2,270 research outputs found
Tropical curves, graph complexes, and top weight cohomology of M_g
We study the topology of a space parametrizing stable tropical curves of
genus g with volume 1, showing that its reduced rational homology is
canonically identified with both the top weight cohomology of M_g and also with
the genus g part of the homology of Kontsevich's graph complex. Using a theorem
of Willwacher relating this graph complex to the Grothendieck-Teichmueller Lie
algebra, we deduce that H^{4g-6}(M_g;Q) is nonzero for g=3, g=5, and g at least
7. This disproves a recent conjecture of Church, Farb, and Putman as well as an
older, more general conjecture of Kontsevich. We also give an independent proof
of another theorem of Willwacher, that homology of the graph complex vanishes
in negative degrees.Comment: 31 pages. v2: streamlined exposition. Final version, to appear in J.
Amer. Math. So
Spectral preorder and perturbations of discrete weighted graphs
In this article, we introduce a geometric and a spectral preorder relation on
the class of weighted graphs with a magnetic potential. The first preorder is
expressed through the existence of a graph homomorphism respecting the magnetic
potential and fulfilling certain inequalities for the weights. The second
preorder refers to the spectrum of the associated Laplacian of the magnetic
weighted graph. These relations give a quantitative control of the effect of
elementary and composite perturbations of the graph (deleting edges,
contracting vertices, etc.) on the spectrum of the corresponding Laplacians,
generalising interlacing of eigenvalues.
We give several applications of the preorders: we show how to classify graphs
according to these preorders and we prove the stability of certain eigenvalues
in graphs with a maximal d-clique. Moreover, we show the monotonicity of the
eigenvalues when passing to spanning subgraphs and the monotonicity of magnetic
Cheeger constants with respect to the geometric preorder. Finally, we prove a
refined procedure to detect spectral gaps in the spectrum of an infinite
covering graph.Comment: 26 pages; 8 figure
Quiver grassmannians, quiver varieties and the preprojective algebra
Quivers play an important role in the representation theory of algebras, with
a key ingredient being the path algebra and the preprojective algebra. Quiver
grassmannians are varieties of submodules of a fixed module of the path or
preprojective algebra. In the current paper, we study these objects in detail.
We show that the quiver grassmannians corresponding to submodules of certain
injective modules are homeomorphic to the lagrangian quiver varieties of
Nakajima which have been well studied in the context of geometric
representation theory. We then refine this result by finding quiver
grassmannians which are homeomorphic to the Demazure quiver varieties
introduced by the first author, and others which are homeomorphic to the
graded/cyclic quiver varieties defined by Nakajima. The Demazure quiver
grassmannians allow us to describe injective objects in the category of locally
nilpotent modules of the preprojective algebra. We conclude by relating our
construction to a similar one of Lusztig using projectives in place of
injectives.Comment: 30 pages. v2: minor corrections and notation changes, some proofs
simplified. v3: Some statements and their proofs corrected. This version
incorporates an erratum to the published version. See Appendix B for detail
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