Symmetric products of two dimensional complexes

Kallel, Sadok; Salvatore, Paolo

doi:10.1090/conm/407/07675

Cited by 8 publications

(23 citation statements)

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“…Because the differential is trivial, the chain groups agree with the homology groups. This completes the computation in [31] of the homology of Sym k (Σ).…”

Section: This Cell Decomposition Restricts To Cell Decompositions Of supporting

confidence: 53%

“…As in the 1-dimensional case, we do not give the symmetric product itself a cell structure, but we produce a cell complex which is homotopy equivalent to the symmetric product. Following [31], given a 2-complex Y whose 1-skeleton X = Sk 1 Y is a CW complex with a single 0-cell, we define Sym k (Y ) as the quotient of Sym k (Y ) which for x, x ′ ∈ Sym l (X) and y ∈ Sym k−l (Y ) identifies x * y with x ′ * y, if x and x ′ have the same image in Sym l (X). The quotient map (12) r…”

Section: This Cell Decomposition Restricts To Cell Decompositions Of mentioning

confidence: 99%

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On the curvature of vortex moduli spaces

Bökstedt

Romão

2014

Math. Z.

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We use algebraic topology to investigate local curvature properties of the moduli spaces of gauged vortices on a closed Riemann surface. After computing the homotopy type of the universal cover of the moduli spaces (which are symmetric products of the surface), we prove that, for genus g > 1, the holomorphic bisectional curvature of the vortex metrics cannot always be nonnegative in the multivortex case, and this property extends to all Kähler metrics on certain symmetric products. Our result rules out an established and natural conjecture on the geometry of the moduli spaces.

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“…Because the differential is trivial, the chain groups agree with the homology groups. This completes the computation in [31] of the homology of Sym k (Σ).…”

Section: This Cell Decomposition Restricts To Cell Decompositions Of supporting

confidence: 53%

Section: This Cell Decomposition Restricts To Cell Decompositions Of mentioning

confidence: 99%

On the curvature of vortex moduli spaces

Bökstedt

Romão

2014

Math. Z.

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“…We identify the hyperbolic plane H with its Poincaré disc model H = D 2 (the open unit disc in R 2 with the hyperbolic metric). By Lemma 5 of [42], there is a homeomorphism of pairs (D 2n , ∂D 2n = S 2n−1 ) ∼ = (Sym n (D 2 ), Sym n (D 2 ) Sym n (D 2 )), where D 2n is an open 2n-dimensional disc. Thus, we can identify Sym n (H) ∼ = D 2n , with the metric induced by the hyperbolic metric on H, so that Sym n (H)/S n = D 2n /S n .…”

Section: Symmetric Products Orbifold Fundamental Group and Orbifoldmentioning

confidence: 99%

Twisted index theory on orbifold symmetric products and the fractional quantum Hall effect

Marcolli

Seipp

2017

Advances in Theoretical and Mathematical Physics

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We extend the noncommutative geometry model of the fractional quantum Hall effect, previously developed by Mathai and the first author, to orbifold symmetric products. It retains the same properties of quantization of the Hall conductance at integer multiples of the fractional Satake orbifold Euler characteristics. We show that it also allows for interesting composite fermions and anyon representations, and possibly for Laughlin type wave functions.

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“…Write X = w S 1 ∪ (D 2 1 ∪ · · · ∪ D 2 r ) and denote by the symmetric product at the chain level. In [22] we constructed a space SP n X homotopy equivalent to SP n (X) and such that SPX n≥0 SP n X has a multiplicative cellular chain complex generated under by a zero dimensional class v 0 , degree one classes e 1 , . .…”

Section: Two Dimensional Complexesmentioning

confidence: 99%

“…Note that Theorem 1.3 is stated for simply connected spaces. To get connectivity results for reduced symmetric products of a compact Riemann surface for example we use geometric input from Kallel and Salvatore [22]. This applies to any two dimensional complex.…”

Section: Introductionmentioning

confidence: 99%

Symmetric products, duality and homological dimension of configuration spaces

Kallel¹

2008

Geometry and Topology Monographs

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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". 55R80; 55S15, 18G20To Fred Cohen on his 60th birthday 500 Sadok Kallel[8]). Its fundamental group written Br n (M) is the "braid group" of M . The object of this paper is to study the homology of braid spaces and the main approach we adopt is that of duality with the symmetric products. In so doing we take the opportunity to refine and elaborate on some classical material. Next is a brief content summary.Section 2 describes the homotopy type of braid spaces of some familiar spaces and discusses orientation issues. Section 3 introduces truncated products, as in Bödigheimer, Cohen and Milgram [6] and Milgram and Löffler [24], states the duality with braid spaces and then proves our first main result on the cohomological dimension of braid spaces. Section 4 uses truncated product constructions to split in an elementary fashion the homology of braid spaces for punctured manifolds. In Section 5 we prove our sharp connectivity result for reduced symmetric products of CW complexes which seems to be new and a significant improvement on work of Nakaoka and Welcher [42]. In Section 5.2 we make the link between the homology of symmetric and truncated products by discussing a spectral sequence introduced by Bödigheimer, Cohen and Milgram and exploited by them to study "braid homology" H * (B(M, n)). Finally Section 6 completes a left out piece from McDuff and Segal's work on configuration spaces [23]. In that paper, H * (B(M, n)), for closed manifolds M , is compared to the homology of some spaces of "compactly supported vector fields" on M and the main theorem there states that these homologies are isomorphic up to a range that increases with n. We make this range more explicit and use it for example to determine the abelianization of the braid groups of a closed Riemann surface. A final appendix collects some homotopy theoretic properties of section spaces that we use throughout.Below are precise statements of our main results which we have divided up into three main parts. Unless explicitly stated, all spaces are assumed to be connected. The n th symmetric group is written S n . Connectivity and cohomological dimensionFor M a manifold, we write H * (M, ±Z) for the cohomology of M with coefficients in the orientation sheaf ±Z; in other words H * (M, ±Z) is the homology ...

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Symmetric products of two dimensional complexes

Cited by 8 publications

References 9 publications

On the curvature of vortex moduli spaces

On the curvature of vortex moduli spaces

Twisted index theory on orbifold symmetric products and the fractional quantum Hall effect

Symmetric products, duality and homological dimension of configuration spaces

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