7 research outputs found
Polynomial-Time Homology for Simplicial Eilenberg-MacLane Spaces
In an earlier paper of Čadek, Vokřínek, 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 π 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 Čadek etal. mentioned abov
Effective homology for homotopy colimit and cofibrant replacement
We extend the notion of simplicial set with effective homology to diagrams of
simplicial sets. Further, for a given finite diagram of simplicial sets such that each simplicial set has
effective homology, we present an algorithm computing the homotopy colimit
as a simplicial set with effective homology. We also give
an algorithm computing the cofibrant replacement of as a
diagram with effective homology. This is applied to computing of equivariant
cohomology operations
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
Computing all maps into a sphere
Given topological spaces X and Y, a fundamental problem of algebraic topology
is understanding the structure of all continuous maps X -> Y . We consider a
computational version, where X, Y are given as finite simplicial complexes, and
the goal is to compute [X,Y], i.e., all homotopy classes of such maps. We solve
this problem in the stable range, where for some d >= 2, we have dim X <= 2d -
2 and Y is (d - 1)-connected; in particular, Y can be the d-dimensional sphere
S^d. The algorithm combines classical tools and ideas from homotopy theory
(obstruction theory, Postnikov systems, and simplicial sets) with algorithmic
tools from effective algebraic topology (locally effective simplicial sets and
objects with effective homology). In contrast, [X,Y] is known to be
uncomputable for general X,Y, since for X = S^1 it includes a well known
undecidable problem: testing triviality of the fundamental group of Y. In
follow-up papers, the algorithm is shown to run in polynomial time for d fixed,
and extended to other problems, such as the extension problem, where we are
given a subspace A of X and a map A -> Y and ask whether it extends to a map X
-> Y, or computing the Z_2-index---everything in the stable range. Outside the
stable range, the extension problem is undecidable.Comment: 42 pages; a revised and substantially updated version (referring to
follow-up papers and results