Circle actions on six dimensional oriented manifolds with isolated fixed points

Abstract: Let the circle group act on a 6-dimensional compact oriented manifold $M$ with isolated fixed points. The fixed point data of $M$ is the collection of signs and weights at the fixed points. To classify such a manifold $M$, the fixed point data is an essential information. Under an assumption that each isotropy submanifold is orientable, we classify the fixed point data of $M$, by showing that we can convert it to the empty collection by performing a combination of a number of types of operations. We do so by proving that we can successively take equivariant connected sums of $M$ at fixed points with $S^6$, $\mathbb{CP}^3$, and 6-dimensional analogue $Z_1$ and $Z_2$ of the Hirzebruch surfaces (and these with opposite orientations), to a fixed point free action on a compact oriented 6-manifold.