Generalized manifolds, normal invariants, and L-homology

Hegenbarth, Friedrich (author), Repovš, Dušan (author)

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.

• Language: English
• Keywords: generalized manifold, Steenrod ▫$\mathbb{L}$▫-homology, Poincaré duality complex, normal invariant of degree, one map, periodic surgery spectrum ▫$\mathbb{L}$▫, fundamental complex, Spivak fibration, Pontryagin-Thom construction, Spanier-Whitehead duality, absolute neighbourhood retract
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Publication version: Author Accepted Manuscript
• Publisher: Cambridge University Press
• Year: 2021
• Number of pages: Str. 574-589
• Numbering: Vol. 64, Iss. 3
• PID: 20.500.12556/RUL-132482
• UDC: 515.14
• ISSN on article: 0013-0915
• DOI: 10.1017/S0013091521000316
• COBISS.SI-ID: 67730691
• Publication date in RUL: 27.10.2021

Record is a part of a journal

• Title: Proceedings of the Edinburgh Mathematical Society
• Shortened title: Proc. Edinb. Math. Soc.
• Publisher: Scottish Academic Press
• ISSN: 0013-0915
• COBISS.SI-ID: 27124480