[Paper Review] An etale approach to the Novikov conjecture

Alexander Dranishnikov, Steven C. Ferry+1 more|ArXiv.org|Sep 27, 2005

Advanced Operator Algebra Research26 references4 citations

TL;DR

This paper establishes a novel etale approach to the Novikov conjecture by proving that the Higson compactification of the classifying space $B\Gamma$ is mod $p$ acyclic for all primes $p$ when $\Gamma$ has finite asymptotic dimension. This acyclicity implies the integral Novikov conjecture holds for such groups, providing a topological alternative to C*-algebraic methods and resolving the prime 2 obstruction via finite prime acyclicity.

ABSTRACT

We show that the rational Novikov conjecture for a group $Γ$ of finite homological type follows from the mod 2 acyclicity of the Higson compactifcation of an E$Γ$. We then show that for groups of finite asymptotic dimension the Higson compactification is mod p acyclic for all p, and deduce the integral Novikov conjecture for these groups.

Motivation & Objective

• To rehabilitate the Higson compactification method for the Novikov conjecture, which had been deemed insufficient due to non-acyclicity in rational cohomology.
• To overcome the failure of Higson compactification to be acyclic in rational cohomology by using finite prime coefficients (mod $p$) instead.
• To establish the integral Novikov conjecture for groups of finite asymptotic dimension by proving mod $p$ acyclicity of the Higson compactification of $B\Gamma$.
• To demonstrate that for groups of finite asymptotic dimension, the Higson compactification is mod $p$ acyclic for all $p$, enabling integral results in L-theory.

Proposed method

• The authors use the Higson compactification of $E\Gamma$ and $B\Gamma$, defined as the closure of $X$ in the space of bounded continuous functions with decaying variation.
• They define the Higson corona as the boundary of this compactification and analyze its cohomological properties using uniformly finite homology analogies.
• They apply quantitative algebraic topology techniques, including asymptotic polyhedra and Lipschitz maps, to control the geometry of maps into Eilenberg-MacLane spaces.
• They use the Higson extendibility condition (*) to extend maps from $X$ to the Higson corona, ensuring homotopy extensions over the compactification.
• They prove that any map into $K(\mathbb{Z}_p, n)$ is null homotopic on the Higson compactification by constructing $\epsilon$-close approximations and using finite homotopy groups.
• They apply Lemma 3.2 to construct a $\lambda$-Lipschitz homotopy to a constant map, leveraging compactness of the space of Lipschitz maps.

Experimental results

Research questions

• RQ1 Can the Higson compactification be used to prove the Novikov conjecture if it is mod $p$ acyclic for all primes $p$?
• RQ2 Does finite asymptotic dimension of a group $\Gamma$ imply that the Higson compactification of $B\Gamma$ is mod $p$ acyclic for all $p$?
• RQ3 Can mod $p$ acyclicity of the Higson compactification be used to deduce the integral Novikov conjecture in L-theory?
• RQ4 Why does the standard Higson compactification fail for rational cohomology, and can finite prime coefficients repair this?
• RQ5 Is there a topological alternative to C*-algebraic methods for proving the Novikov conjecture, particularly for integral results?

Key findings

• The Higson compactification of $B\Gamma$ is mod $p$ acyclic for all $p$ when $\Gamma$ has finite asymptotic dimension and $B\Gamma$ is a finite complex.
• For such groups, the integral assembly map $H_*(B\Gamma; \mathbf{L}) \to L_*(\mathbb{Z}\Gamma)$ is injective.
• The mod 2 acyclicity of the Higson compactification of $E\Gamma$ implies the Novikov conjecture at the prime 2.
• The cohomology $\check{H}^n(\bar{X}; \mathbb{Z}_p)$ vanishes for all $n > 0$ and all $p$, when $X$ is a metric space of finite asymptotic dimension.
• The proof relies on constructing $\epsilon$-close approximations to maps into $K(\mathbb{Z}_p, n)$ and using finite homotopy groups to ensure null homotopy.
• The method avoids reliance on the Baum-Connes conjecture and provides a new topological route to the integral Novikov conjecture for finite asymptotic dimension groups.