146 research outputs found
Eliminating higher-multiplicity intersections in the metastable dimension range
The -fold analogues of Whitney trick were `in the air' since 1960s.
However, only in this century they were stated, proved and applied to obtain
interesting results. Here we prove and apply a version of the -fold Whitney
trick when general position -tuple intersections have positive dimension.
Theorem. Assume that is disjoint union of
-dimensional disks, , and a proper PL (smooth)
map such that . If the map
extends to
, then there is a proper PL (smooth) map such that on and .Comment: 13 pages, 2 figures, exposition improve
A new invariant and parametric connected sum of embeddings
We define an isotopy invariant of embeddings N -> R^m of manifolds into
Euclidean space. This invariant together with the \alpha-invariant of
Haefliger-Wu is complete in the dimension range where the \alpha-invariant
could be incomplete. We also define parametric connected sum of certain
embeddings (analogous to surgery). This allows to obtain new completeness
results for the \alpha-invariant and the following estimation of isotopy
classes of embeddings.
For the piecewise-linear category, a (3n-2m+2)-connected n-manifold N and
(4n+4)/3 < m < (3n+3)/2 each preimage of \alpha-invariant injects into a
quotient of H_{3n-2m+3}(N), where the coefficients are Z for m-n odd and Z_2
for m-n even.Comment: 13 pages, to appear in Fundamenta Mathematica
Discrete field theory: symmetries and conservation laws
We present a general algorithm constructing a discretization of a classical
field theory from a Lagrangian. We prove a new discrete Noether theorem
relating symmetries to conservation laws and an energy conservation theorem not
based on any symmetry. This gives exact conservation laws for several discrete
field theories: electrodynamics, gauge theory, Klein-Gordon and Dirac ones. In
particular, we construct a conserved discrete energy-momentum tensor,
approximating the continuum one at least for free fields. The theory is stated
in topological terms, such as coboundary and products of cochains.Comment: 40 pages, 7 figures; exposition improve
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