Friedrich Hegenbarth (Author), Dušan Repovš (Author)

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.

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;

Data

Language: English
Year of publishing:
Typology: 1.01 - Original Scientific Article
Organization: UL FMF - Faculty of Mathematics and Physics
Publisher: Cambridge University Press
UDC: 515.14
COBISS: 67730691 Link will open in a new window
ISSN: 0013-0915
Views: 200
Downloads: 51
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Other data

Type (COBISS): Article
Embargo end date (OpenAIRE): 2021-12-16
Pages: str. 574-589
Volume: ǂVol. ǂ64
Issue: ǂiss. ǂ3
Chronology: Aug. 2021
DOI: 10.1017/S0013091521000316
ID: 13582417