40,722 research outputs found
Lifting generic maps to embeddings
Given a generic PL map or a generic smooth fold map , where
and , we prove that lifts to a PL or smooth
embedding if and only if its double point locus
admits an equivariant map to
. As a corollary we answer a 1990 question of P. Petersen on whether
the universal coverings of the lens spaces , odd, lift to
embeddings in . We also show that if a non-degenerate
PL map lifts to a topological embedding in then
it lifts to a PL embedding in there.
The Appendix extends the 2-multi-0-jet transversality over the usual
compactification of and Section 3 contains an
elementary theory of stable PL maps.Comment: 37 pages. v4: Added a discussion of stable PL maps (in Section 3) and
the general case of the extended 2-multi-0-jet transversality theorem (in the
end of the Appendix
Hodge cohomology of gravitational instantons
We study the space of L^2 harmonic forms on complete manifolds with metrics
of fibred boundary or fibred cusp type. These metrics generalize the geometric
structures at infinity of several different well-known classes of metrics,
including asymptotically locally Euclidean manifolds, the (known types of)
gravitational instantons, and also Poincar\'e metrics on Q-rank 1 ends of
locally symmetric spaces and on the complements of smooth divisors in K\"ahler
manifolds. The answer in all cases is given in terms of intersection cohomology
of a stratified compactification of the manifold. The L^2 signature formula
implied by our result is closely related to the one proved by Dai [dai] and
more generally by Vaillant [Va], and identifies Dai's tau invariant directly in
terms of intersection cohomology of differing perversities. This work is also
closely related to a recent paper of Carron [Car] and the forthcoming paper of
Cheeger and Dai [CD]. We apply our results to a number of examples,
gravitational instantons among them, arising in predictions about L^2 harmonic
forms in duality theories in string theory.Comment: 45 pages; corrected final version. To appear in Duke Math. Journa
Excitation of a Kaluza-Klein mode by parametric resonance
In this paper we investigate a parametric resonance phenomenon of a
Kaluza-Klein mode in a -dimensional generalized Kaluza-Klein theory. As the
origin of the parametric resonance we consider a small oscillation of a scale
of the compactification around a today's value of it. To make our arguments
definite and for simplicity we consider two classes of models of the
compactification: those by () and those by (, ). For these models we show that
parametric resonance can occur for the Kaluza-Klein mode. After that, we give
formulas of a creation rate and a number of created quanta of the Kaluza-Klein
mode due to the parametric resonance, taking into account the first and the
second resonance band. By using the formulas we calculate those quantities for
each model of the compactification. Finally we give conditions for the
parametric resonance to be efficient and discuss cosmological implications.Comment: 36 pages, Latex file, Accepted for publication in Physical Review
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