We extend Y. Eliashberg's h-principle to smooth maps of surfaces which are allowed to have cusp singularities, as well as folds. More precisely, we prove a necessary and sufficient condition for a given map of surfaces to be homotopic to one with given loci of folds and cusps. Then we use these results to obtain a necessary and sufficient condition for a subset of a surface M to be realizable as the critical set of some generic smooth map from M to a given surface N .arXiv:1810.00205v2 [math.GT]
We present a motivated exposition of the proof of the following Tverberg Theorem: For every integers d, r any (d + 1)(r − 1) + 1 points in R d can be decomposed into r groups such that all the r convex hulls of the groups have a common point. The proof is by wellknown reduction to the Bárány Theorem. However, our exposition is easier to grasp because additional constructions (of an embedding R d ⊂ R d+1 , of vectors ϕ j,i and statement of the Barańy Theorem) are not introduced in advance in a non-motivated way, but naturally appear in an attempt to construct the required decomposition. This attempt is based on rewriting several equalities between vectors as one equality between vectors of higher dimension.
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