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Action minimizing solutions of the Newtonian n-body problem: from homology to symmetry
An action minimizing path between two given configurations, spatial or
planar, of the -body problem is always a true -- collision-free -- solution.
Based on a remarkable idea of Christian Marchal, this theorem implies the
existence of new "simple" symmetric periodic solutions, among which the Eight
for 3 bodies, the Hip-Hop for 4 bodies and their generalizations
Homology operations on homology of quandles
We consider various homological operations on homology of quandles. We
introduce the notion of quandle partial derivatives, and extreme chains on
which appropriate partial derivatives vanish. Extreme chains yield homological
operations. We also consider the degree one homology operations created using
elements of the quandle satisfying the so-called -condition.Comment: 26 pages, 1 figur
Rabinowitz Floer homology and symplectic homology
The Rabinowitz-Floer homology groups are associated to an exact
embedding of a contact manifold into a symplectic manifold
. They depend only on the bounded component of .
We construct a long exact sequence in which symplectic cohomology of maps
to symplectic homology of , which in turn maps to Rabinowitz-Floer homology
, which then maps to symplectic cohomology of . We compute
, where is the unit cosphere bundle of a closed
manifold . As an application, we prove that the image of an exact contact
embedding of (endowed with the standard contact structure) cannot be
displaced away from itself by a Hamiltonian isotopy, provided and
the embedding induces an injection on . In particular, does not
admit an exact contact embedding into a subcritical Stein manifold if is
simply connected. We also prove that Weinstein's conjecture holds in symplectic
manifolds which admit exact displaceable codimension 0 embeddings.Comment: 59 pages, 8 figure
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