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Elmas Irmak - One of the best experts on this subject based on the ideXlab platform.

  • injective Simplicial Maps of the arc complex on nonorientable surfaces
    arXiv: Geometric Topology, 2008
    Co-Authors: Elmas Irmak
    Abstract:

    We prove that each injective Simplicial Map from the arc complex of a compact, connected, nonorientable surface with nonempty boundary to itself is induced by a homeomorphism of the surface. We also prove that the automorphism group of the arc complex is isomorphic to the quotient of the Mapping class group of the surface by its center.

  • injective Simplicial Maps of the arc complex
    arXiv: Geometric Topology, 2006
    Co-Authors: Elmas Irmak, John D Mccarthy
    Abstract:

    In this paper, we prove that each injective Simplicial Map of the arc complex of a compact, connected, orientable surface with nonempty boundary is induced by a homeomorphism of the surface. We deduce, from this result, that the group of automorphisms of the arc complex is naturally isomorphic to the extended Mapping class group of the surface, provided the surface is not a disc, an annulus, a pair of pants, or a torus with one hole. We also show, for each of these special exceptions, that the group of automorphisms of the arc complex is naturally isomorphic to the quotient of the extended Mapping class group of the surface by its center.

  • superinjective Simplicial Maps of complexes of curves and injective homomorphisms of subgroups of Mapping class groups
    Topology, 2004
    Co-Authors: Elmas Irmak
    Abstract:

    Abstract Let S be a closed, connected, orientable surface of genus at least 3, C (S) be the complex of curves on S and Mod S ∗ be the extended Mapping class group of S. We prove that a Simplicial Map, λ : C (S)→ C (S) , preserves nondisjointness (i.e. if α and β are two vertices in C (S) and i(α,β)≠0, then i(λ(α),λ(β))≠0) iff it is induced by a homeomorphism of S. As a corollary, we prove that if K is a finite index subgroup of Mod S ∗ and f : K→Mod S ∗ is an injective homomorphism, then f is induced by a homeomorphism of S and f has a unique extension to an automorphism of Mod S ∗ .

  • superinjective Simplicial Maps of complexes of curves and injective homomorphisms of subgroups of Mapping class groups
    arXiv: Geometric Topology, 2002
    Co-Authors: Elmas Irmak
    Abstract:

    Let R be a compact, connected, orientable surface of genus g with p boundary components. Let C(R) be the complex of curves on R and Mod_R^* be the extended Mapping class group of R. Suppose that either g = 2 and p > 1 or g > 2 and p >= 0. We prove that a Simplicial Map lambda from C(R) to C(R) is superinjective if and only if it is induced by a homeomorphism of R. As a corollary, we prove that if K is a finite index subgroup of Mod_R^* and f is an injective homomorphism from K to Mod_R^*, then f is induced by a homeomorphism of R and f has a unique extension to an automorphism of Mod_R^*. This extends the author's previous results about closed connected orientable surfaces of genus at least 3, to the surface R.

Paris Luis - One of the best experts on this subject based on the ideXlab platform.

Ilbira Sabahattin - One of the best experts on this subject based on the ideXlab platform.

Zhechev, Stephan Y - One of the best experts on this subject based on the ideXlab platform.

  • IST Austria Thesis
    IST Austria, 2019
    Co-Authors: Zhechev, Stephan Y
    Abstract:

    The first part of the thesis considers the computational aspects of the homotopy groups πd(X) of a topological space X. It is well known that there is no algorithm to decide whether the fundamental group π1(X) of a given finite Simplicial complex X is trivial. On the other hand, there are several algorithms that, given a finite Simplicial complex X that is simply connected (i.e., with π1(X) trivial), compute the higher homotopy group πd(X) for any given d ≥ 2. However, these algorithms come with a caveat: They compute the isomorphism type of πd(X), d ≥ 2 as an abstract finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of πd(X). We present an algorithm that, given a simply connected space X, computes πd(X) and represents its elements as Simplicial Maps from suitable triangulations of the d-sphere Sd to X. For fixed d, the algorithm runs in time exponential in size(X), the number of simplices of X. Moreover, we prove that this is optimal: For every fixed d ≥ 2, we construct a family of simply connected spaces X such that for any Simplicial Map representing a generator of πd(X), the size of the triangulation of S d on which the Map is defined, is exponential in size(X). In the second part of the thesis, we prove that the following question is algorithmically undecidable for d < ⌊3(k+1)/2⌋, k ≥ 5 and (k, d) ̸= (5, 7), which covers essentially everything outside the meta-stable range: Given a finite Simplicial complex K of dimension k, decide whether there exists a piecewise-linear (i.e., linear on an arbitrarily fine subdivision of K) embedding f : K ↪→ Rd of K into a d-dimensional Euclidean space

  • Computing Simplicial representatives of homotopy group elements
    'Springer Science and Business Media LLC', 2018
    Co-Authors: Filakovský Marek, Franek Peter, Wagner Uli, Zhechev, Stephan Y
    Abstract:

    A central problem of algebraic topology is to understand the homotopy groups () of a topological space X. For the computational version of the problem, it is well known that there is no algorithm to decide whether the fundamental group 1() of a given finite Simplicial complex X is trivial. On the other hand, there are several algorithms that, given a finite Simplicial complex X that is simply connected (i.e., with 1() trivial), compute the higher homotopy group () for any given ≥2 . However, these algorithms come with a caveat: They compute the isomorphism type of () , ≥2 as an abstract finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of () . Converting elements of this abstract group into explicit geometric Maps from the d-dimensional sphere to X has been one of the main unsolved problems in the emerging field of computational homotopy theory. Here we present an algorithm that, given a simply connected space X, computes () and represents its elements as Simplicial Maps from a suitable triangulation of the d-sphere to X. For fixed d, the algorithm runs in time exponential in size() , the number of simplices of X. Moreover, we prove that this is optimal: For every fixed ≥2 , we construct a family of simply connected spaces X such that for any Simplicial Map representing a generator of () , the size of the triangulation of on which the Map is defined, is exponential in size()

Korkmaz Mustafa - One of the best experts on this subject based on the ideXlab platform.