A comparison of the Georgescu and Vasy spaces associated to the N-body problems and applications

Abstract

We show that the space introduced by Vasy in order to construct a pseudodifferential calculus adapted to the N -body problem can be obtained as the primitive ideal spectrum of one of the N -body algebras considered by Georgescu. In the process, we provide an alternative description of the iterated blow-up space of a manifold with corners with respect to a clean semilattice of adapted submanifolds (i.e. p-submanifolds). Since our constructions and proofs rely heavily on manifolds with corners and their submanifolds, we found it necessary to clarify the various notions of submanifolds of a manifold with corners. CONTENTSIntroduction 1. Manifolds with corners and their submanifolds 1.1. Manifolds with corners 1.2. The boundary and boundary faces 1.3. Submanifolds of manifolds with corners 2. The blow-up for manifolds with corners 2.1. Definition of the blow-up and its smooth structure 2.2. Exploiting the local structure of the blow-up 2.3. Cleanly intersecting families and liftings 3. The graph blow-up 3.1. Definition of the graph blow-up 3.2. Disjoint submanifolds 4. Iterated blow-ups 4.1. Definition of the iterated blow-up 4.2. Clean semilattices 4.3. The pair blow-up lemma 5. Applications to the N -body problem 5.1. Spherical compactifications 5.2. Quotients and compactifications 5.3. Induced maps on C * -algebras 5.4. Identification of the Georgescu and Vasy spaces Appendix A. Proper maps Appendix B. Submanifold Criteria References B.A. has been partially supported by SPP 2026 (Geometry at infinity) and the SFB 1085 (Higher Invariants), both funded by the DFG (German Science Foundation). J.M. and V.N. have been partially supported by ANR-14-CE25-0012-01 (SINGSTAR) funded by ANR (French Science Foundation).

Products of manifolds with fibered corners

“…Many body spaces. Many body compactifications of vector spaces go back at least to [31], and have been discussed more recently in [21,6]. Using notation from [21], recall that given a finite dimensional vector space V and a linear system S V , meaning a finite set of subspaces of V which is closed under intersection and contains {0} and V , the many body compactification M (V ) of V is the manifold with corners…”

“…Fibered corners structures arise in particular in two settings: the first setting is the resolution of (smoothly) stratified spaces [5,3,4,1], which are often equipped with wedge (or 'iterated incomplete edge') metrics, Riemannian metrics degenerating conically along the strata in an iterated fashion. The second setting is typified in simplest form by many body spaces, which are vector spaces which have been radially compactified and subsequently lown up along the boundaries of a family of linear subspaces [31,21,6]. Euclidean metrics on the original vector spaces become quasi-asymptotically conic (QAC) metrics on the associated many body spaces, and this asymptotic geometry generalizes to the manifold setting in the form of QAC manifolds [12] and even more generally in the form of 'quasi-fibered boundary' (QFB, aka Φ) manifolds [10], extending the scattering and fibered boundary structures of [27] and [26] on manifolds with boundary, respectively, as well as the quasi-asymptotically locally euclidean (QALE) metrics introduced by Joyce [17,19,8].…”

“…The paper [20] extends the notion of refinement and generalized blow-up to the category [18] of manifolds with generalized corners. 6 Or more generally, a generalized blow-up.…”