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Coarse homological invariants of metric spaces

Alexander Margolis

Abstract

Inspired by group cohomology, we define several coarse topological invariants of metric spaces. We define the coarse cohomological dimension of a metric space, and demonstrate that if G is a countable group, then the coarse cohomological dimension of G as a metric space coincides with the cohomological dimension of $G$ as a group whenever the latter is finite. Extending a result of Sauer, it is shown that coarse cohomological dimension is monotone under coarse embeddings, and hence is invariant under coarse equivalence. We characterise unbounded quasi-trees as quasi-geodesic metric spaces of coarse cohomological dimension one. A classical theorem of Hopf and Freudenthal states that if G is a finitely generated group, then the number of ends of G is either 0, 1, 2 or $\infty$. We prove a higher-dimensional analogue of this result, showing that if F is a field, G is countable, and $H^k(G,FG)=0$ for k<n, then $\dim H^n(G,FG)$=0,1 or $\infty$, significantly extending a result of Farrell from 1975. Moreover, in the case $\dim H^n(G,FG)=1$, then G must be a coarse Poincaré duality group. We prove an analogous result for metric spaces.

Coarse homological invariants of metric spaces

Abstract

Inspired by group cohomology, we define several coarse topological invariants of metric spaces. We define the coarse cohomological dimension of a metric space, and demonstrate that if G is a countable group, then the coarse cohomological dimension of G as a metric space coincides with the cohomological dimension of as a group whenever the latter is finite. Extending a result of Sauer, it is shown that coarse cohomological dimension is monotone under coarse embeddings, and hence is invariant under coarse equivalence. We characterise unbounded quasi-trees as quasi-geodesic metric spaces of coarse cohomological dimension one. A classical theorem of Hopf and Freudenthal states that if G is a finitely generated group, then the number of ends of G is either 0, 1, 2 or . We prove a higher-dimensional analogue of this result, showing that if F is a field, G is countable, and for k<n, then =0,1 or , significantly extending a result of Farrell from 1975. Moreover, in the case , then G must be a coarse Poincaré duality group. We prove an analogous result for metric spaces.

Paper Structure

This paper contains 15 sections, 119 theorems, 179 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coarse geometry of metric spaces
  4. Group cohomology
  5. Modules over metric spaces
  6. Projective modules over metric spaces
  7. Hom-sets and duality
  8. Projective resolutions over a metric space
  9. Coarse cohomology and homology
  10. Coarse cohomological dimension of a metric space
  11. Uniform acyclicity and finite type resolutions
  12. Spaces of coarse cohomological dimension one
  13. Cap and cup products
  14. Coarse Poincaré duality spaces
  15. Some results about derived limits

Key Result

Theorem 1.1

If $G$ is a finitely generated group, then exactly one of the following holds:

Figures (6)

  • Figure 1: Each map in the right-hand diagram has finite displacement over the corresponding map in the left-hand diagram.
  • Figure 2: The lifting property for projective $R$-modules over a metric space.
  • Figure 3: A commutative diagram showing naturality of $\Psi$.
  • Figure 4: Naturality of the isomorphism $\Lambda_M$.
  • Figure 5: A commutative diagram showing the chain maps $f_\#$ and $g_\#$.
  • ...and 1 more figures

Theorems & Definitions (278)

  • Theorem 1.1: hopf1944endenfreudenthal1945uber
  • Theorem 1: Theorem \ref{['thm:main_field']}
  • Theorem 2: Corollary \ref{['cor:main_finiteness_space']}
  • Theorem 3: Theorems \ref{['thm:ccd_monotonicity']} and \ref{['thm:gp_ccd']}
  • Remark 1.3
  • Corollary 4: sauer2006cdqi
  • Theorem 5: Corollary \ref{['cor:bdry']}
  • Theorem 6: Theorem \ref{['thm:ccd1']}
  • Corollary 1.4: stallings1968torsionfreedunwoody1979accessibility
  • Remark 1.5
  • ...and 268 more