Let $X^{n}$ be an oriented closed generalized $n$-manifold, $n\ge 5$. In our recent paper (Proc. Edinb. Math. Soc. (2) 63 (2020), no. 2, 597–607), we have constructed a map $t:\mathcal {N}(X^{n}) \to H^{st}_{n} ( X^{n}; \mathbb{L}^{+})$ which extends the normal invariant map for the case when $X^{n}$ is a topological $n$-manifold. Here, $\mathcal {N}(X^{n})$ denotes the set of all normal bordism classes of degree one normal maps $(f,\,b): M^{n} \to X^{n},$ and $H^{st}_{*} ( X^{n}; \mathbb{E})$ denotes the Steenrod homology of the spectrum $\mathbb{E}$. An important non-trivial question arose whether the map $t$ is bijective (note that this holds in the case when $X^{n}$ is a topological $n$-manifold). It is the purpose of this paper to prove that the answer to this question is affirmative.

中文翻译：

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广义流形、正态不变量和 𝕃-同调

让$X^{n}$是一个有向的封闭广义$n$-歧管，$n\ge 5$. 在我们最近的论文（Proc. Edinb. Math. Soc. (2) 63 (2020), no. 2, 597–607）中，我们构建了一张地图$t:\mathcal {N}(X^{n}) \to H^{st}_{n} ( X^{n}; \mathbb{L}^{+})$它扩展了正常不变量映射的情况$X^{n}$是一个拓扑$n$-歧管。这里，$\mathcal {N}(X^{n})$表示一阶法线贴图的所有法线边界类的集合$(f,\,b): M^{n} \to X^{n},$和$H^{st}_{*} ( X^{n}; \mathbb{E})$表示谱的 Steenrod 同源性$\mathbb{E}$. 一个重要的不平凡的问题出现了，地图是否$t$是双射的（请注意，这在以下情况下成立$X^{n}$是一个拓扑$n$-歧管）。本文的目的是证明这个问题的答案是肯定的。