178 research outputs found

    Manifolds of isospectral arrow matrices

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    An arrow matrix is a matrix with zeroes outside the main diagonal, first row, and first column. We consider the space MStn,λM_{St_n,\lambda} of Hermitian arrow (n+1)×(n+1)(n+1)\times (n+1)-matrices with fixed simple spectrum λ\lambda. We prove that this space is a smooth 2n2n-manifold, and its smooth structure is independent on the spectrum. Next, this manifold carries the locally standard torus action: we describe the topology and combinatorics of its orbit space. If n⩾3n\geqslant 3, the orbit space MStn,λ/TnM_{St_n,\lambda}/T^n is not a polytope, hence this manifold is not quasitoric. However, there is a natural permutation action on MStn,λM_{St_n,\lambda} which induces the combined action of a semidirect product Tn⋊ΣnT^n\rtimes\Sigma_n. The orbit space of this large action is a simple polytope. The structure of this polytope is described in the paper. In case n=3n=3, the space MSt3,λ/T3M_{St_3,\lambda}/T^3 is a solid torus with boundary subdivided into hexagons in a regular way. This description allows to compute the cohomology ring and equivariant cohomology ring of the 6-dimensional manifold MSt3,λM_{St_3,\lambda} using the general theory developed by the first author. This theory is also applied to a certain 66-dimensional manifold called the twin of MSt3,λM_{St_3,\lambda}. The twin carries a half-dimensional torus action and has nontrivial tangent and normal bundles.Comment: 29 pages, 8 figure

    Embedding calculus knot invariants are of finite type

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    We show that the map on components from the space of classical long knots to the n-th stage of its Goodwillie-Weiss embedding calculus tower is a map of monoids whose target is an abelian group and which is invariant under clasper surgery. We deduce that this map on components is a finite type-(n-1) knot invariant. We also compute the second page in total degree zero for the spectral sequence converging to the components of this tower as Z-modules of primitive chord diagrams, providing evidence for the conjecture that the tower is a universal finite-type invariant over the integers. Key to these results is the development of a group structure on the tower compatible with connect-sum of knots, which in contrast with the corresponding results for the (weaker) homology tower requires novel techniques involving operad actions, evaluation maps, and cosimplicial and subcubical diagrams.Comment: Revised maps to the infinitesimal mapping space model in Sections 3 and 4 and analysis of cubical diagrams in Section 5. Minor expository and organizational changes throughout. Now 28 pages, 4 figure

    Rational Homotopy Type of Complements of Submanifold Arrangements

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    We will provide an explicit cdga controlling the rational homotopy type of the complement to a smooth arrangement X−∪iZiX-\cup_i Z_i in a smooth compact algebraic variety XX over C\mathbb{C}. This generalizes the corresponding result of Morgan in case of a divisor with normal crossings to arbitrary smooth arrangements. The model is given in terms of the arrangement ZiZ_i and agrees with a model introduced by Chen-L\"u-Wu for computing the cohomology. As an application we reprove a formality theorem due to Feichtner-Yuzvinksy. Then we show that the Kritz-Totaro model computes the rational homotopy type in case of chromatic configuration spaces of smooth compact algebraic varieties

    Noetherianity of representation categories with applications to configuration spaces of graphs

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    In questo lavoro si introduce la teoria della rappresentazione delle categorie e la si applica allo studio dei gruppi di omologia di spazi di configurazioni di grafi. Si conclude il lavoro studiando alcuni aspetti di spazi di configurazioni di alberi con un'azione di un gruppo fissatoope
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