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

## TL;DR

This paper offers a new geometric approach to controlled surgery obstructions using the $L$-spectrum, providing explicit descriptions of the obstruction and assembly map in algebraic topology.

## Contribution

It introduces an alternative geometric proof for controlled surgery obstructions and details the construction and explicit description of related algebraic maps.

## Key findings

- Provides a geometric proof using the $L$-spectrum
- Explicitly describes the assembly map in terms of forms
- Determines the canonical map to $H_n(B, L_0)$

## Abstract

This paper presents an alternative approach to controlled surgery obstructions. The obstruction for a degree one normal map $(f,b): M^n \rightarrow X^n$ with control map $q: X^n \rightarrow B$ 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 $n \geq 5$. 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 $H_n (B, \mathbb{L}) \rightarrow L_n (\pi_1 (B))$ in terms of forms in the case $n \equiv 0 (4)$. Finally, we explicitly determine the canonical map $H_n (B, \mathbb{L}) \rightarrow H_n (B, L_0)$.