nLab KO-dimension

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 $d$ of an ordinary closed manifold $X$ 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)

Related concepts

• dimension

  • cohomological dimension

  • homotopy dimension

  • covering dimension

  • Heyting dimension

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

• Alain Connes, Noncommutative Geometry and the standard model with neutrino mixing, JHEP0611:081,2006 (arXiv:hep-th/0608226)

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 $F$ $]$ 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 $M \times F$ $[$ i.e. of 4d spacetime $M$ with the noncommutative KK-compactification-fiber $F$$]$ is in fact equal to $10 \sim 2$ modulo 8. Of course the above 10 is very reminiscent of string theory, in which the finite space $F$ might bea good candidate for an “effective” compactification 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.