9 research outputs found
Plato's cave and differential forms
In the 1970s and again in the 1990s, Gromov gave a number of theorems and
conjectures motivated by the notion that the real homotopy theory of compact
manifolds and simplicial complexes influences the geometry of maps between
them. The main technical result of this paper supports this intuition: we show
that maps of differential algebras are closely shadowed, in a technical sense,
by maps between the corresponding spaces. As a concrete application, we prove
the following conjecture of Gromov: if and are finite complexes with
simply connected, then there are constants and such that
any two homotopic -Lipschitz maps have a -Lipschitz homotopy (and
if one of the maps is a constant, can be taken to be .) We hope that it
will lead more generally to a better understanding of the space of maps from
to in this setting.Comment: 39 pages, 1 figure; comments welcome! This is the final version to be
published in Geometry & Topolog
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
IST Austria Thesis
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