THEOREM. The cobordism group of orientable 4k-dimensional LSF spaces is isomorphic to z/(4) and the invariant is given by 7t. (d) i-duality spaces. Of course, LSF spaces may be combined with local orientability to produce a host of characteristic numbers involving u '. Instead, we prefer to strengthen the LSF condition by demanding not just that Sq' vanish on the intersection homology of links of even codimension strata, but that the whole group vanish: Definition. A normal pseudomanifold X is an S-duality space if, for each stratum of odd codimension c = 2k + 1, we have ZHr(L; i?/(2)) = 0 and for each stratum of even codimension c = 2k we have ZH:(L; Z/(2)) = ZHF-1 (L; Z/(2)) = 0 where L is the link of the stratum. For such spaces the middle intersection homology groups admit a Bockstein operation /I: ZH,"(X; Z/(2)) + ZHF-1 (X; Z) whose mod 2 reduction is a Steenrod operation, Sq': ZHF(X; iZ/(2)) + ZHFi (X; Z/(2)). It even turns out that there is a universal coefficient theorem for I-duality spaces, and that the intersection homology of F-duality spaces satisfies Poincare duality over the localization Z,,,. THEOREM 16.5. An orientable n dimensional i-duality space X of odd dimension n = 2k + 1 is the boundary of an orientable S-duality space Y ifand only if the characteristic number vk Sq' t.'(X) vanishes.
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