Tightness is a generalisation of the notion of convexity: a space is tight if and only if it is as convex as possible, given its topological constraints. For a simplicial complex, deciding tightness has a straightforward exponential time algorithm, but efficient methods to decide tightness are only known in the trivial setting of triangulated surfaces. In this article, we present a new polynomial time procedure to decide tightness for triangulations of $3$-manifolds -- a problem which previously was thought to be hard. Furthermore, we describe an algorithm to decide general tightness in the case of $4$-dimensional combinatorial manifolds which is fixed parameter tractable in the treewidth of the $1$-skeletons of their vertex links, and we present an algorithm to decide $mathbb{F}_2$-tightness for weak pseudomanifolds $M$ of arbitrary but fixed dimension which is fixed parameter tractable in the treewidth of the dual graph of $M$.
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