382 research outputs found
Polynomial-Time Homology for Simplicial Eilenberg-MacLane Spaces
In an earlier paper of ÄŚadek, VokĹ™Ănek, Wagner, and the present authors, we investigated an algorithmic problem in computational algebraic topology, namely, the computation of all possible homotopy classes of maps between two topological spaces, under suitable restriction on the spaces. We aim at showing that, if the dimensions of the considered spaces are bounded by a constant, then the computations can be done in polynomial time. In this paper we make a significant technical step towards this goal: we show that the Eilenberg-MacLane space , represented as a simplicial group, can be equipped with polynomial-time homology (this is a polynomial-time version of effective homology considered in previous works of the third author and co-workers). To this end, we construct a suitable discrete vector field, in the sense of Forman's discrete Morse theory, on . The construction is purely combinatorial and it can be understood as a certain procedure for reducing finite sequences of integers, without any reference to topology. The Eilenberg-MacLane spaces are the basic building blocks in a Postnikov system, which is a "layered” representation of a topological space suitable for homotopy-theoretic computations. Employing the result of this paper together with other results on polynomial-time homology, in another paper we obtain, for every fixed k, a polynomial-time algorithm for computing the kth homotopy group Ď€ k (X) of a given simply connected space X, as well as the first k stages of a Postnikov system forX, and also a polynomial-time version of the algorithm of ÄŚadek etal. mentioned abov
Constructive Algebraic Topology
The classical ``computation'' methods in Algebraic Topology most often work
by means of highly infinite objects and in fact +are_not+ constructive. Typical
examples are shown to describe the nature of the problem. The Rubio-Sergeraert
solution for Constructive Algebraic Topology is recalled. This is not only a
theoretical solution: the concrete computer program +Kenzo+ has been written
down which precisely follows this method. This program has been used in various
cases, opening new research subjects and producing in several cases significant
results unreachable by hand. In particular the Kenzo program can compute the
first homotopy groups of a simply connected +arbitrary+ simplicial set.Comment: 24 pages, background paper for a plenary talk at the EACA Congress of
Tenerife, September 199
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
Simplicial commutative algebras with vanishing Andre-Quillen homology
In this paper, we study the Andr\'e-Quillen homology of simplicial
commutative -algebras, a field, having certain vanishing
properties. When has non-zero characteristic, we obtain an algebraic
version of a theorem of J.-P. Serre and Y. Umeda that characterizes such
simplicial algebras having bounded homotopy groups. We further discuss how this
theorem fails in the rational case and, as an application, indicate how the
algebraic Serre theorem can be used to resolve a conjecture of D. Quillen for
algebras of finite type over Noetherian rings, which have non-zero
characteristic.Comment: 11 page
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