2004/04/21 by Hendryk Pfeiffer, Pfeiffer, Hendryk · 1 citation
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology #Noncommutative and Quantum Gravity Theories #gr-qc #hep-th
paper · pdf · doi:10.48550/arxiv.gr-qc/0404088
41 pages, LaTeX2e with combined eps/PicTeX figures; v2: inaccuracies fixed
openalex publication_date 2004/04/21 · arxiv created 2004/05/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The gauge symmetry of classical general relativity under space-time diffeomorphisms implies that any path integral quantization which can be interpreted as a sum over space-time geometries, gives rise to a formal invariant of smooth manifolds. This is an opportunity to review results on the classification of smooth, piecewise-linear and topological manifolds. It turns out that differential topology distinguishes the space-time dimension d=3+1 from any other lower or higher dimension and relates the sought-after path integral quantization of general relativity in d=3+1 with an open problem in topology, namely to construct non-trivial invariants of smooth manifolds using their piecewise-linear structure. In any dimension d<=5+1, the classification results provide us with triangulations of space-time which are not merely approximations nor introduce any physical cut-off, but which rather capture the full information about smooth manifolds up to diffeomorphism. Conditions on refinements of these triangulations reveal what replaces block-spin renormalization group transformations in theories with dynamical geometry. The classification results finally suggest that it is space-time dimension rather than absence of gravitons that renders pure gravity in d=2+1 a `topological' theory.