Fundamental Groups of Blow-ups

By Michael W. Davis, Tadeusz Januszkiewicz, R. Scott. 2002

76 citations recorded by the source corpus.

Abstract Many examples of nonpositively curved closed manifolds arise as real blow-ups of projective hyperplane arrangements. If the hyperplane arrangement is associated to a finite reflection group W and if the blow-up locus is W -invariant, then the resulting manifold will admit a cell decomposition whose maximal cells are all combinatorially isomorphic to a given convex polytope P . In other words, M admits a tiling with tile P . The universal covers of such examples yield tilings of R n whose symmetry groups are generated by involutions but are not, in general, reflection groups. We begin a study of these “mock reflection groups”, and develop a theory of tilings that includes the examples coming from blow-ups and that generalizes the corresponding theory of reflection tilings. We apply our general theory to classify the examples coming from blow-ups in the case where the tile P is either the permutohedron or the associahedron.

Subfields: Discrete Mathematics and Combinatorics, Electrical and Electronic Engineering, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a50b330434f02e0f1581b43fc5a1e60577ade932

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In Discrete Mathematics and Combinatorics

In Electrical and Electronic Engineering

In Geometry and Topology

Citations in the corpus

Permutohedra, Associahedra, and Beyond (2005, methodology): “Various graph associahedra, especially those graph associahedra that correspond to Dynkin diagrams and extended Dynkin diagrams, came up earlier in the work of De Concini and Procesi [12] on wonderful models of subspace arrangements and then in the work on Davis–Januszkiewitz–Scott [9, 10].”
The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points (2005, background): “Furthermore, it is explained in [Dev, DJS , HK] that the group Jn has the following presentation: it is generated by elements sp,q, 1 ≤ p < q ≤ n, with defining relations (i) s 2 = 1;”
The Geometry and Topology of Coxeter Groups (2025, background): “It is also the basic mechanism behind the proofs in [84, 85] that various blowups of real projective space are nonpositively curved.”
Crystals and coboundary categories (2004, methodology): “cording to the number of double points. We call a connected component of a stratum a cell. Figure 2 shows a picture of this cell complex2 for n= 3. 3.3. Partial bracketings. Davis-Januszkiewicz-Scott [DJS], Devadoss [De], and Kapranov [Kap] have all studied this cell structure on Mfn+1 0 . Actually,”
Finite index subgroups of graph products (2008, unknown): “[9].”
Generalized cluster complexes and Coxeter combinatorics (2005, unknown): “On the other hand, the fact that the n -cycle is combi-natorially related to the type-B n cluster complex (the n -dimensional cyclohedron) is well known: there is a diﬀerent generalization of associahedra [7, 10, 21] which produces the cyclohedron in case of an n -cycle.”