545 research outputs found
d-Representability of simplicial complexes of fixed dimension
Let K be a simplicial complex with vertex set V = {v_1,..., v_n}. The complex
K is d-representable if there is a collection {C_1,...,C_n} of convex sets in
R^d such that a subcollection {C_{i_1},...,C_{i_j}} has a nonempty intersection
if and only if {v_{i_1},...,v_{i_j}} is a face of K.
In 1967 Wegner proved that every simplicial complex of dimension d is
(2d+1)-representable. He also suggested that his bound is the best possible,
i.e., that there are -dimensional simplicial complexes which are not
2d-representable. However, he was not able to prove his suggestion.
We prove that his suggestion was indeed right. Thus we add another piece to
the puzzle of intersection patterns of convex sets in Euclidean space.Comment: 6 pages, 2 figure
Good covers are algorithmically unrecognizable
A good cover in R^d is a collection of open contractible sets in R^d such
that the intersection of any subcollection is either contractible or empty.
Motivated by an analogy with convex sets, intersection patterns of good covers
were studied intensively. Our main result is that intersection patterns of good
covers are algorithmically unrecognizable.
More precisely, the intersection pattern of a good cover can be stored in a
simplicial complex called nerve which records which subfamilies of the good
cover intersect. A simplicial complex is topologically d-representable if it is
isomorphic to the nerve of a good cover in R^d. We prove that it is
algorithmically undecidable whether a given simplicial complex is topologically
d-representable for any fixed d \geq 5. The result remains also valid if we
replace good covers with acyclic covers or with covers by open d-balls.
As an auxiliary result we prove that if a simplicial complex is PL embeddable
into R^d, then it is topologically d-representable. We also supply this result
with showing that if a "sufficiently fine" subdivision of a k-dimensional
complex is d-representable and k \leq (2d-3)/3, then the complex is PL
embeddable into R^d.Comment: 22 pages, 5 figures; result extended also to acyclic covers in
version
Derived Algebraic Geometry
This text is a survey of derived algebraic geometry. It covers a variety of
general notions and results from the subject with a view on the recent
developments at the interface with deformation quantization.Comment: Final version. To appear in EMS Surveys in Mathematical Science
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