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We produce a rational homology 3-sphere that does not smoothly bound either apositive or negative definite 4-manifold. Such a 3-manifold necessarily cannotbe rational homology cobordant to a Seifert fibered space or any 3-manifoldobtained by Dehn surgery on a knot. The proof requires an analysis of shortcharacteristic covectors in bimodular lattices.

3-manifolds that bound no definite 4-manifold