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I study topology of 3 and 4 dimensional manifolds, using moduli spaces of solutions to partial differential equations which come from gauge theory. We can construct differential topological invariants of 4-manifold from the moduli spaces. In general, it is difficult to calculate the invariants. But if you cut the 4-manifold along 3-manifolds and split it into smaller 4-manifolds, the calculation of the invariants becomes easier. In the procedure, Floer theory of 3-manofolds plays an important role. Currently I study Seiberg-Witten-Floer stable homotopy type of 3-manifolds.