Current implementation
Mathematical scope
The complete pipeline treats marked rational elliptic surfaces for which Q is globally narrow; P may be non-narrow. It computes the topology from the selected local models by integral Mayer–Vietoris and van Kampen.
Implemented hypotheses
- The section pair satisfies the displayed local component conditions.
- The linearization divisor is supported on semistable Ik fibers, including added smooth I0 fibers.
- Log vectors are exact rational vectors fixed by the corresponding 4 × 4 monodromy.
- The log order divides the minimal semistable-reduction degree.
- Mumford compactifications use the prescribed A2 tiling or rank-one wheel.
- Quotient compactifications use the selected minimal resolution.
None and explicit zero
At an Ik fiber, including I0, the two choices give the same original or prescribed Mumford filling. At a potentially good additive fiber, None selects the original O(P−O) filling, while an explicit zero selects the good-reduction quotient with zero added twist. At a positive-index In* fiber the analogous distinction is between the original filling and the quadratic semistable-reduction quotient.
Result qualification
When the program reports “Topology of S6,” the supplied smooth geometric model has trivial computed fundamental group and the integral cohomology of S6. Realization of every abstract Oguiso–Shioda marking as a specified analytic family is a separate question.
Runtime database
The application uses a read-only table of 487 content-addressed local models. Each entry contains only the marked local cochain pair, the boundary comparison used by Mayer–Vietoris, peripheral relations for van Kampen, compact stalk data, and lookup metadata.
Expanded cellular models, reduction certificates, comparison homotopies, detailed derivations, and private validation inputs are maintained separately and are not published with the applet.
Outside the current entry point
- non-narrow Q;
- torsion fixed only modulo the lattice;
- log order not dividing the semistable-reduction degree;
- linearization zeros or poles on additive fibers;
- arbitrary Mumford subdivisions;
- general recognition of every finitely presented fundamental group.