45 research outputs found
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
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 denote the minimal number of critical points (not necessarily
non-degenerate) on a closed smooth manifold . We are interested in the
evaluation of . It is worth noting that we do not know yet whether is a homotopy invariant of . This makes the research of a
challenging problem.
In particular, we pose the following question: given a map of
degree 1 of closed manifolds, is it true that ? 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 ; this simple observation is a good motivation for
the research