Tight complexes in 3-space admit perfect discrete Morse functions
  In 1967, Chillingworth proved that all convex simplicial 3-balls are collapsible. Using the classical notion of tightness, we generalize this to arbitrary manifolds: We show that all tight simplicial 3-manifolds admit some perfect discrete Morse function. We also strengthen Chillingworth's theorem by proving that all convex simplicial 3-balls are non-evasive. In contrast, we show that many non-evasive 3-balls are not convex. 
Karim Adiprasito and Bruno Benedetti
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