Generalized manifolds, normal invariants, and $\mathbb{L}$-homology

Let $X^n$ be an arbitrary 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}; $\mathbb{L}$^+)$whichextendsthenormalinvariantmapforthecasewhen$X^{n}$isatopological$n$-manifold\mathrm{.}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 $H_{\ast }^{st}(X^n;\mathbb{E})$ denotes the Steenrod homology of the spectrum $\mathbb{E}$. An important nontrivial question arose whether the map $t$ is bijective (note that this holds in the case that $X^n$ is a topological $n$-manifold). It is the purpose of this paper to prove that the answer to this question is affirmative.