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 $t:\mathcal{N}(X^n)\to H_n^{st}(X^n;{\mathbb{L}}^{+})$ 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 $H_{\ast }^{st}(X^n;\mathbb{E})$ 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.

References in corpus (1)

• On Steenrod $\mathbb{L}$-homology, generalized manifolds, and surgery

Cited by in corpus (1)

• Limits of manifolds in the Gromov-Hausdorff metric space