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.
Zhechev SY. Algorithmic aspects of homotopy theory and embeddability. 2019. doi:10.15479/AT:ISTA:6681
Zhechev, S. Y. (2019). Algorithmic aspects of homotopy theory and embeddability. IST Austria. https://doi.org/10.15479/AT:ISTA:6681
Zhechev, Stephan Y. “Algorithmic Aspects of Homotopy Theory and Embeddability.” IST Austria, 2019. https://doi.org/10.15479/AT:ISTA:6681.
S. Y. Zhechev, “Algorithmic aspects of homotopy theory and embeddability,” IST Austria, 2019.
Zhechev SY. 2019. Algorithmic aspects of homotopy theory and embeddability. IST Austria.
Zhechev, Stephan Y. Algorithmic Aspects of Homotopy Theory and Embeddability. IST Austria, 2019, doi:10.15479/AT:ISTA:6681.
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