Floer homology
One conceivable way to construct a Floer homology theory of some object would be to construct a related spectrum whose ordinary homology is the desired Floer homology. Applying other homology theories to such a spectrum could yield other interesting invariants. This strategy was proposed by Ralph Cohen, John Jones, and Graeme Segal, and carried out in certain cases for Seiberg–Witten–Floer homology by and for the symplectic Floer homology of cotangent bundles by Cohen. This approach was the basis of Manolescu's 2013 construction of Pin (2)-equivariant Seiberg-Witten Floer homology, with which he disproved the Triangulation Conjecture for manifolds of dimension 5 and higher.