178 research outputs found
Manifolds of isospectral arrow matrices
An arrow matrix is a matrix with zeroes outside the main diagonal, first row,
and first column. We consider the space of Hermitian arrow
-matrices with fixed simple spectrum . We prove
that this space is a smooth -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
, the orbit space is not a polytope, hence
this manifold is not quasitoric. However, there is a natural permutation action
on which induces the combined action of a semidirect product
. The orbit space of this large action is a simple
polytope. The structure of this polytope is described in the paper.
In case , the space 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 using the general theory developed by
the first author. This theory is also applied to a certain -dimensional
manifold called the twin of . 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
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
We will provide an explicit cdga controlling the rational homotopy type of
the complement to a smooth arrangement in a smooth compact
algebraic variety over . 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 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
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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