129 research outputs found
The topology of closed manifolds with quasi-constant sectional curvature
We prove that closed manifolds admitting a generic metric whose sectional
curvature is locally quasi-constant are graphs of space forms. In the more
general setting of QC spaces where sets of isotropic points are arbitrary,
under suitable positivity assumption and for torsion-free fundamental groups
they are still diffeomorphic to connected sums of spherical space forms and
spherical bundles over the circle.Comment: 56p, JEP to appea
Cubulations, immersions, mappability and a problem of Habegger
The aim of this paper (inspired from a problem of Habegger) is to describe
the set of cubical decompositions of compact manifolds mod out by a set of
combinatorial moves analogous to the bistellar moves considered by Pachner,
which we call bubble moves. One constructs a surjection from this set onto the
the bordism group of codimension one immersions in the manifold. The connected
sums of manifolds and immersions induce multiplicative structures which are
respected by this surjection. We prove that those cubulations which map
combinatorially into the standard decomposition of for large enough
(called mappable), are equivalent. Finally we classify the cubulations of
the 2-sphere.Comment: Revised version, Ann.Sci.Ecole Norm. Sup. (to appear
Surface cubications mod flips
Let be a compact surface. We prove that the set of surface
cubications modulo flips, up to isotopy, is in one-to-one correspondence with
.Comment: revised version, 18
On the TQFT representations of the mapping class groups
We prove that the image of the mapping class group by the representations
arising in the SU(2)-TQFT is infinite, provided that the genus is bigger than 2
and the level r of the theory is different from 2,3,4,6. In particular the
quotient of the mapping class group by the normaizer of the r-th power of a
Dehn twist is infinite if the genus is at least 3 and r is bigger than 12.Comment: 21 pages, 6 eps figures, Latex figures (to appear Pacific.J.Math.
Global classification of isolated singularities in dimensions and
We characterize those closed -manifolds admitting smooth maps into
-manifolds with only finitely many critical points, for .
We compute then the minimal number of critical points of such smooth maps for
and, under some fundamental group restrictions, also for . The main
ingredients are King's local classification of isolated singularities,
decomposition theory, low dimensional cobordisms of spherical fibrations and
3-manifolds topology.Comment: 31p, revised version, Ann. Scuola Norm. Sup. Pisa Cl. Sci., to appea
On the groupoid of transformations of rigid structures on surfaces
We prove that the groupoid of transformations of rigid structures on surfaces
has a finite presentation as a 2-groupoid establishing a result first
conjectured by G.Moore and N.Seiberg. An alternative proof was given by
B.Bakalov and A.Kirillov Jr. We present some applications to TQFTs. This is
also related to recent work on the Grothendieck-Teichmuller groupoid by
P.Lochak, A.Hatcher and L.Schneps.Comment: 38 pages, 35 eps figure
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