Topological Torsion, Chirality and the Hopf Map

The Hopf Map has been used to describe the relationship between a sphere S3 in 4D mapped to a sphere S2 in 3D. It was a surprise to find that the Hopf Map has two different "chirality" formats, and that each produces the same relationships between the spheres of different dimensions. However, the maps may be used to understand properties of the neighborhoods of the two spheres as well.

An extraordinary result is that the evolutionary stability in the neighborhood of the spheres depends upon the choice of the chirality index, which is either plus or minus 1!

Can this be the raison d'etre to explain the handedness of nature?

In the downloads that follow, Maple is used to compute the quantities needed, and to handle the formidable algebraic manipulations. If you have Maple, you can vary the quantities and try various special cases that may be of interest to you.

The fundamental 1-form on the 4D manifold is of a classic Zhitomarski form,

A = (-YdX+XdY)+(rot)(SdZ-ZdS)

with the chirality coefficient,

rot = plus or minus 1

The three vector fields which are associated (orthogonal) with the 1-form A, given above, are perfect differentials of the three functions that describe the Hopf Map. The three associated fields and the 1-form coefficients can be used to describe a global basis frame (mod the origin) in R4.

The 1-form can be scaled by a divisor in the form of an arbitrary Holder norm, beta, of homogeneity index n. The Pfaff sequence is computed to yield the 4-form of topological parity

K = -4(dX^dY^dZ^dS)(n-2)(ch)/(beta)^2

This result implies that for n=2, the Zhitomarski 1-form has closed integrals which are "quantized". They are topological deformation invariants!

The function K is like a Liapunov exponent and its sign depends upon the Holder homogeneity index, n, and the chiral factor, ch.

In the usual gaussian type norm - based upon a quadratic form, n = 1. Hence, the stability of evolutionary processes depends upon the choice of chirality index, ch.

If the scaling denominator is expanded in terms of factors involving the homogeneous exponent, then only the n=2 exponent will remain after long evolutionary times. The resultant manifold is not a symplectic sphere of non-zero radius, but it is a contact manifold which now supports a unique conservative (Hamiltonian) evolutionary process.

The download also computes the repere mobile (the basis frame), the Cartan matrix, the curvature and torsion two-forms for each choice of chirality index.

Of related interest is the Instanton (differential) map. In the instanton case, three of the 1-forms induced be the map are non-holonomic. Yet the Basis Frame has an induced metric form which is the same as for the Hopf map.