Floer homology and its continuity for non-compact Lagrangian submanifolds (Q2719821)

One form of the Arnold conjecture states that, for a Hamiltonian diffeomorphism \(\psi\) on \(T^*M\), NEWLINE\[NEWLINE\#(\psi(M) \cap M) \geq \text{crit}(M),NEWLINE\]NEWLINE where \(M\) is embedded as the zero section in \(T^*M\). The conjecture has been proved for lower bounds which are bounded above by \(\text{crit}(M)\); for instance, \(\text{cup}(M)+1\) (in the degenerate case). In this paper, the author considers a pair of compact submanifolds \(S_1\), \(S_2\) with conormal bundles \(\nu^*S_1\), \(\nu^*S_2\) and shows that, for a restricted class of Hamiltonian diffeomorphisms and \(\nu^*S_1\) transverse to \(\psi(\nu^*S_2)\), NEWLINE\[NEWLINE\#(\nu^*S_1 \cap \psi(\nu^*S_2)) \geq \text{rank}(H_*(S_1 \cap S_2)),NEWLINE\]NEWLINE where coefficients are \(\mathbb Z\) in the oriented case and \(\mathbb Z/2\) in general. The proof uses an appropriate Floer homology and one essential ingredient is the construction of a certain chain map induced by Lagrangian cobordant submanifolds (which the author defines).