Periodic Symplectic Cohomologies

Symplectic homology and the Eilenberg-Steenrod axioms

“…It also indicates the analogy to the S 1 -equivariant cohomology groups defined by Jones and Petrack [54]. Indeed, it is proved in [73,4] that for a Liouville domain W and with rational coefficients the second group satisfies fixed point localization…”

“…Each of the these flavors of S 1 -equivariant symplectic homology groups obeys the same set of Eilenberg-Steenrod type axioms as their nonequivariant counterparts. Transfer maps and invariance for the case of Liouville domains were previously discussed in [70,73,48]. Moreover, it follows from the construction that the Gysin and fundamental exact triangles are functorial with respect to the tautological exact triangles and also with respect to the exact triangles of pairs, see also [17,18] for a basic instance of this phenomenon.…”

“…As such, one can define S 1 -equivariant flavors of symplectic homology groups. In the case of Liouville domains relevant instances have been defined in [70,67,18,73,4]. Following Seidel [67] and [18,73], the relevant structure is that of an S 1 -complex, meaning a Z-graded chain complex (C * , ∂) together with a sequence of maps ∂ i :…”

“…Recall that we work with coefficients in a field K. Denote by u a formal variable of degree −2. Given an S 1 -complex C = (C * , {∂ i } i≥0 ) we define following Jones [53] and Zhao [73] the periodic cyclic chain complex…”

“…In order to define S 1 -equivariant symplectic homology and cohomology groups, we use the structure of an S 1 -complex on each truncated Floer chain complex C := FC (a,b) * (H) and cochain complex C ∨ := FC * (a,b) (H) constructed in [18,73]. We set…”