Abstract
We present three equivalent definitions of $S^1$-equivariant symplectic homology. We show that, using rational coefficients, the positive part of $S^1$-equivariant symplectic homology is isomorphic to linearized contact homology, when the latter is defined. We present several computations and applications, as well as a rigorous definition of cylindrical/linearized contact homology based on an $S^1$-equivariant construction.
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