Dotterrer, Dominic; Kaufman, Tali; Wagner, UliIST Austria
We give a detailed and easily accessible proof of Gromov’s Topological Overlap Theorem. Let X be a finite simplicial complex or, more generally, a finite polyhedral cell complex of dimension d. Informally, the theorem states that if X has sufficiently strong higher-dimensional expansion properties (which generalize edge expansion of graphs and are defined in terms of cellular cochains of X) then X has the following topological overlap property: for every continuous map (Formula presented.) there exists a point (Formula presented.) that is contained in the images of a positive fraction (Formula presented.) of the d-cells of X. More generally, the conclusion holds if (Formula presented.) is replaced by any d-dimensional piecewise-linear manifold M, with a constant (Formula presented.) that depends only on d and on the expansion properties of X, but not on M.
Dotterrer D, Kaufman T, Wagner U. On expansion and topological overlap. Geometriae Dedicata. 2018;195(1):307–317. doi:10.1007/s10711-017-0291-4
Dotterrer, D., Kaufman, T., & Wagner, U. (2018). On expansion and topological overlap. Geometriae Dedicata. Springer. https://doi.org/10.1007/s10711-017-0291-4
Dotterrer, Dominic, Tali Kaufman, and Uli Wagner. “On Expansion and Topological Overlap.” Geometriae Dedicata. Springer, 2018. https://doi.org/10.1007/s10711-017-0291-4.
D. Dotterrer, T. Kaufman, and U. Wagner, “On expansion and topological overlap,” Geometriae Dedicata, vol. 195, no. 1. Springer, pp. 307–317, 2018.
Dotterrer D, Kaufman T, Wagner U. 2018. On expansion and topological overlap. Geometriae Dedicata. 195(1), 307–317.
Dotterrer, Dominic, et al. “On Expansion and Topological Overlap.” Geometriae Dedicata, vol. 195, no. 1, Springer, 2018, pp. 307–317, doi:10.1007/s10711-017-0291-4.
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