Hyperbolic manifolds with injectivity radius exceeding 50 log((n+1)!) have fibers of maps to R^m whose k-dimensional cells exceed n in number for any cell structure, when 0 < k < d-m.
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Cellular waists of hyperbolic spaces
Hyperbolic manifolds with injectivity radius exceeding 50 log((n+1)!) have fibers of maps to R^m whose k-dimensional cells exceed n in number for any cell structure, when 0 < k < d-m.