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We introduce properties of metric spaces and, specifically, finitely generated groups with word metrics, which we call coarse coherence and coarse regular coherence. They are geometric counterparts of the classical algebraic notion of coherence and the regular coherence property of groups defined and studied by Waldhausen. The new properties can be defined in the general context of coarse metric geometry and are coarse invariants. In particular, they are quasi-isometry invariants of spaces and groups. The new framework allows us to prove structural results by developing permanence properties, including the particularly important fibering permanence property, for coarse regular coherence. [ABSTRACT FROM AUTHOR]

In [17], we proved that, among 85 families of ℚ-Fano threefold weighted complete intersections of codimension two, 19 families consist of birationally rigid varieties and the remaining families consists of birationally non-rigid varieties. The aim of this paper is to study systematically the remaining families and prove that every quasismooth member of 14 families is birational to another ℚ-Fano threefold but not birational to any other Mori fiber space. [ABSTRACT FROM AUTHOR]

Let k be non–negative integer. The unoriented bordism classes, which can be represented as [RP(ξ k)] where ξ k is a k–plane bundle, form an ideal of the unoriented bordism ring MO*. A group of generators of this ideal expressed by a base of MO* and a necessary and sufficient condition for a bordism class to belong to this ideal are given. [ABSTRACT FROM AUTHOR]