Lattices in R^2 and finite subsets of a circle

Abstract

An elementary geometric construction is used to relate the space of lattices in a plane to the space exp_3(S^1) of the subsets of a circle of cardinality at most 3. As a consequence we obtain new proofs of a theorem of Bott which says that exp_3(S^1) is homeomorphic to a 3-sphere and a theorem of Shchepin which says that points of exp_3(S^1) that correspond to one-point subsets form a trefoil knot in this 3-sphere.