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In 1995 the speaker, Jones, and Segal introduced the notion of "FloerÌýhomotopy theory."ÌýThe proposal was to attach a (stable) homotopy typeÌýto the geometric data given in a version of Floer homology. More to theÌýpoint, the question was asked,"When is the Floer homology isomorphicÌýto the (singular) homology of a naturally occurring spectrum definedÌýfromÌýthe properties of the moduli spaces inherent in the Floer theory?"ÌýYearsÌýpassed before this notion found some genuine applications to symplecticÌýgeometry and low dimensional topology. However in recent years severalÌýstriking applications have been found, and the theory has been developedÌýon a much deeper level. In this talk I will sketch the theory and talk aboutÌýapplications by Lipshitz-Sarkar, Manolescu, and others.
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