Controlled surgery and ▫$\mathbb{L}$▫-homology

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

This paper presents an alternative approach to controlled surgery obstructions. The obstruction for a degree one normal map ▫$(f,b) \colon M^n \to X^n$▫ with control map ▫$q \colon X^n \to B$▫ to complete controlled surgery is an element ▫$\sigma^c (f,b) \in H_n(B, \mathbb{L})$▫, where ▫$M^n, \, X^n$▫ are topological manifolds of dimension ▫$n \ge 5$▫. Our proof uses essentially the geometrically defined ▫$\mathbb{L}$▫-spectrum as described by Nicas (going back to Quinn) and some well-known homotopy theory. We also outline the construction of the algebraically defined obstruction, and we explicitly describe the assembly map ▫$H_n(B,L) \to L_n(\pi_1(B))$▫ in terms of forms in the case ▫$n \equiv 0(4)$▫. Finally, we explicitly determine the canonical map ▫$H_n(B,L) \to H_n(B, \, L_0)$▫.

• Language: English
• Keywords: generalized manifold, resolution obstruction, controlled surgery, controlled structure set, ▫$\mathbb{L}_q$▫-surgery, Wall obstruction
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Year: 2019
• Number of pages: Str. 1-22
• Numbering: art. 79, iss. 3
• PID: 20.500.12556/RUL-108784
• UDC: 515.1
• ISSN on article: 1660-5446
• DOI: 10.1007/s00009-019-1354-6
• COBISS.SI-ID: 18630745
• Publication date in RUL: 25.07.2019

Record is a part of a journal

• Title: Mediterranean journal of mathematics
• Shortened title: Mediterr. j. math.
• Publisher: Springer Nature, Birkhäuser
• ISSN: 1660-5446
• COBISS.SI-ID: 13561433