Modules of topological spaces, applications to homotopy limits andE ? structures

but also the Sylow p-subgroup of G together with all fusion among its subgroups, are determined up to isomorphism by the homotopy type of BG ∧ p . Our goal here is to give a direct link between p-local structures and homotopy types which arise from them. This theory tries to make explicit the essence of what it means to be the p-completed classifying space of a finite group, and at the same time yields new spaces which are not of this type, but which still enjoy most of the properties a space of the form BG ∧ p would have. We hope that the ideas presented here will have further applications and generalizations in algebraic topology. But this theory also fits well with certain aspects of modular representation theory. In particular, it may give a way of constructing classifying spaces for blocks in the group ring of a finite group over an algebraically closed field of characteristic p.A saturated fusion system F over a p-group S consists of a set Hom F (P, Q) of monomorphisms, for each pair of subgroups P, Q ≤ S, which form a category under composition, include all monomorphisms induced by conjugation in S, and satisfy certain other axioms formulated by Puig (Definitions 1.1 and 1.2 below). In particular, these axioms are satisfied by the conjugacy homomorphisms in a finite group. We refer to [Pu] and [Pu2] for more details of Puig's work on saturated

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“…Since |L| ∼ = hocolim − −−−− →L ( * ), (1) holds by [HV,Theorem 5.5] (and the basic idea is due to Segal [Se,Proposition B.1] …”

Section: Homotopy Decompositions Of Classifying Spacesmentioning

confidence: 99%

“…Let τ : L − − → (F e ) op be the functor which sends an object (P, E) to E. Then by [HV,Theorem 5.5],…”

Section: And Only If P ≥ Z(q) and P Q Is F -Centric; And If This Holdmentioning

confidence: 99%

The homotopy theory of fusion systems

Broto

Levi

Oliver³

2003

J. Amer. Math. Soc.

267

439

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“…A concise reference is [16]. Our approach will also use the definitions and calculations in [8]. All the functors will be covariant.…”

Section: Preliminariesmentioning

confidence: 99%

“…Let X : C → Spaces and Y : C op → Spaces be two continuous functors. A simplicial set B(Y, C, X) (called the bar construction) is defined in [8] such that the n-cells are…”

Section: Preliminariesmentioning

confidence: 99%

On the exponent of the cokernel of the forget-control map on K₀-groups

Connolly

Prassidis

2002

Fund. Math.

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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%

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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Modules of topological spaces, applications to homotopy limits andE ? structures

Cited by 69 publications

References 18 publications

The homotopy theory of fusion systems

The homotopy theory of fusion systems

On the exponent of the cokernel of the forget-control map on K₀-groups

Topological representations of matroid maps

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Modules of topological spaces, applications to homotopy limits andE ? structures

Cited by 69 publications

References 18 publications

The homotopy theory of fusion systems

The homotopy theory of fusion systems

On the exponent of the cokernel of the forget-control map on K0-groups

Topological representations of matroid maps

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On the exponent of the cokernel of the forget-control map on K₀-groups