On Geometric Representation of 𝕃 $$\mathbb {L}$$ -Homology Classes

Friedrich Hegenbarth () and Dušan D. Repovš ()
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Friedrich Hegenbarth: Università degli studi di Milano, Dipartimento di Matematica “Federigo Enriques”
Dušan D. Repovš: University of Ljubljana & Institute of Mathematics, Physics and Mechanics, Faculty of Education and Faculty of Mathematics and Physics

Chapter Chapter 19 in Essays on Topology, 2025, pp 429-436 from Springer

Abstract: Abstract In this chapter we give a geometric representation of H n ( B ; 𝕃 ) $$H_{n}(B;\mathbb {L})$$ classes, where 𝕃 $$\mathbb {L}$$ is the 4-periodic surgery spectrum, by establishing a relationship between the normal cobordism classes N n H ( B , ∂ ) $${\mathcal {N}}^{H}_{n}(B,\partial)$$ and the n-th 𝕃 $$\mathbb {L}$$ -homology of B, representing the elements of H n ( B ; 𝕃 ) $$H_{n}(B;\mathbb {L})$$ by normal degree one maps with a reference map to B. More precisely, we prove that for every n ≥ 6 $$n \ge 6$$ and every finite complex B , $$B,$$ there exists a map Γ : H n ( B ; 𝕃 ) → N n H ( B , ∂ ) . $$\Gamma: H_n(B;\mathbb {L}) \longrightarrow \mathcal {N}^{H}_{n}(B,\partial).$$

Keywords: Generalized manifold; Cell-like map; Normal degree one map; Steenrod 𝕃 $$\mathbb {L}$$ -homology; Poincaré duality complex; Periodic surgery spectrum 𝕃 $$\mathbb {L}$$; Geometric representation; 𝕃 $$\mathbb {L}$$ -homology class; Primary: 55R20, 57P10, 57R65 57R67; Secondary: 55M05, 55N99, 57P05, 57P99 (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-81414-3_19

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DOI: 10.1007/978-3-031-81414-3_19

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