Abstract.A generic smooth map of a closed 2k-manifold into (3k − 1)-space has a finite number of cusps ( 1,1 -singularities). We determine the possible numbers of cusps of such maps. A fold map is a map with singular set consisting of only fold singularities ( 1,0 -singularities). Two fold maps are left-right fold bordant if there are cobordisms between their source and target manifolds with a fold map extending the two maps between the boundaries. If the two targets agree and the target cobordism can be taken as a product with a unit interval, then the maps are fold cobordant. Cobordism classes of fold maps are known to form groups. We compute these groups for fold maps of (2k − 1)-manifolds into (3k − 2)-space. Analogous cobordism semigroups for arbitrary closed (3k − 2)-dimensional target manifolds are endowed with Abelian group structures and described. Left-right fold bordism groups in the same dimensions are also described.
In frozen percolation, i.i.d. uniformly distributed activation times are assigned to the edges of a graph. At its assigned time, an edge opens provided neither of its endvertices is part of an infinite open cluster; in the opposite case, it freezes. Aldous (2000) showed that such a process can be constructed on the infinite 3-regular tree and asked whether the event that a given edge freezes is a measurable function of the activation times assigned to all edges. We give a negative answer to this question, or, using an equivalent formulation and terminology introduced by Aldous and Bandyopadhyay (2005), we show that the recursive tree process associated with frozen percolation on the oriented binary tree is nonendogenous. An essential role in our proofs is played by a frozen percolation process on a continuous-time binary Galton Watson tree that has nice scale invariant properties.
In frozen percolation, i.i.d. uniformly distributed activation times are assigned to the edges of a graph. At its assigned time, an edge opens provided neither of its endvertices is part of an infinite open cluster; in the opposite case, it freezes. Aldous (2000) showed that such a process can be constructed on the infinite 3-regular tree and asked whether the event that a given edge freezes is a measurable function of the activation times assigned to all edges. We give a negative answer to this question, or, using an equivalent formulation and terminology introduced by Aldous and Bandyopadhyay ( 2005), we show that the recursive tree process associated with frozen percolation on the oriented binary tree is nonendogenous. An essential role in our proofs is played by a frozen percolation process on a continuous-time binary Galton Watson tree that has nice scale invariant properties.
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