We consider arrangements of n connected codimensional one submanifolds in closed d-dimensional manifold M. Let f be the number of connected components of the complement in M to the union of submanifolds. We prove the sharp lower bound for f via n and homology group H_{d-1}(M). The sets of all possible f-values for given n are studied for hyperplane arrangements in real projective spaces and for subtori arrangements in d-dimensional tori.
Cycles, submanifolds, and structures on normal bundles
Connected components of Isom($\mathbb{H}^3$)-representations of non-orientable surfaces
Connected components of strata of Abelian differentials over Teichmüller space
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