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

Abstract

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.