Controlled surgery and L-homology

Friedrich Hegenbarth (Author), Dušan Repovš (Author)

Abstract

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 ▫$H_n(B,L) \to H_n(B, \, L_0)$▫.

Keywords

generalized manifold;resolution obstruction;controlled surgery;controlled structure set;▫$\mathbb{L}_q$▫-surgery;Wall obstruction;

Data

Language:	English
Year of publishing:	2019
Typology:	1.01 - Original Scientific Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	515.1
COBISS:	18630745
ISSN:	1660-5446
Views:	532
Downloads:	220
Average score:	0 (0 votes)
Metadata:

Other data

Type (COBISS):	Article
Pages:	art. 79 (22 str.)
Volume:	ǂVol. ǂ16
Issue:	ǂiss. ǂ3
Chronology:	June 2019
DOI:	10.1007/s00009-019-1354-6
ID:	11193862