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Suppose, just for example, that you have a smooth manifold $M$ of dimension greater than $8$,and a cohomology class $q$ in $H^3(M,{\Bbb R})$. Suppose further that one can represent $q$ with a differential form $w$ on $M$ (at least modulo exact forms, and in certain contexts even with a canonical form) that doesn't vanish at any point. So at each point on $M$, $w$ gives a non-trivial alternating trilinear form on tangent space. Now $GL(8,R)$ acts on the space of alternating trilinear forms and one has an infinite orbit space, a moduli space $S$ for alternating trilinear forms. So $w$ determines a map $v$ from $M$ to $S$.

1) Does $q$ determines this map up to homotopy?

If so, next one might

2) Does $S$ has interesting cohomology? Could describing the cohomology of moduli spaces like $S$ turn out simpler than describing the spaces themselves?

If so, finally, one might as

3) What relation exists between $q$ and the $v^* H(S)$, viewed as characteristic classes classes of some sort.

I mean this example to suggest in an obvious way a general question. I chose $3$ as in $H^3(M,{ \Bbb R})$ because alternating $2$-forms up to equivalence form a finite set; I chose dimension $8$, because up to dimension $6$, anyway, alternating $3$-forms up to equivalence may also form a finite set, but by dimension $9$, a simple dimension count will guarantee that one gets an infinite moduli space. My rudimentary search of the literature indicates that the invariant theory of alternating forms of many variables has not progressed far. I'm referring here to: Cohen, Arjeh M., Helminck, Aloysius G. Trilinear alternating forms on a vector space of dimension $7$. Comm. Algebra 16 (1988), no. 1, 1–25.

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The dimension count is not right. 64 dimensions of automorphisms, 56 dimensions of forms. There are only finitely many orbits over $\C$. – moonface Jan 31 '11 at 18:10
Thanks Tyler, yes that's what I meant, will edit. – David Feldman Jan 31 '11 at 18:27
Right moonface, I'll make it nine. This is an edit of an old email I sent a colleague...I don't know how or when that mistake slipped in. – David Feldman Jan 31 '11 at 18:43

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