1 results
Generalized manifolds, normal invariants, and 𝕃-homology
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 64 / Issue 3 / August 2021
- Published online by Cambridge University Press:
- 16 June 2021, pp. 574-589
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- Article
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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.
