On higher analogs of topological complexity

Farber introduced a notion of topological complexity \TC(X) that is related to robotics. Here we introduce a series of numerical invariants \TC_n(X), n=1,2, ... such that \TC_2(X)=\TC(X) and \TC_n(X)\le \TC_{n+1}(X). For these higher complexities, we define their symmetric versions that can also be regarded as higher analogs of the symmetric topological complexity.Comment: LATEX, 8 page

On analytical applications of stable homotopy (the Arnold conjecture, critical points)

We prove the Arnold conjecture for closed symplectic manifolds with $\pi_2(M)=0$ and \cat M=\dim M. Furthermore, we prove an analog of the Lusternik-Schnirelmann theorem for functions with ``generalized hyperbolicity'' property.Comment: AMSTEX, 12 pages, submitted to Math. Zeitschrift, improvement (correction) of the line of the proof of the Arnold conjectur

Systoles of 2-complexes, Reeb graph, and Grushko decomposition

Let X be a finite 2-complex with unfree fundamental group. We prove lower bounds for the area of a metric on X, in terms of the square of the least length of a noncontractible loop in X. We thus establish a uniform systolic inequality for all unfree 2-complexes. Our inequality improves the constant in M. Gromov's inequality in this dimension. The argument relies on the Reeb graph and the coarea formula, combined with an induction on the number of freely indecomposable factors in Grushko's decomposition of the fundamental group. More specifically, we construct a kind of a Reeb space ``minimal model'' for X, reminiscent of the ``chopping off long fingers'' construction used by Gromov in the context of surfaces. As a consequence, we prove the agreement of the Lusternik-Schnirelmann and systolic categories of a 2-complex.Comment: 29 pages; to appear in Int. Math. Res. Notice

Maps of Degree One, Lusternik Schnirelmann Category, and Critical Points

Let $Crit M$ denote the minimal number of critical points (not necessarily non-degenerate) on a closed smooth manifold $M$. We are interested in the evaluation of $Crit$. It is worth noting that we do not know yet whether $Crit M$ is a homotopy invariant of $M$. This makes the research of $Crit$ a challenging problem. In particular, we pose the following question: given a map $f: M \to N$ of degree 1 of closed manifolds, is it true that $Crit M \geq Crit N$? We prove that this holds in dimension 3 or less. Some high dimension examples are considered. Note also that an affirmative answer to the question implies the homotopy invariance of $Crit$; this simple observation is a good motivation for the research