Homotopy colimits – comparison lemmas for combinatorial applications

Welker, Volkmar; Ziegler, Günter M.; Živaljević, Rade T.

doi:10.1515/crll.1999.509.117

Cited by 56 publications

(90 citation statements)

References 33 publications

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“…In particular, this expresses any quasitoric manifold M as a homotopy colimit, and extends the corresponding result [97,Proposition 5.3] for toric varieties, where homotopy colimits and toric geometry were first associated. The close relationship between derived forms and homotopy colimits actually hinges on the fact that the nerve of cat(K) is the cone on the barycentric subdivision of K, and may therefore be identified with the polyhedral complex P K .…”

Section: Vertex Four -Toric Topologysupporting

confidence: 69%

An invitation to toric topology: vertex four of a remarkable tetrahedron

Buchstaber¹,

Ray²

2008

Toric Topology

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Motivation. Sometime around the turn of the recent millennium, those of us in Manchester and Moscow who had been collaborating since the mid-1990s began using the term toric topology to describe our widening interests in certain well-behaved actions of the torus. Little did we realise that, within seven years, a significant international conference would be planned with the subject as its theme, and delightful Japanese hospitality at its heart.When first asked to prepare this article, we fantasised about an authoritative and comprehensive survey; one that would lead readers carefully through the foothills above which the subject rises, and provide techniques for gaining sufficient height to glimpse its extensive mathematical vistas. All this, and more, would be illuminated by references to the wonderful Osaka lectures! Soon afterwards, however, reality took hold, and we began to appreciate that such a task could not be completed to our satisfaction within the timescale available. Simultaneously, we understood that at least as valuable a service could be rendered to conference participants by an invitation to a wider mathematical audience -an invitation to savour the atmosphere and texture of the subject, to consider its geology and history in terms of selected examples and representative literature, to glimpse its exciting future through ongoing projects; and perhaps to locate favourite Osaka lectures within a novel conceptual framework. Thus was born the Toric Tetrahedron TT , which identifies aspects of algebraic, combinatorial, and symplectic geometry as the precursors of toric topology, and symbolises the powerful mathematical bonds between all four areas.The Tetrahedron is the convex hull of these vertex disciplines, and every point has barycentric coordinates that measure the extent of their respective contributions. We introduce the vertices in chronological order (a mere two years separates

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Section: Vertex Four -Toric Topologysupporting

confidence: 69%

An invitation to toric topology: vertex four of a remarkable tetrahedron

Buchstaber¹,

Ray²

2008

Toric Topology

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“…We content ourselves with listing some facts we'll have use for and refer to [37] or [20] for an introduction to the subject and proofs.…”

Section: Preparationsmentioning

confidence: 99%

The integer cohomology algebra of toric arrangements

Callegaro

Delucchi

2017

Advances in Mathematics

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Abstract. We compute the cohomology ring of the complement of a toric arrangement with integer coefficients and investigate its dependency from the arrangement's combinatorial data. To this end, we study a morphism of spectral sequences associated to certain combinatorially defined subcomplexes of the toric Salvetti category in the complexified case, and use a technical argument in order to extend the results to full generality. As a byproduct we obtain:-a "combinatorial" version of Brieskorn's lemma in terms of Salvetti complexes of complexified arrangements, -a uniqueness result for realizations of arithmetic matroids with at least one basis of multiplicity 1.

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“…The next two results are taken from [22], but are commonly known in the theory of homotopy colimits, see [11,18], and [21]. For the remainder of this section, let D : P → Top and E : Q → Top be diagrams of spaces over posets P and Q.…”

Section: Definition 10 For Anymentioning

confidence: 99%

“…Since homotopy colimits of diagrams of spaces can be used to construct toric varieties [22], one might hope to extend the work of Huh and Katz to non-representable matroids.…”

Section: The Unimodality Of the Whitney Numbersmentioning

confidence: 99%

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Topological representations of matroid maps

Stamps

2012

J Algebr Comb

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The Topological Representation Theorem for (oriented) matroids states that every (oriented) matroid arises from the intersection lattice of an arrangement of codimension one homotopy spheres on a homotopy sphere. In this paper, we use a construction of Engström to show that structure-preserving maps between matroids induce topological mappings between their representations; a result previously known only in the oriented case. Specifically, we show that weak maps induce continuous maps and that this process is a functor from the category of matroids with weak maps to the homotopy category of topological spaces. We also give a new and conceptual proof of a result regarding the Whitney numbers of the first kind of a matroid.

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Homotopy colimits – comparison lemmas for combinatorial applications

Cited by 56 publications

References 33 publications

An invitation to toric topology: vertex four of a remarkable tetrahedron

An invitation to toric topology: vertex four of a remarkable tetrahedron

The integer cohomology algebra of toric arrangements

Topological representations of matroid maps

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