Generalized manifolds, normal invariants, and 𝕃-homology

Abstract

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.

Limits of Manifolds in the Gromov–Hausdorff Metric Space

“…This paper is a continuation of our systematic study of the characterization problem for generalized n-manifolds, n ≥ 5, (see Cavicchioli et al [5,6] and Hegenbarth and Repov² [23,24,25,26,27,28]). This is a very important class of spaces which in the algebraic sense strongly resemble topological manifolds, whereas in the geometric sense they can fail to be locally Euclidean at any point (see e.g., Cannon [4], Edwards [11], and Repov² [42,43,44]).…”

Limits of manifolds in the Gromov-Hausdorff metric space

“…This paper is a continuation of our systematic study of the characterization problem for generalized n-manifolds, n ≥ 5, (see Cavicchioli et al [5,6] and Hegenbarth and Repovš [23,24,25,26,27,28]). This is a very important class of spaces which in the algebraic sense strongly resemble topological manifolds, whereas in the geometric sense they can fail to be locally Euclidean at any point (see e.g., Cannon [4], Edwards [11], and Repovš [42,43,44]).…”