FINITE LENGTH DATA DO NOT UNIVERSALLY DETERMINE FLAT THREE-TORI EXPLICIT CONSTRUCTIONS, PROOFS, AND A CAUTION FOR COSMIC TOPOLOGY
Abstract. A flat three-torus is determined by a Euclidean lattice up to orthogonal changes of coordinates, or equivalently by a positive-definite Gram matrix up to integral change of basis. A finite list of short lattice lengths is a much weaker datum. This paper gives an explicit and fully elementary construction showing that the weakness can persist over a large range of scales. For every real parameter A ≥ 2, we exhibit two positive-definite Gram matrices G+(A) and G−(A) whose associated flat tori have different volumes and hence are not isometric, while their length multisets agree below…
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