Singular Homology Theory

Massey, William S.

doi:10.1007/978-1-4684-9231-6

Cited by 170 publications

(91 citation statements)

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“…However, 0 ∈ R n/2 , so A is a proper closed subset of R n/2 and hence it is a proper closed subspace of a compact manifold (sphere) of dimension n/2 as well. Since n/2 ≤ m − 1 for n ≥ 8, by [4,Proposition 6.5], H m−1 (A) = H m (A) = 0 (we are using that A is a CW-complex soCech cohomology coincides with singular cohomology). Hence the exactness of the sequence in (16) implies…”

Section: 2mentioning

confidence: 99%

The structure of symmetric n-player games when influence and independence collide

Molitor¹,

Steel²,

Taylor³

2015

Advances in Applied Mathematics

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Abstract. We study the mathematical properties of probabilistic processes in which the independent actions of n players ('causes') can influence the outcome of each player ('effects'). In such a setting, each pair of outcomes will generally be statistically correlated, even if the actions of all the players provide a complete causal description of the players' outcomes, and even if we condition on the outcome of any one player's action. This correlation always holds when n = 2, but when n = 3 there exists a highly symmetric process, recently studied, in which each cause can influence each effect, and yet each pair of effects is probabilistically independent (even upon conditioning on any one cause). We study such symmetric processes in more detail, obtaining a complete classification for all n ≥ 3. Using a variety of mathematical techniques, we describe the geometry and topology of the underlying probability space that allows independence and influence to coexist.

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Section: 2mentioning

confidence: 99%

The structure of symmetric n-player games when influence and independence collide

Molitor¹,

Steel²,

Taylor³

2015

Advances in Applied Mathematics

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“…It is well known that classical singular homology of topological spaces (usually defined using singular simplices) can be also obtained by means of singular cubes (see e.g., [17], [21], where the cubical approach is used). A question arises whether other homologies that are special cases of Tierney-Vogel theory admit a cubical description.…”

Section: ) (102)])mentioning

confidence: 99%

Cubical approach to derived functors

Patchkoria

2012

Homology, Homotopy and Applications

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“…. , ξ t ) corresponding to the maximal -ideal m; in symbols, [13], the intuitively obvious fact that V cannot be homeomorphic to the k-ball is confirmed by the verification that the singular homology groups [11] of V and of [e i 1 , . .…”

Section: T) Then η Is Surjective Because B Generates Gmentioning

confidence: 99%

Lattice-ordered Abelian groups and Schauder bases of unimodular fans

Manara

Marra

Mundici

2006

Trans. Amer. Math. Soc.

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Abstract. Baker-Beynon duality theory yields a concrete representation of any finitely generated projective Abelian lattice-ordered group G in terms of piecewise linear homogeneous functions with integer coefficients, defined over the support |Σ| of a fan Σ. A unimodular fan ∆ over |Σ| determines a Schauder basis of G: its elements are the minimal positive free generators of the pointwise ordered group of ∆-linear support functions. Conversely, a Schauder basis H of G determines a unimodular fan over |Σ|: its maximal cones are the domains of linearity of the elements of H. The main purpose of this paper is to give various representation-free characterisations of Schauder bases. The latter, jointly with the De Concini-Procesi starring technique, will be used to give novel characterisations of finitely generated projective Abelian lattice ordered groups. For instance, G is finitely generated projective iff it can be presented by a purely lattice-theoretical word.

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Singular Homology Theory

Cited by 170 publications

References 0 publications

The structure of symmetric n-player games when influence and independence collide

The structure of symmetric n-player games when influence and independence collide

Cubical approach to derived functors

Lattice-ordered Abelian groups and Schauder bases of unimodular fans

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