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

Abstract

We find lower bounds on the topological complexity of the critical (values) sets Σ(F ) ⊂ Y of generic smooth maps F : X → Y , as well as on the complexity of the fibers F −1 (y) ⊂ 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.

Using simplicial volume to count multi-tangent trajectories of traversing vector fields

“…The main theorem is a variant of the theorem "Δ-Inequality for Generic Maps" in Section 3.3 of Gromov's paper [5]. It requires the notion of simplicial volume, which was introduced in [4] and is defined as follows.…”

“…In Sect. 2 we summarize which properties of traversally generic vector fields are needed in order to apply the methods of Gromov from [5]. In Sect.…”

Applications of Almgren-Pitts Min-max theory

“…With that in mind, Gromov [17,18,19] and Guth [20] studied psweepouts of M . Heuristically, given a simplicial complex X, a continuous map Φ :…”

“…Denoting byλ p the cup product ofλ with itself p-times, Guth [20] and Gromov [17,18,19] studied continuous maps Φ from a cellular complex X into Z n (M ; Z 2 ) that detectλ p i.e., Φ * (λ p ) = 0. We now describe some of their results.…”

Stable maps and branched shadows of 3-manifolds

“…Costantino-Thurston [12] and Gromov [15] gave, independently, a linear lower bound of the number of non-simple crossings of stable maps that a closed 3-manifold admits in terms of Gromov norm.…”