17,772 research outputs found
A classification of smooth embeddings of 3-manifolds in 6-space
We work in the smooth category. If there are knotted embeddings S^n\to R^m,
which often happens for 2m<3n+4, then no concrete complete description of
embeddings of n-manifolds into R^m up to isotopy was known, except for disjoint
unions of spheres. Let N be a closed connected orientable 3-manifold. Our main
result is the following description of the set Emb^6(N) of embeddings N\to R^6
up to isotopy.
The Whitney invariant W : Emb^6(N) \to H_1(N;Z) is surjective. For each u \in
H_1(N;Z) the Kreck invariant \eta_u : W^{-1}u \to Z_{d(u)} is bijective, where
d(u) is the divisibility of the projection of u to the free part of H_1(N;Z).
The group Emb^6(S^3) is isomorphic to Z (Haefliger). This group acts on
Emb^6(N) by embedded connected sum. It was proved that the orbit space of this
action maps under W bijectively to H_1(N;Z) (by Vrabec and Haefliger's
smoothing theory). The new part of our classification result is determination
of the orbits of the action. E. g. for N=RP^3 the action is free, while for
N=S^1\times S^2 we construct explicitly an embedding f : N \to R^6 such that
for each knot l:S^3\to R^6 the embedding f#l is isotopic to f.
Our proof uses new approaches involving the Kreck modified surgery theory or
the Boechat-Haefliger formula for smoothing obstruction.Comment: 32 pages, a link to http://www.springerlink.com added, to appear in
Math. Zei
Non-linear finite -symmetries and applications in elementary systems
In this paper it is stressed that there is no {\em physical} reason for
symmetries to be linear and that Lie group theory is therefore too restrictive.
We illustrate this with some simple examples. Then we give a readable review on
the theory finite -algebras, which is an important class of non-linear
symmetries. In particular, we discuss both the classical and quantum theory and
elaborate on several aspects of their representation theory. Some new results
are presented. These include finite coadjoint orbits, real forms and
unitary representation of finite -algebras and Poincare-Birkhoff-Witt
theorems for finite -algebras. Also we present some new finite -algebras
that are not related to embeddings. At the end of the paper we
investigate how one could construct physical theories, for example gauge field
theories, that are based on non-linear algebras.Comment: 88 pages, LaTe
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