Unitary $S^1$-Manifolds with Four Fixed Points and Kosniowski's Conjecture

We study unitary $S^1$-manifolds with four isolated fixed points in relation to Kosniowski’s conjectural inequality $4\chi (M)>\dim M$. For every integer $m\ge 3$, we show that there exists a unitary $S^1$-manifold $M_m^{18}$ with exactly four isolated fixed points, and these manifolds are pairwise distinct and nonzero in ordinary complex bordism. Since $\chi (M_m^{18})=4$, they give an infinite family of counterexamples to $4\chi (M)>\dim M$. By gluing two blocks formed by $G_2\mathrm{/}SU(3)\cong S^6$ and the unit disk $D(V)$ of 12- dimensional real $S^1$-representation $V$ along their equivariantly identified free boundaries, we explicitly construct such a manifold $M_3^{18}$. We also prove that the four-point case is null-bordant in real dimension $16$ and in every real dimension greater than $18$; hence $18$ is the smallest dimension in which such a counterexample occurs.