Singularities, Expanders and Topology of Maps. Part 1: Homology Versus Volume in the Spaces of Cycles

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.

[1]  A. Szűcs.  Cobordism of singular maps , 2006 .

[2]  G. Besson,et al.  Minimal entropy and Mostow's rigidity theorems , 1996, Ergodic Theory and Dynamical Systems.

[3]  Topological Invariants for Discrete Group Actions , 2004 .

[4]  Crossing number of links formed by edges of a triangulation , 2001, math/0110174.

[5]  P. Papasoglu.  Cheeger constants of surfaces and isoperimetric inequalities , 2007, 0706.4449.

[6]  Shing-Tung Yau,et al.  Eigenvalues of the laplacian of compact Riemann surfaces and minimal submanifolds , 1980 .

[7]  J. Èliašberg.  SURGERY OF SINGULARITIES OF SMOOTH MAPPINGS , 1972 .

[8]  Simplicial volume of locally symmetric spaces covered by SL3R/SO(3) , 2006 .

[9]  M. Gromov.  Volume and bounded cohomology , 1982 .

[10]  John R Gilbert,et al.  A Separator Theorem for Graphs of Bounded Genus , 1984, J. Algorithms.

[11]  Nicholas J. Korevaar.  Upper bounds for eigenvalues of conformal metrics , 1993 .

[12]  Misha Gromov,et al.  Spaces and Questions , 2010 .

[13]  M. Katz.  Systolic geometry and topology , 2007 .

[14]  Bruce Kleiner,et al.  An isoperimetric comparison theorem , 1992 .

[15]  B. Kalmár.  Cobordism invariants of fold maps , 2007, 0709.0350.

[16]  M. Gromov,et al.  Random walk in random groups , 2003 .

[17]  J. Milnor,et al.  Prospects in Mathematics , 1971 .

[18]  Anton Petrunin,et al.  Metric minimizing surfaces , 1999 .

[19]  Li-Yeng Sung,et al.  Square Integrability and Uniqueness of the Solutions of the Kadomtsev–Petviashvili-I Equation , 1999 .

[20]  Nielsen Methods and Groups Acting on Hyperbolic Spaces , 2002, math/0203014.

[21]  Misha Gromov,et al.  Topological Invariants of Dynamical Systems and Spaces of Holomorphic Maps: I , 1999 .

[22]  M. Gromov.  Carnot-Carathéodory spaces seen from within , 1996 .

[23]  Osamu Saeki,et al.  Singular fibers of stable maps and signatures of 4-manifolds , 2006, 0902.4576.

[24]  F. Ronga.  On multiple points of smooth immersions , 1980 .

[25]  James Lepowsky,et al.  Functional Analysis on the Eve of the 21st Century , 1995 .

[26]  S. Gersten.  Essays in Group Theory , 2011 .

[27]  M. Gromov.  Dimension, non-linear spectra and width , 1988 .

[28]  M. Gromov,et al.  Partial Differential Relations , 1986 .

[29]  D. Sullivan,et al.  String Topology , 1999, math/9911159.

[30]  M. Gromov,et al.  Isoperimetry of Waists and Concentration of Maps , 2003 .

[31]  M. Gromov.  Metric Structures for Riemannian and Non-Riemannian Spaces , 1999 .

[32]  R. Herbert.  Multiple Points of Immersed Manifolds , 1981 .

[33]  M. Gromov.  Filling Riemannian manifolds , 1983 .

[34]  N. Alon,et al.  A separator theorem for nonplanar graphs , 1990 .

[35]  Graham A. Niblo,et al.  Asymptotic invariants of infinite groups , 1993 .

[36]  Osamu Saeki,et al.  Topology of Singular Fibers of Differentiable Maps , 2004, Lecture notes in mathematics.

[37]  N. Monod,et al.  Ideal bicombings for hyperbolic groups and applications , 2003, math/0304278.

[38]  M. Gromov,et al.  Positive Curvature, Macroscopic Dimension, Spectral Gaps and Higher Signatures , 1996 .

[39]  M. Kazarian.  Multisingularities, cobordisms, and enumerative geometry , 2003 .

[40]  L. Clozel.  Démonstration de la conjecture τ , 2003 .

[41]  A. Nabutovsky,et al.  Volume, diameter and the minimal mass of a stationary 1-cycle , 2002, math/0201269.

[42]  Masamichi Takase.  An Ekholm–Szűcs‐type formula for codimension one immersions of 3‐manifolds up to bordism , 2007 .

[43]  Simplicial volume of locally symmetric spaces covered by $${\rm SL}_{3}{\mathbb{R}}/{\rm SO}(3)$$ , 2007 .

[44]  L. Guth.  Minimax Problems Related to Cup Powers and Steenrod Squares , 2007, math/0702066.

[45]  Christopher B. Croke.  A sharp four dimensional isoperimetric inequality , 1984 .

[46]  F. Almgren,et al.  Optimal isoperimetric inequalities , 1985 .

[47]  D. Thurston,et al.  3‐manifolds efficiently bound 4‐manifolds , 2005, math/0506577.