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

TL;DR

This paper proves that a specific map relating normal invariants and $ ext{L}$-homology is bijective for arbitrary oriented closed generalized $n$-manifolds, extending known results from topological manifolds.

Contribution

It establishes the bijectivity of the map $t$ for generalized manifolds, generalizing previous results from topological manifolds.

Findings

The map $t$ is bijective for generalized $n$-manifolds.

Extension of normal invariant theory to generalized manifolds.

Confirms the conjecture for the case $n \\ge 5$.

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.