1,271 research outputs found
Computing simplicial representatives of homotopy group elements
A central problem of algebraic topology is to understand the homotopy groups
of a topological space . For the computational version of the
problem, it is well known that there is no algorithm to decide whether the
fundamental group of a given finite simplicial complex is
trivial. On the other hand, there are several algorithms that, given a finite
simplicial complex that is simply connected (i.e., with
trivial), compute the higher homotopy group for any given .
%The first such algorithm was given by Brown, and more recently, \v{C}adek et
al.
However, these algorithms come with a caveat: They compute the isomorphism
type of , as an \emph{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 -dimensional sphere
to 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 ,
computes and represents its elements as simplicial maps from a
suitable triangulation of the -sphere to . For fixed , the
algorithm runs in time exponential in , the number of simplices of
. Moreover, we prove that this is optimal: For every fixed , we
construct a family of simply connected spaces 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
Homotopy Type of the Boolean Complex of a Coxeter System
In any Coxeter group, the set of elements whose principal order ideals are
boolean forms a simplicial poset under the Bruhat order. This simplicial poset
defines a cell complex, called the boolean complex. In this paper it is shown
that, for any Coxeter system of rank n, the boolean complex is homotopy
equivalent to a wedge of (n-1)-dimensional spheres. The number of such spheres
can be computed recursively from the unlabeled Coxeter graph, and defines a new
graph invariant called the boolean number. Specific calculations of the boolean
number are given for all finite and affine irreducible Coxeter systems, as well
as for systems with graphs that are disconnected, complete, or stars. One
implication of these results is that the boolean complex is contractible if and
only if a generator of the Coxeter system is in the center of the group. of
these results is that the boolean complex is contractible if and only if a
generator of the Coxeter system is in the center of the group.Comment: final version, to appear in Advances in Mathematic
The Relative Power of Composite Loop Agreement Tasks
Loop agreement is a family of wait-free tasks that includes set agreement and
simplex agreement, and was used to prove the undecidability of wait-free
solvability of distributed tasks by read/write memory. Herlihy and Rajsbaum
defined the algebraic signature of a loop agreement task, which consists of a
group and a distinguished element. They used the algebraic signature to
characterize the relative power of loop agreement tasks. In particular, they
showed that one task implements another exactly when there is a homomorphism
between their respective signatures sending one distinguished element to the
other. In this paper, we extend the previous result by defining the composition
of multiple loop agreement tasks to create a new one with the same combined
power. We generalize the original algebraic characterization of relative power
to compositions of tasks. In this way, we can think of loop agreement tasks in
terms of their basic building blocks. We also investigate a category-theoretic
perspective of loop agreement by defining a category of loops, showing that the
algebraic signature is a functor, and proving that our definition of task
composition is the "correct" one, in a categorical sense.Comment: 18 page
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