Thus, a question about exceptional Spin(7)-structures on 8-manifolds studied by D. Joyce and others is motivated by a problem in systolic geometry. Let us state the question in more detail.
The problem of the computation of the optimal middle-dimensional stable systolic ratio of certain 8-manifolds (see preprint by Bangert, Shnider, Katz, and Weinberger on the arxiv) leads to the following question.
The problem: one is looking for a quotient of an 8-dimensional flat torus T^8 by a finite group G with the following properties:
(a) the quotient T/G is simply connected;
(b) the fourth Betti number b_4 is equal to one, b_4(T/G)=1;
(c) the space T/G is a Poincare duality space.
(d) G is a subgroup of Spin(7) inside SO(8).
The homology in question is the ordinary singular (or de Rham) homology of a topological space. The finite group G acting on the torus has to preserve the structure of a Spin(7) space. Namely, it has to preserve a certain 4-form called the Cayley form. This form is the basic building block for Joyce manifolds with exceptional Spin(7) holonomy. Equivalently, G is a subgroup of Spin(7) sitting inside SO(8) in the standard way.
It is interesting to note that the densest lattice in dimension 7 is the root lattice of the Lie algebra E_7, closely related to the Cayley form, see the preprint mentioned above. Is there a way of exploiting this fact so as to obtain a suitable quotient of a suitable 8-torus with the properties enumerated above?
One natural source of examples of flat orbifolds is actions of Weyl
groups on the maximal tori. One could formulate the following
question. Is there a Lie algebra of rank 8 such that the (orientation
preserving part of the) action of its Weyl group is a subgroup of the
8-dimensional representation of Spin(7)? Is this the case for E_8?
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