---
res:
bibo_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.\r\nHowever, 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,\r\nwe 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).\r\nIn 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.@eng"
bibo_authorlist:
- foaf_Person:
foaf_givenName: Stephan Y
foaf_name: Zhechev, Stephan Y
foaf_surname: Zhechev
foaf_workInfoHomepage: http://www.librecat.org/personId=3AA52972-F248-11E8-B48F-1D18A9856A87
bibo_doi: 10.15479/AT:ISTA:6681
dct_date: 2019^xs_gYear
dct_isPartOf:
- http://id.crossref.org/issn/2663-337X
dct_language: eng
dct_publisher: IST Austria@
dct_title: Algorithmic aspects of homotopy theory and embeddability@
...