To classify a group action on a manifold, the data associated with the fixed point set is essential. In this paper, we classify the fixed point data of a circle action on a 6-dimensional compact connected oriented manifold with isolated fixed points, where the fixed point data consists of the collection of signs and weights at the fixed points. We show that this fixed point data can be reduced to the empty collection by performing a sequence of operations. Specifically, we prove that one can successively take equivariant connected sums at fixed points with $S^6$, $\mathbb{CP}^3$, or 6-dimensional analogues of the Hirzebruch surfaces (and their oppositely oriented counterparts), resulting in a fixed-point-free action on a compact connected oriented 6-manifold.
The author proved that if the circle acts symplectically on a compact, connected symplectic manifold $M$ with three fixed points, then $M$ is equivariantly symplectomorphic to some standard action on $\mathbb{CP}^2$. In this paper, we extend the result to a circle action on an almost complex manifold; if the circle acts on a compact, connected almost complex manifold $M$ with exactly three fixed points, then $\dim M=4$. Moreover, we deal with the cases of one fixed point and two fixed points.
Whitehead's Integral Formula, Isolated Critical Points, and the Enhancement of the Milnor Number
J. H. C. Whitehead给出了从3-球面到2-球面的光滑映射p的Hopf不变量H（p）的一个优美的积分公式。给定一个在3-球面上的开卷结构b（或者，本质上等价地，一个从4-空间到平面的映射F的孤立临界点），怀特海公式可以“沿着纤维积分”，将H（p）表示为某个1-形式在圆上的积分。在p与b（或F）几何相关的情况下-例如，如果p是映射（F的纤维广义高斯映射的一个分量），其Hopf不变量lambda（K）是与F（或b）相关联的3-球面中纤维链K的“Milnor数的增强”，作者和其他人以前研究过-可能希望这种1-形式具有几何意义。本说明以几个猜测和问题的形式使这一希望更加具体。
