Circle actions with two fixed points

Abstract

We prove that if the circle group acts smooth and unitary on 2n-dimensional stably complex manifold with two isolated fixed points and it is not bound equivariantly, then n = 1 or 3. Our proof relies on the rigid Hirzebruch genera.

Circle actions on 6-dimensional oriented manifolds with 4 fixed points

“…Then the sign of every weight at a fixed point is well-defined. While for oriented manifolds on any even dimension there exists a circle action with 2 fixed points (a rotation of S 2n ), for almost complex manifolds if the circle acts on a compact almost complex manifold with 2 fixed points, then the dimension of the manifold must be either 2 or 6; see [J2], [K1], [K2], [Mu2], [PT]. In addition, an almost complex manifold with 3 fixed points only exists in dimension 4, see [J2]; also see [J1].…”

Rigidity of powers and Kosniowski's conjecture

“…The following theorem has been proved in [19]. Note that its proof is included here for a local completeness.…”

“…x,y (z) = T x,y (M). For the case m = 2, Theorem 2.2 (see also [19] or [12,Theorem 5]) tells us that there are the following possibilities for the unitary S 1 -manifold M 2n : (Z) : n is a arbitrary integer, w 1i = w 2i = a i where a i ∈ Z, a i = 0, for all i = 1, . .…”