indeterminates is closely related to the homology of lattices of equivalence relations (again these are geometric lattices).

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Thanks, Todd! The first application was mentioned by Quillen in his paper I cited above as well. It's related to Cohen-Macaulay posets (i.e. posets whose Stainley-Reisner rings are Cohen-Macaulay). Quillen claimed to generalized Folkman's proof (to prove something else). But your second one is new to me. :) –  Jizhan Hong Aug 15 '11 at 13:34
    
Sorry, my mistake. Quillen didn't claim to have generalized it. I misread his statement. He just sketched a proof of a special case of it. –  Jizhan Hong Aug 15 '11 at 18:37
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I like the following application in group cohomology: Let $G$ be a group of finite virtual cohomological dimension and let $$H^\ast_G(\Delta;M) = H^\ast(EG \times_G \Delta;M)$$ be the $G$-equivariant cohomology of $\Delta$ where $\Delta$ is the classifying space of the poset of the non-trivial elementary abelian $p$-subgroups of $G$. Then the projection $\Delta \to \ast$ induces an isomorphism $$H^i(G;\mathbb{F}_p) = H^i_G(\ast;\mathbb{F}_p) \to H^i_G(\Delta;\mathbb{F}_p)$$ for $i > \text{dim } \Delta + \text{ vcd } G$. This can be used to cleanly state finite generation results: $H^\ast_G(\Delta;\mathbb{F}_p)$ is a finitely generated $\mathbb{F}_p$-algebra (under some weak conditions) while $H^\ast(G;\mathbb{F}_p)$ is usually not finitely generated (as can be seen by taking $G$ a free group of infinite rank). This can be found in the following 1998 paper

Lee: On the finite generation of high dimensional cohomology ring of virtually torsion-free groups.

As another application in group cohomology may serve the formula in Corollary V.3.3 in Adem's book (Cohomology of Finite Groups) that has been used in the computation of the cohomology of sporaid simple groups.

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