We give a brief introduction to Floer homology and homotopy, from the Seiberg-Witten point of view. We will then discuss Manolescu's version of finite-dimensional approximation for rational homology spheres. We prove that a version of finite-dimensional approximation for the Seiberg-Witten equations associates equivariant spectra to a large class of three-manifolds. We give some applications to the the study of four-manifolds. This is joint work with Hirofumi Sasahira.

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