## Computing simplicial representatives of homotopy group elements.(English)Zbl 1430.55014

Summary: 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 $$d\ge 2$$. However, these algorithms come with a caveat: They compute the isomorphism type of $$\pi _d(X), d\ge 2$$ 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 $$\text{size}(X)$$, the number of simplices of $$X$$. Moreover, we prove that this is optimal: For every fixed $$d\ge 2$$, 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 $$\text{size}(X)$$.

### MSC:

 55U10 Simplicial sets and complexes in algebraic topology 55Q05 Homotopy groups, general; sets of homotopy classes 68W30 Symbolic computation and algebraic computation

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### References:

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