Floer Homology via Semi-Infinite Dimensional Cycles

The basic idea behind Floer homology is to define homology groups of a functional on an infinite-dimensional space by mimicking the construction of Morse homology for finite dimensional manifolds. Various flavors of Floer homology have been successfully applied to the study of knots and 3-manifolds and are increasingly often being applied to the study of 4-manifolds, leveraging their TQFT properties. In the talk, I will describe an approach to defining Floer homology imitating the construction of singular homology, and thus aiming to be more general and widely applicable than the Morse homology approach by avoiding some of the analytical problems arising in the latter. Potentially, these ideas could be used in the future to define counterparts of other invariants coming from algebraic topology in the setting of Floer theory without the need to construct a homotopy type first.