  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 $t:\mathcal{N}(X^{n}) \to H^{st}_{n} ( X^{n};
\mathbb{L}^+)$ 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 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.
