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Coarse Baum-Connes and warped cones: failure of surjectivity in odd degree

Christos Kitsios, Thomas Schick, Federico Vigolo

TL;DR

The paper studies coarse index theory for unified warped cones $\mathcal{O}_\Gamma M$ and resolves Roe's conjecture by showing a nuanced obstruction to the coarse Baum–Connes map occurs in odd degree. It proves that generalized Drutu–Nowak projections vanish in $K$-theory ($[\mathfrak G]=0$ in $K_0$) via an Eilenberg swindle and a Mayer–Vietoris strategy, while simultaneously constructing an explicit odd-degree $K$-theory class outside the image of the coarse assembly map when $\Gamma$ has property A and acts freely and strongly ergodically on $M$. The core method combines a dyadic Mayer–Vietoris decomposition of the warped cone with compatibility between the MV sequence and the coarse assembly map, allowing the translation of dynamical information into $K$-theoretic obstructions. This yields a natural, odd-degree counterexample to surjectivity, enriching the landscape of coarse Baum–Connes phenomena beyond previously known even-degree cases and highlighting the subtle role of dynamical properties and warped cones in large-scale index theory.

Abstract

We prove a conjecture of Roe by constructing unified warped cones that violate the coarse Baum-Connes conjecture. Interestingly, the reason for this is probably not what Roe expected, as the obstruction arises in odd rather than even degree.

Coarse Baum-Connes and warped cones: failure of surjectivity in odd degree

TL;DR

The paper studies coarse index theory for unified warped cones and resolves Roe's conjecture by showing a nuanced obstruction to the coarse Baum–Connes map occurs in odd degree. It proves that generalized Drutu–Nowak projections vanish in -theory ( in ) via an Eilenberg swindle and a Mayer–Vietoris strategy, while simultaneously constructing an explicit odd-degree -theory class outside the image of the coarse assembly map when has property A and acts freely and strongly ergodically on . The core method combines a dyadic Mayer–Vietoris decomposition of the warped cone with compatibility between the MV sequence and the coarse assembly map, allowing the translation of dynamical information into -theoretic obstructions. This yields a natural, odd-degree counterexample to surjectivity, enriching the landscape of coarse Baum–Connes phenomena beyond previously known even-degree cases and highlighting the subtle role of dynamical properties and warped cones in large-scale index theory.

Abstract

We prove a conjecture of Roe by constructing unified warped cones that violate the coarse Baum-Connes conjecture. Interestingly, the reason for this is probably not what Roe expected, as the obstruction arises in odd rather than even degree.
Paper Structure (9 sections, 16 theorems, 52 equations, 1 figure)

This paper contains 9 sections, 16 theorems, 52 equations, 1 figure.

Table of Contents

  1. Background and positive results
  2. Vanishing of K-class of projections
  3. Proof of vanishing by a direct Eilenberg swindle
  4. A Mayer--Vietoris argument
  5. Setup
  6. Mayer--Vietoris on warped cones
  7. Consequences of \ref{['thm:LES']}
  8. Vanishing revisited
  9. Failure of coarse Baum--Connes

Key Result

Theorem 2

If $\Gamma\curvearrowright M$ is ergodic and $\mathfrak G\in C^*_{\rm Roe}(\mathcal{O}_\Gamma M)$ is a generalized Drut̨u--Nowak projection, then $[\mathfrak G]=0$ in $K_0(C^*_{\rm Roe}(\mathcal{O}_\Gamma M))$.

Figures (1)

  • Figure 1: Mappings among intervals and their extremities. All the solid arrows represent 1- or 2-Lipschitz maps, while the dashed one is 4-Lipschitz.

Theorems & Definitions (42)

  • Conjecture 1: Roe, Drut̨u--Nowak DrutuNowak2019KazhdanProjections*Conjecture 7.7
  • Theorem 2: \ref{['thm:vanishing']}
  • Theorem 3
  • Remark
  • Remark
  • Theorem 4
  • Corollary 5
  • Remark
  • Example 1.1
  • Remark 1.2
  • ...and 32 more