arXiv subjects

Sadok Kallel

Publications and source records attributed to Sadok Kallel.

30 records · Page 2Linked to original sources

arXiv2009

Symmetric products, duality and homological dimension of configuration spaces

We discuss various aspects of `braid spaces' or configuration spaces of unordered points on manifolds. First we describe how the homology of these spaces is affected by puncturing the underlying manifold, hence extending some results of Fred Cohen, Goryunov and Napolitano. Next we obtain a precise bound for the cohomological dimension of braid spaces. This is related to some sharp and useful connectivity bounds that we establish for the reduced symmetric products of any simplicial complex. Our methods are geometric and exploit a dual version of configuration spaces given in terms of truncated symmetric products. We finally refine and then apply a theorem of McDuff on the homological connectivity of a map from braid spaces to some spaces of `vector fields'.

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arXiv2009

Rational maps and string topology

We apply a version of the Chas-Sullivan-Cohen-Jones product on the higher loop homology of a manifold in order to compute the homology of the spaces of continuous and holomorphic maps of the Riemann sphere into a complex projective space. This product makes sense on the homology of maps from a co-H space to a manifold, and comes from a ring spectrum. We also build a holomorphic version of the product for maps of the Riemann sphere into homogeneous spaces. In the continuous case we define a related module structure on the homology of maps from a mapping cone into a manifold, and then describe a spectral sequence that can compute it. As a consequence we deduce a periodicity and dichotomy theorem when the source is a compact Riemann surface and the target is a complex projective space.

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arXiv2008

Symmetric Products, Duality and Homological Dimension of Configuration Spaces

We discuss various aspects of "braid spaces'' or configuration spaces of unordered points on manifolds. First we describe how the homology of these spaces is affected by puncturing the underlying manifold, hence extending some results of Fred Cohen, Goryunov and Napolitano. Next we obtain a precise bound for the cohomological dimension of braid spaces. This is related to some sharp and useful connectivity bounds that we establish for the reduced symmetric products of any simplicial complex. Our methods are geometric and exploit a dual version of configuration spaces given in terms of truncated symmetric products. We finally refine and then apply a theorem of McDuff on the homological connectivity of a map from braid spaces to some spaces of ``vector fields''.

math.AT↗

arXiv2006

Invariant Spin Structures on Riemann Surfaces

We investigate the action of the automorphism group of a closed Riemann surface on its set of theta characteristics (or spin structures). We give criteria for when an automorphism fixes all spin structures, or when it fixes just one. The case of hyperelliptic curves and of the Klein quartic are discussed in detail.

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arXiv2005

Symmetric Products of Two Dimensional Complexes

This paper exhibits a multiplicative and minimal cellular complex which allows explicit and complete (co)homological calculations for the symmetric products of a finite two dimensional CW complex. By considering cohomology, we observe that a classical theorem of Clifford on the dimension of various linear series on a projective curve has a purely topological statement. We give a "real" analog of this theorem for unoriented topological surfaces.

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arXiv2004

Some Remarks on Symmetric Products of Curves

Symmetric products of curves are important spaces for both geometers and topologists, and increasingly useful objects for physicists. We summarize in this note some of their basic homotopy theoretic properties and derive a handful of known and less well-known results about them.

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arXiv2003

On the Group of Automorphisms of Cyclic Covers of the Riemann Sphere

This note uses some recent calculations of Conder and Bujalance (on classifying finite index group extensions of Fuchsian groups with abelian quotient and torsion free kernel) in order to determine the full automorphism groups of some cylic coverings of the line (including curves of Fermat and Lefschetz type). The answer is complete for cyclic covers that branch over three points.

math.AG↗

arXiv2001

Configuration Spaces and the Topology of Curves in Projective Space

We survey and expand on the work of Segal, Milgram and the author on the topology of spaces of maps of positive genus curves into $n$-th complex projective space, $n\geq 1$ (in both the holomorphic and continuous categories). Both based and unbased maps are studied and in particular we compute the fundamental groups of the spaces in question. The relevant case when $n=1$ is given by a non-trivial extension which we fully determine.

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arXiv2000

On the Topology of Fibrations with Section and Free Loop Spaces

We relate the brace products of a fibration with section to the differentials in its serre spectral sequence. In the particular case of free loop fibrations, we establish a link between these differentials and browder operations in the fiber. Applications and several calculations (for the particular case of spheres and wedges of spheres) are given.

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arXiv1999

Genus Zero Actions on Riemann Surfaces

In this paper we determine all finite groups G that can act on some compact Riemann surface M with the property that if H is any non-trivial subgroup of G, then the orbit surface M/H is the Riemann sphere. The idea is to look at the induced action on the vector space of holomorphic differentials on M (in the positive genus case) and then use the old-known (Wolf) classification of groups admitting fixed point-free linear actions. A description of the corresponding group actions is given in terms of Fuchsian representations.

math.AG↗

arXiv1998

An Interpolation between Homology and Stable Homotopy

By considering labeled configurations of ``bounded multiplicity'', one can construct a functor that fits between homology and stable homotopy. Based on previous work, we are able to give an equivalent description of this labeled construction in terms of loop space functors and symmetric products. This yields a direct generalization of the May-Milgram model for iterated loop spaces, and answers questions of Carlsson and Milgram posed in the handbook. We give a classifying space formulation of our results hence extending an older result of Segal. We finally relate our labeled construction to a theory of Lesh and give a generalization of a well-known theorem of Quillen, Barratt and Priddy.

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arXiv1998

Particle Spaces on Manifolds and Generalized Poincaré Dualities

It is quite an interesting phenomenon in Topology that configuration spaces on a manifold M are intrinsically related to certain mapping spaces from M. In this paper we interpret and greatly expand on this relationship. Building (mainly) on work of Segal, we introduce a new class of configuration spaces; the particle spaces, and these include the classical configuration spaces of distinct points, symmetric products, truncated products, divisor spaces, positive and negative particles of McDuff, as well as many others. We show that there are "scanning" maps that relate these spaces of particles on smooth manifolds to section spaces of appropriate bundles. These correspondences we prove are homology equivalences always and in some identifiable cases homotopy equivalences. Applications of this to both algebraic geometry and algebraic topology are given. We recover old as well as prove some new results on configuration spaces, bounded multiplicity polynomials and loop spaces. Our methods provide interesting insight into the stability phenomenon of holomorphic mapping spaces discovered in the early eighties and asserting that some spaces of rational maps give good topological approximations to second loop spaces. Finally, we rediscover a beautiful space level description of the classical duality theorems of Poincaré, Alexander and Lefshetz (first given by Gajer), and further extend it to include Spanier-Whitehead duality.

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