Any knot in $S^3$ may be reduced to a slice knot by crossing changes. Indeed, this slice knot can be taken to be the unknot. In this paper we study the question of when the same holds for knots in homology spheres. We show that a knot in a homology sphere is nullhomotopic in a smooth homology ball if and only if that knot is smoothly concordant to a knot which is homotopic to a smoothly slice knot. As a consequence, we prove that the equivalence relation on knots in homology spheres given by cobounding immersed annuli in a homology cobordism is generated by concordance in homology cobordisms together with homotopy in a homology sphere.

中文翻译：

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同源领域中的一致性、交叉变化和结

$S^3$ 中的任何结都可以通过交叉变化简化为切片结。事实上，这个切片结可以被认为是未结。在本文中，我们研究了同源球中的结何时同样成立的问题。我们证明同源球中的结在光滑同源球中是零同伦的，当且仅当该结与与光滑切片结同伦的结平滑一致。因此，我们证明了通过将浸入环在同源协调中的共同约束而给出的同源球中结的等价关系是由同源协调中的一致性和同源球中的同伦产生的。