A central problem of algebraic topology is to understand the homotopy groups \pi_d(X) of a topological space X. For the computational version of the problem, it is well known that there is no algorithm to decide whether the fundamental group \pi_1(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 \pi_1(X) trivial), compute the higher homotopy group \pi_d(X) for any given d\geq 2. \%The first such algorithm was given by Brown, and more recently, Cadek et al. However, these algorithms come with a caveat: They compute the isomorphism type of \pi_d(X), d\geq 2 as an abstract finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of \pi_d(X). Converting elements of this abstract group into explicit geometric maps from the d-dimensional sphere S^d to X 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 X, computes \pi_d(X) and represents its elements as simplicial maps from a suitable triangulation of the d-sphere S^d 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\geq 2, we construct a family of simply connected spaces X such that for any simplicial map representing a generator of \pi_d(X), the size of the triangulation of S^d on which the map is defined, is exponential in size(X).
A central problem of algebraic topology is to understand the homotopy groups \pi_d(X) of a topological space X. For the computational version of the problem, it is well known that there is no algorithm to decide whether the fundamental group \pi_1(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 \pi_1(X) trivial), compute the higher homotopy group \pi_d(X) for any given d\geq 2. \%The first such algorithm was given by Brown, and more recently, Cadek et al. However, these algorithms come with a caveat: They compute the isomorphism type of \pi_d(X), d\geq 2 as an abstract finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of \pi_d(X). Converting elements of this abstract group into explicit geometric maps from the d-dimensional sphere S^d to X 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 X, computes \pi_d(X) and represents its elements as simplicial maps from a suitable triangulation of the d-sphere S^d 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\geq 2, we construct a family of simply connected spaces X such that for any simplicial map representing a generator of \pi_d(X), the size of the triangulation of S^d on which the map is defined, is exponential in size(X).
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homotopy groups
homotopy groups
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fundamental group
fundamental group
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simply connected
simply connected