Cohomology in algebra, geometry, physicsand statistics

Floer-Novikov (co)homology associated with non-abelian coverings and symplectic fixed points

In my talk   I shall explain   our    with Kaoru Ono   construction   of  Floer-Novikov  cohomology  groups $HFN^* (M^{\Gamma_\xi \times H},\xi, Q)$ defined on a regular covering $M^{\Gamma_\xi \times H}$ of a  compact   symplectic  manifold   $(M, \omega)$ with  transformation group  $\Gamma_\xi \times  H$  and associated  to  a    locally symplectic isotopy ${\{\varphi_t\}}$ of $(M, \omega)$ with  flux $\xi \in H ^1 (M, R)$. Then  $H$ acts naturally on $HFN^* (M^{\Gamma_\xi \times H},\xi, Q)$.  For a subgroup $G \subset H$  denote  by $(HFN^* (M^{\Gamma_\xi \times H},\xi,  Q))^G$  the   subgroup of $HFN^* (M^{\Gamma_\xi \times H}, \xi, Q)$  consisting   of the fixed  points of the $G$-action.  We  prove that  the   rank   of  $(HFN^* (M ^{\Gamma_\xi \times H},\xi,  Q))^G$     equals   the rank  of the   subgroup  $(HN^* (M^{\Gamma_\xi \times  H}, Q))^G$  of  the fixed points  of the  $G$-action in the Novikov  cohomology  group $HN^* (M^{\Gamma_\xi \times  H}, \Q)$. If $H$ is  trivial,  this implies  our   previous  result   asserting that  the sum of the  Betti  numbers   of $HFN^* (M ^{\Gamma_\xi}, \xi, Q)$ equals the  sum  of the  Betti numbers  of the Novikov   cohomology group  $HN_* (M, \xi, Q)$.  This  equality  leads  to the classical cohomological  estimate  of the  numbers  of    the fixed points  of  a nondegenerate  locally Hamiltonian symplectomorphism.   If  $H$ is nontrivial,   we  obtain a new   lower  bound  for the number  of the  fixed  points of   non-degenerate  locally Hamiltonian   symplectomorphisms  of $(M, \omega)$. 

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We shall open the seminar room + ZOOM meeting at 13.15  for coffee



Join Zoom Meeting

https://cesnet.zoom.us/j/99598413922?pwd=YXNFbk50aVhleXhWSGtISFViLytRUT09

Meeting ID: 995 9841 3922

Passcode: Galois