355 research outputs found
A note on approximate limits
The main theorems of this paper are the following theorems. THEOREM 2.7. Let X={Xn, Pmn, IN} be an approximate sequence of non - empty Čech-complete paracompact spaces Xn such that each Pnm(Xm) is dense in Xn, then limX is non-empty and Čech-complete. Moreover, Pn(limX) is dense in Xn for each n∈IN . THEOREM 2.11. Let X={Xn, Pmn, IN} be an approximate inverse sequence of absolute Gδ - space. Then there exist: a) a cofinal subset M = {ni : i∈IN} of IN, b) a usual inverse sequence Y = {Yi , qij , M} such that Yi = Xni and qij = Pii+1 Pi+1j+2 ... pj-1j for each i,j∈IN, c) a homeomorphism H : limX→limY
On real analytic Banach manifolds
Let be a real Banach space with an unconditional basis (e.g.,
Hilbert space), open, a closed split real
analytic Banach submanifold of , a real analytic Banach vector
bundle, and {\Cal A}^E\to M the sheaf of germs of real analytic sections of
. We show that the sheaf cohomology groups H^q(M,{\Cal A}^E) vanish
for all , and there is a real analytic retraction from an
open set with such that for all . Some applications are also given, e.g., we show that any infinite
dimensional real analytic Hilbert submanifold of separable affine or projective
Hilbert space is real analytically parallelizable
Embedding variables in finite dimensional models
Global problems associated with the transformation from the Arnowitt, Deser
and Misner (ADM) to the Kucha\v{r} variables are studied. Two models are
considered: The Friedmann cosmology with scalar matter and the torus sector of
the 2+1 gravity. For the Friedmann model, the transformations to the Kucha\v{r}
description corresponding to three different popular time coordinates are shown
to exist on the whole ADM phase space, which becomes a proper subset of the
Kucha\v{r} phase spaces. The 2+1 gravity model is shown to admit a description
by embedding variables everywhere, even at the points with additional symmetry.
The transformation from the Kucha\v{r} to the ADM description is, however,
many-to-one there, and so the two descriptions are inequivalent for this model,
too. The most interesting result is that the new constraint surface is free
from the conical singularity and the new dynamical equations are linearization
stable. However, some residual pathology persists in the Kucha\v{r}
description.Comment: Latex 2e, 29 pages, no figure
On orientations for gauge-theoretic moduli spaces
Let be a compact manifold, a real elliptic operator on , a Lie
group, a principal -bundle, and the
infinite-dimensional moduli space of all connections on modulo
gauge, as a topological stack. For each , we can
consider the twisted elliptic operator on X. This is a
continuous family of elliptic operators over the base , and so
has an orientation bundle , a principal -bundle parametrizing orientations of
KerCoker at each . An
orientation on is a trivialization .
In gauge theory one studies moduli spaces of connections
on satisfying some curvature condition, such as anti-self-dual
instantons on Riemannian 4-manifolds . Under good conditions is a smooth manifold, and orientations on pull back to
orientations on in the usual sense under the inclusion . This is important in areas such as Donaldson
theory, where one needs an orientation on to define enumerative
invariants.
We explain a package of techniques, some known and some new, for proving
orientability and constructing canonical orientations on ,
after fixing some algebro-topological information on . We use these to
construct canonical orientations on gauge theory moduli spaces, including new
results for moduli spaces of flat connections on 2- and 3-manifolds,
instantons, Kapustin-Witten and Vafa-Witten equations on 4-manifolds, and the
Haydys-Witten equations on 5-manifolds.
Two sequels arXiv:1811.02405, arXiv:1811.09658 discuss orientations in 7 and
8 dimensions.Comment: 60 pages. (v2) sections 2.3-2.5 rewritte
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