Arcadian Functor

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Marni D. Sheppeard

Thursday, February 14, 2008

Ternary Geometry II

Recall [1] that the pullback $U \cap V$ of open sets in a manifold $M$ is essential to the definition of homology, which begins with the differential forms functor $\Omega^{\ast}$ acting on the arrows relating $M$ to the disjoint union of $U$ and $V$, in which the set $U \cap V$ is included. From the point of view of logic, it is a nice feature of manifolds that they are defined in terms of glued sets.

Let us view the Euler characteristic of a space as a homological entity (which it is). For a compact oriented manifold, under geometric Poincare duality the homology groups, and cohomology groups in the dual dimension, are isomorphic. Since the Euler characteristic is an alternating sum of dimensions of homology groups, it behaves simply under duality. Considering a compact hypersurface, in odd dimension $d$ we see that

$\chi = n_0 - n_1 + n_2 \cdots + n_d$

goes to $- \chi$ under duality. These two can only be equal if $\chi = 0$, which is the general result. Thus $\chi$ is not always a useful invariant. Its essence is already captured in dimension one, where $\chi = n_0 - n_1 = P - L$.

Observe that here we see only 2-logos (binary) logic, rather than ternary logic. Moreover, quantum invariants need not take values as scalars, but rather as q-numbers, perhaps represented by matrices. Now let us reinterpret the 2-logos $\chi$ as a combination of the Pauli MUB operators $1$ and $\sigma_{x}$ (the swap circulant, interpreted as -1).

A ternary analogue for $\chi$ is then naturally the $3 \times 3$ quantum Fourier transform, an example of which are the neutral and charged lepton mass matrices. Under triality, these matrices are invariant, at least up to equivalence.

[1] R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, Springer (1982)


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