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There is an algebraic theory that I'm thinking of trying to develop and I wanted to know if it had any real world prevalence --- I'd like to know an example of a generator L of a Fukaya category on a compact symplectic manifold where this Lagrangian has the properties that :

1. Simply connected,but not a sphere or a product of spheres 2) The homology of the Floer complex HF*(L,L) is isomorphic as an algebra to H*(L). Extra points if the Floer theory can in a reasonable sense be defined over C. I'd also be really interested to know why this situation can't happen.

Edit: I know it is kind of rare for a single Lagrangian(impossible?) to generate a Fukaya category. I'd be happy to have a bunch of such Lagrangians L_i where at least one of them was as above as long as Hom(L_i, L_j)=0 for i not equal to j...

There is an algebraic theory that I'm thinking of trying to develop and I wanted to know if it had any real world prevalence --- I'd like to know an example of a generator L of a Fukaya category on a compact symplectic manifold where this Lagrangian has the properties that :

1. Simply connected,but not a sphere or a product of spheres 2) The homology of the Floer complex HF*(L,L) is isomorphic as an algebra to H*(L). (Edit: as an ordinary algebra, the higher operations induced by the perturbation lemma can be different.) Extra points if the Floer theory can in a reasonable sense be defined over C. I'd also be really interested to know why this situation can't happen.

Edit: I know it is kind of rare for a single Lagrangian(impossible?) to generate a Fukaya category. I'd be happy to have a bunch of such Lagrangians L_i where at least one of them was as above as long as Hom(L_i, L_j)=0 for i not equal to j...

There is an algebraic theory that I'm thinking of trying to develop and I wanted to know if it had any real world prevalence --- I'd like to know an example of a generator L of a Fukaya category on a compact symplectic manifold where this Lagrangian has the properties that :

1. Simply connected,but not a sphere or a product of spheres 2) The homology of the Floer complex HF*(L,L) is isomorphic as an algebra to H*(L). Extra points if the Floer theory can in a reasonable sense be defined over C. I'd also be really interested to know why this situation can't happen.

There is an algebraic theory that I'm thinking of trying to develop and I wanted to know if it had any real world prevalence --- I'd like to know an example of a generator L of a Fukaya category on a compact symplectic manifold where this Lagrangian has the properties that :

1. Simply connected,but not a sphere or a product of spheres 2) The homology of the Floer complex HF*(L,L) is isomorphic as an algebra to H*(L). Extra points if the Floer theory can in a reasonable sense be defined over C. I'd also be really interested to know why this situation can't happen.

Edit: I know it is kind of rare for a single Lagrangian(impossible?) to generate a Fukaya category. I'd be happy to have a bunch of such Lagrangians L_i where at least one of them was as above as long as Hom(L_i, L_j)=0 for i not equal to j...