Factorization for stacks and boundary complexes

Harper, Alicia

doi:10.48550/arxiv.1706.07999

Cited by 9 publications

(12 citation statements)

References 15 publications

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“…Many applications involve the fact that the homotopy types (and even simple homotopy types) of boundary complexes, the dual complexes of boundary divisors in simple normal crossings compactifications, are independent of the choice of compactification. The same is also true for Deligne-Mumford (DM) stacks [Har17]. Boundary complexes were introduced and studied by Danilov in the 1970s [Dan75], and have become an important focus of research activity in the past few years, with new connections to Berkovich spaces, singularity theory, geometric representation theory, and the minimal model program.…”

Section: Boundary Complexesmentioning

confidence: 94%

Tropical curves, graph complexes, and top weight cohomology of M_g

Chan,

Galatius,

Payne

2018

Preprint

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We study the topology of a space ∆ g 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-Teichmüller Lie algebra, we deduce that H 4g−6 (M g ; Q) is nonzero for g = 3, g = 5, and g ≥ 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. Contents 1. Introduction 1 2. Graphs, tropical curves, and moduli 6 3. Symmetric semi-simplicial objects 13 4. Symmetric ∆-complexes 19 5. Graph complexes and cellular chains on ∆ g 24 6. Boundary complexes 26 7. Applications 34 8. Generalizations of abelian cycles 35 References 37

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Section: Boundary Complexesmentioning

confidence: 94%

Tropical curves, graph complexes, and top weight cohomology of M_g

Chan,

Galatius,

Payne

2018

Preprint

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“…for the (p + 1)-fold iterated fiber product. Define D([p]) ⊂ D [p] as the open subvariety consisting of (p + 1)-tuples of pairwise distinct points in D [p] that all lie over the same point of D. We can then define ∆(D)( If X is proper then the simple homotopy type of this dual complex depends only on the open complement X D [Pay13,Har17], and its reduced rational homology is naturally identified with the top weight cohomology of X D. More precisely, if X has pure dimension d, then (6.1.1)…”

Section: 3mentioning

confidence: 99%

Topology of moduli spaces of tropical curves with marked points

Chan,

Galatius,

Payne

2019

Preprint

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This article is a sequel to [CGP18]. We study a space ∆ g,n of genus g stable, n-marked tropical curves with total edge length 1. Its rational homology is identified both with top-weight cohomology of the complex moduli space M g,n and with the homology of a marked version of Kontsevich's graph complex, up to a shift in degrees.We prove a contractibility criterion that applies to various large subspaces of ∆ g,n . From this we derive a description of the homotopy type of ∆ 1,n , the top weight cohomology of M 1,n as an S n -representation, and additional calculations of H i (∆ g,n ; Q) for small (g, n). We also deduce a vanishing theorem for homology of marked graph complexes from vanishing of cohomology of M g,n in appropriate degrees, by relating both to ∆ g,n . We comment on stability phenomena, or lack thereof.Marked graph complexes. In [CGP18] we gave a cellular chain complex C * (X; Q) associated to any symmetric ∆-complex X, and showed that it computes the reduced rational homology of (the geometric realization of) X. In the case X = ∆ g,n , we are able to deduce that C * (∆ g,n ; Q) is quasi-isomorphic to the marked graph complex G (g,n) , using Theorem 1.1. We shall define G (g,n) precisely in Section 2. Briefly, it is generated by isomorphism classes of connected, loopless, n-marked stable graphs Γ together with the choice of one of the two possible orientations of R E (Γ) . The main difference between G (g,n) and the hairy graph complexes in the existing literature [CKV13, CKV15, KW Ž16, TW17] is that we consider graphs with n labeled markings, rather than n unlabeled hairs or half-edges. 1 We use the terminology "marked" rather than "hairy" for this reason. Kontsevich's graph complex G (g) [Kon93, Kon94] occurs as the special case n = 0.Theorem 1.4. For 2g − 2 + n > 0, there is a natural split injection of chain complexes1 We expect that there may be natural maps between the S n -coinvariants of our marked graph complex and some existing hairy graph complexes, but have been unable to find such precise relations.

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“…This result is briefly mentioned in [Ber17] as Corollary 1.5. Since then, a stronger result by Harper [Har17] on weak factorization of Deligne-Mumford stacks based on destackification has appeared, which makes our result less relevant. Instead of discussing our previous result further, we simply restate Harper's theorem as Theorem D. We also give a minor simplification of its proof.…”

Section: Introductionmentioning

confidence: 93%

Functorial destackification and weak factorization of orbifolds

Bergh¹,

Rydh²

2019

Preprint

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Let X be a smooth and tame stack with finite inertia. We prove that there is a functorial sequence of blow-ups with smooth centers after which the stabilizers of X become abelian. Using this result, we can extend the destackification results of the first author to any smooth tame stack. We give applications to resolution of tame quotient singularities, prime-to-ℓ alterations of singularities and weak factorization of Deligne-Mumford stacks. We also extend the abelianization result to infinite stabilizers in characteristic zero, generalizing earlier work of Reichstein-Youssin.

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Factorization for stacks and boundary complexes

Cited by 9 publications

References 15 publications

Tropical curves, graph complexes, and top weight cohomology of M_g

Tropical curves, graph complexes, and top weight cohomology of M_g

Topology of moduli spaces of tropical curves with marked points

Functorial destackification and weak factorization of orbifolds

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