nLab
KO-dimension

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Duality

Contents

Idea

What is called KO-dimension (Connes 95) is one notion of dimension which may be associated with a space that is given by a spectral triple with “real structure”.

The definition is motivated by the fact that the dimension dd of an ordinary closed manifold XX is seen in the grading shift involved in the Poincaré duality that the manifold induces on its ordinary homology/ordinary cohomology. Now since in spectral geometry (“noncommutative geometry”) a space is represented by a spectral triple and hence by a kind of Dirac operator which naturally defines a class not in ordinary homology but in K-homology, so the idea of KO-dimension is that it is the shift in the grading on K-theory which is involved in a Poincaré duality for spectral triples.

Now since complex K-theory is 2-periodic this sees such a dimension only modulo 2, and hence only sees whether the dimension is even or odd. But KO-theory is 8-periodic and hence sees dimension at least modulo 8.

The exact definition of KO-dimension given in (Connes 95, def. 3) moreover requires that the Poincaré duality is exhibited by a class in KR-homology.

Examples

For classical Riemannian manifolds KO-dimension coincides with the traditional concept of dimension of manifolds, modulo 8.

The Podlés spheres? have KO-dimension 2, but classical dimension 0.

The spectral Kaluza-Klein compactification considered in the Connes-Lott-Chamseddine model (Connes 06) is taken to be along fibers with KO-dimension 6 and classical dimension 0 (just as perturbative superstring vacua)

References

General

The original source is def. 3 in

  • Alain Connes, Noncommutative geometry and reality, J. Math. Phys. 36 (11), 1995 (pdf)

In Connes-Lott models and superstring vacua

With an eye towards phenomenological considerations of the spectral action (the Connes-Lott-Chamseddine model) this is recalled as def. 7.2 in

From p. 8 there:

When one looks at the table (7.2) of Appendix 7 giving the KO-dimension of the finite space [[ i.e. the noncommutative KK-compactification-fiber FF ]] one then finds that its KO-dimension is now equal to 6 modulo 8 (!). As a result we see that the KO-dimension of the product space M×FM \times F [[ i.e. of 4d spacetime MM with the noncommutative KK-compactification-fiber FF]] is in fact equal to 10210 \sim 2 modulo 8. Of course the above 10 is very reminiscent of string theory, in which the finite space FF might bea good candidate for an “effectivecompactification at least for low energies. But 10 is also 2 modulo 8 which might be related to the observations of Lauscher-Reuter 06 about gravity.

For more on this see at 2-spectral triple.

Last revised on August 31, 2019 at 15:28:35. See the history of this page for a list of all contributions to it.