Moduli Space
Definition
The moduli space M_n is the space of all CompleteType instances over R_n at a given Witt level. Its geometry is governed by the DeformationComplex at each point, and its stratification is indexed by holonomy classes from Monodromy.
A ModuliSpace carries:
- moduliWittLevel — the Witt level of the space
- moduliPoint — CompleteType members
- moduliDimension — the dimension of the space
Deformation Complex
The DeformationComplex of a CompleteType T is a ChainComplex whose cohomology groups control the local geometry of M_n at T:
- H^0 = automorphisms of T
- H^1 = first-order deformations (tangent space)
- H^2 = obstructions to extending deformations
Properties: deformationBase, tangentDimension, obstructionDimension.
Holonomy Stratification
The moduli space decomposes into strata indexed by conjugacy classes of subgroups of D_{2^n}:
- HolonomyStratum — a single stratum, with stratumHolonomyClass linking to the indexing MonodromyClass and stratumCodimension recording its codimension in M_n.
- The flat stratum (trivial holonomy, codimension 0) is open and dense.
- Twisted strata have codimension >= 1.
Deformation Families
A DeformationFamily is a one-parameter family {C_t} parameterizing a path in M_n. The property familyPreservesCompleteness records whether every member remains a CompleteType.
The VersalDeformation is the formally universal deformation of a CompleteType T: every other deformation factors through it (possibly non-uniquely). Properties: versalBase, versalDimension.
Moduli Tower
The ModuliTowerMap M_n -> M_{n+1} is induced by WittLift. Its site over T is the space of completions of the partial lift. Property: towerMapSource.
This connects to the Witt Spectral Sequence and incremental completeness pipeline.
Key Identities
| Identity | Statement |
|---|---|
| MD_1 | dim(M_n) = basisSize(T) |
| MD_2 | H^0(Def(T)) = Aut(T) intersect D_{2^n} |
| MD_3 | H^1(Def(T)) = tangent space |
| MD_4 | H^2(Def(T)) = LiftObstruction space |
| MD_5 | FlatType stratum is open (codimension 0) |
| MD_6 | TwistedType strata have codimension >= 1 |
| MD_7 | VersalDeformation exists for every CompleteType |
| MD_8 | Completeness-preserving iff H^2 = 0 along path |
| MD_9 | Site dim = 1 when obstructionTrivial |
| MD_10 | Site empty iff TwistedType at every level |