Abstract
We find lower bounds on the topological complexity of the critical (values) sets \({\Sigma(F) \subset Y}\) of generic smooth maps F : X → Y, as well as on the complexity of the fibers \({F^{-1}(y) \subset X}\) in terms of the topology of X and Y, where the relevant topological invariants of X are often encoded in the geometry of some Riemannian metric supported by X.
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Gromov, M. Singularities, Expanders and Topology of Maps. Part 1: Homology Versus Volume in the Spaces of Cycles. Geom. Funct. Anal. 19, 743–841 (2009). https://doi.org/10.1007/s00039-009-0021-7
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DOI: https://doi.org/10.1007/s00039-009-0021-7