Manifold arrangements

Ehrenborg, Richard; Readdy, Margaret

doi:10.1016/j.jcta.2014.03.003

Cited by 7 publications

(9 citation statements)

References 37 publications

(82 reference statements)

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“…Given a manifold M and a finite collection of submanifolds A = {N i }, where M and each N i are smooth without boundaries. As defined in [11], A is said to be a manifold arrangement if it satisfies the Bott's clean intersection property that for every x ∈ M, there exist a neighborhood U of x, a neighborhood W of the origin in R n , a subspace arrangement {V i } in R n and a diffeomorphism φ : U → W such that φ maps x to the origin and maps…”

Section: Manifold Arrangementsmentioning

confidence: 99%

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Cohomology ring of manifold arrangements

Chen¹,

Lü²,

Wu³

2020

Preprint

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Let L be a geometric lattice. We first give the concept of monoidal cosheaf on L, and then define the generalized Orlik-Solomon algebra A * (L, C) over a commutative ring with unit, which is built by the classical Orlik-Solomon algebra and a monoidal cosheaf C as coefficients. Furthermore, we study the cohomology of the complement M(A) of a manifold arrangement A with geometric intersection lattice L in a smooth manifold M without boundary. Associated with A, we construct a monoidal cosheaf Ĉ(A), so that the generalized Orlik-Solomon algebra A * (L, Ĉ(A)) becomes a double complex with suitable multiplication structure and the associated total complex T ot(A * (L, Ĉ(A))) is a differential algebra, also regarded as a cochain complex. Our main result is that H * (T ot(A * (L, Ĉ(A)))) is isomorphic to H * (M(A)) as algebras. Our argument is of topological with the use of a spectral sequence induced by a geometric filtration associated with A. As an application, we calculate the cohomology of chromatic configuration spaces, which agrees with many known results in some special cases. In addition, some explicit formulas are also given in some special cases.

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Section: Manifold Arrangementsmentioning

confidence: 99%

“…In this paper we are concerned with manifold arrangements, introduced in [11]. The notion of manifold arrangements is a generalization for some classical arrangements, such as hyperplane (or subspace) arrangements and the configuration spaces of manifolds.…”

mentioning

confidence: 99%

Cohomology ring of manifold arrangements

Chen¹,

Lü²,

Wu³

2020

Preprint

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“…Also recall that the Euler characteristic with compact support has the following properties (see and for instance): ○

χ_{C} false({p o i n t} false) = 1

; ○

χ_{C} false(X false) = χ_{C} false(Y false)

if X is homeomorphic to Y ; ○

χ_{C} false(X false) = χ_{C} false(Y false)

for any homotopic compact spaces X and Y ; ○

χ_{C} false(X false) = χ_{C} false(A false) + χ_{C} false(X ∖ A false)

for every closed subset

A \subset X

; ○

χ_{C} false(X \times Y false) = χ_{C} false(X false) χ_{C} false(Y false)

; ○If M is an n ‐dimensional manifold (not necessarily compact), then

χ_{C} false(M false) = {(- 1)}^{n} χ false(M false)

, where

χ (M)

is the Euler characteristic of M . …”

Section: On the Euler Characteristic With Compact Support Of The Linkmentioning

confidence: 99%

“…where H • C (X, R) denotes the cohomology with compact supports and real coefficients of the space X . Also recall that the Euler characteristic with compact support has the following properties (see [6] and [7] for instance):…”

Section: On the Euler Characteristic With Compact Support Of The Linkmentioning

confidence: 99%

On the Lê–Milnor fibration for real analytic maps

Menegon

Seade

2016

Mathematische Nachrichten

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“…Remark 1.1. Ehrenborg and Readdy ask in [18] for a natural class of submanifold arrangements where an "arithmetic" Tutte polynomial can be meaningfully defined.…”

mentioning

confidence: 99%

Finitary affine oriented matroids

Delucchi,

Knauer

2020

Preprint

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We initiate the axiomatic study of affine oriented matroids (AOMs) on arbitrary ground sets, obtaining fundamental notions such as minors, reorientations and a natural embedding into the frame work of Complexes of Oriented Matroids. The restriction to the finitary case (FAOMs) allows us to study tope graphs and covector posets, as well as to view FAOMs as oriented finitary semimatroids. We show shellability of FAOMs and single out the FAOMs that are affinely homeomorphic to R n . Finally, we study group actions on AOMs, whose quotients in the case of FAOMs are a stepping stone towards a general theory of affine and toric pseudoarrangements. Our results include applications of the multiplicity Tutte polynomial of group actions of semimatroids, generalizing properties of arithmetic Tutte polynomials of toric arrangements. Contents 1. Introduction 1 2. Affine oriented matroids (AOM) 6 3. Finitary affine oriented matroids (FAOM) 14 4. Basis frames and embeddings into Euclidean space 22 5. Group actions 28 6. Open questions 34 Appendix A. Appendix 36 References 43

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Manifold arrangements

Cited by 7 publications

References 37 publications

Cohomology ring of manifold arrangements

Cohomology ring of manifold arrangements

On the Lê–Milnor fibration for real analytic maps

Finitary affine oriented matroids

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