We investigate multiframings of a closed oriented 3-manifold. We show that multiframings give a geometric realization of the tensor product of the homotopy set of framings and [Formula: see text]. We prove that the Hirzebruch defect defines a bijection from the homotopy set of multiframings to [Formula: see text] for any connected closed oriented 3-manifold, and we prove that any multiframing defined near the boundary of a compact oriented 3-manifold extends to the bounded manifold.

中文翻译：

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3-歧管的多帧

我们研究了一个封闭的定向 3 流形的多帧。我们证明了多帧给出了同伦组的张量积的几何实现和[公式：见正文]。我们证明了 Hirzebruch 缺陷定义了从多帧的同伦集到 [公式：见文本] 的双射，用于任何连接的封闭定向 3 流形，并且我们证明任何定义在紧凑定向 3 流形边界附近的多帧扩展到有界流形。