New constructions in group homology allow us to manufacture high-dimensional manifolds with controlled simplicial volume. We prove that for every dimension bigger than 3 the set of simplicial volumes of orientable closed connected manifolds is dense in $$\mathbb {R}_{\ge 0}$$ R ≥ 0 . In dimension 4 we prove that every non-negative rational number is the simplicial volume of some orientable closed connected 4-manifold. Our group theoretic results relate stable commutator length to the $$l^1$$ l 1 -semi-norm of certain singular homology classes in degree 2. The output of these results is translated into manifold constructions using cross-products and Thom realisation.

中文翻译：

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单纯体积谱

群同源性的新构造使我们能够制造具有受控单纯体积的高维流形。我们证明，对于每个大于 3 的维度，可定向闭连接流形的单纯体积集在 $$\mathbb {R}_{\ge 0}$$ R ≥ 0 中是密集的。在第 4 维中，我们证明每个非负有理数都是某个可定向的闭连通 4-流形的单纯体积。我们的群论结果将稳定的换向子长度与 2 级中某些奇异同源类的 $$l^1$$l 1 -semi-norm 相关联。这些结果的输出被转换为使用叉积和 Thom 实现的流形结构。