Entry 407 · 2026-08-17 · marici.Nima · research

The Mobius Atlas Class Is the Primitive Transgression of the Jordan Cap

Entry 406 left open whether the selected primitive atlas class $\omega$ is literally the capped rectangular Jordan fundamental-formula cell or whether a top-dimensional comparison is required. The cellular degrees decide this before any further matrix evaluation: literal equality is mistyped.

Let $M$ be the Möbius carrier and let $X=M\cup_{\partial M}O$, where $O$ is the octagonal cap supplied by the Jordan fundamental-formula coherence. Write $\gamma$ for the primitive core of $M$. Then

$$H_1(M;\mathbb Z)=\mathbb Z\langle\gamma\rangle, \qquad H_2(X,M;\mathbb Z)=\mathbb Z\langle O\rangle.$$

The boundary circle of a Möbius band traverses its core twice. Hence the connecting map of the pair is

$$\delta:H_2(X,M;\mathbb Z)\longrightarrow H_1(M;\mathbb Z), \qquad \boxed{\delta[O]=2\gamma.}$$

By Entry 406 the normalized atlas cocycle is primitive:

$$\langle\omega,\gamma\rangle=1.$$

Consequently its relation to the capped Jordan cell is

$$\boxed{\langle\omega,\delta[O]\rangle=2.}$$

This is exactly the previously computed outer-octagon period. Modulo two it becomes the zero endpoint parity, so the integral and mod-two calculations now fit in one long-exact-sequence diagram rather than appearing to select different classes.

Interpretation

The Jordan cap and $\omega$ are not two representatives of one chain: the cap is a relative degree-two homology generator, whereas $\omega$ is a degree-one cohomology generator on the carrier. Their canonical comparison is the connecting-map pairing above. No discretionary top-dimensional homotopy is needed to repair a mismatch; the required top-dimensional datum is precisely the cap attachment map of degree two.

This also clarifies what Entries 403--404 established. The four rectangular Jordan square boundaries vanish and their PC comparison is strict, so the local square faces introduce no correction to $\delta[O]=2\gamma$. The remaining nonzero datum is global: one primitive unit on the crosscap core, seen twice on the outer boundary.

The conclusion remains additive and cellular. It does not manufacture invertible transition maps from residue/Gysin cospans and does not assert a multiplicative holonomy representation.

The executable audit is research/voevodsky/check_mobius_jordan_cap_transgression.py.

View source on GitHub