Except for the work of Torsten I am not aware of any paper which discusses these topics.(adsbygoogle = window.adsbygoogle || []).push({});

Some ideas:

1) all theories for QG I am aware of do either use manifolds and smootheness (string theory, geometrodynamics, shape dynamics, ...) or are constructed from them (LQG, CDT, ...)

2) in some theories the number of dimensions is fixed, whereas in other theories they become dynamical (spectral dimension)

3) we know that piecewise-wise linear and smooth manifolds are not necessarily the same

4) we know that for dim > 3 the classification of manifolds is not decidable, that there is no algorithm to distinguish between arbitrary manifolds, and that manifolds cannot be "listed"

5) we know that in dim = 4 there are in general uncountably many smoothness structures for a given (non-compact) manifold

5') however if we restrict the domain to manifolds which allow for a global foliation M^{4}= R * M^{3}then (3-5) become trivial

6) due to (4) we are not able to define a "sum over all manifolds" and due to (5) we are not able to define a "sum over all smoothness structures for a given manifold"

7) in all approaches I am aware of a kind of gauge fixing is required; therefore we need to know the diffeomorphism group / mapping class group and the fibre bundle topology (Gribov ambiguities) for local Lorentz invariance (for the connections and n-Beins)

Is there any paper discussing these topics and their relevance in the contex of QG?

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Classification of manifolds and smoothness structures

Loading...

Similar Threads - Classification manifolds smoothness | Date |
---|---|

I Calabi-Yau manifold + ideal gas + point disturbance? | Jan 27, 2018 |

B String theory, Calabi–Yau manifolds, complex dimensions | Jan 9, 2018 |

A Einstein manifolds as asymptotic background spacetimes in Type IIB string theory | May 31, 2017 |

B Metric for Calabi-Yau manifolds dynamic? | Mar 31, 2017 |

Classification of conformal anomalies | Jul 26, 2014 |

**Physics Forums - The Fusion of Science and Community**