In this paper we consider $4$-dimensional steady soliton singularity models, i.e., complete steady gradient Ricci solitons that arise as the rescaled limit of a finite time singular solution of the Ricci flow on a closed $4$-manifold. In particular,
we study the geometry at infinity of such Ricci solitons under the assumption that their tangent flow at infinity is the product of $mathbb{R}$ with a $3$-dimensional spherical space form. We also classify the tangent flows at infinity of $4$-dimensional steady soliton singularity models in general.
As indicated by the third author in [19], there is a gap in the previous version of this paper by the first two authors [5]. We provide in this version an argument to fix the aforementioned gap. The main proposition, whose proof uses Perelmans techni
ques, is implied by Ding [9] and is covered by [19]. Our approach, however, is different from theirs. In addition, we prove a necessary and sufficient condition for a three-dimensional $kappa$-solution to form a forward singularity. We hope that this condition is helpful in the classification of all three-dimensional $kappa$-solutions. Up to now, the only main progress on such a classification, as conjectured by Perelman, is by Brendle [2].
