Open-dense classification and the equivariant blow-up obstruction

Definitions and mathematical statement

This record is connective survey prose, not a separately stated conjecture. The supplied source assessment restores its opening to "Partial results on Conjecture 5 were obtained," removes a page-header interruption, and places it after Conjecture 5 and before subsection 2.4 and Question 6. Neither neighboring problem is reproduced here.

The historical paragraph notes that, without partial hyperbolicity, classification may need restriction to an open dense set. It cites nonstandard lattice actions, transitivity or Gromov-rigid-structure hypotheses in a broader classification program, and applications of Ratner measure rigidity to continuous volume-preserving actions of higher-rank semisimple groups and lattices on compact manifolds. Arithmetic quotients motivate the following question, but that question is absent. These comments are not formal universal statements that every such action has a particular quotient or classification.

The aid supplies a background blow-up diagnostic. Let a group act smoothly on a smooth $d$-manifold $M$, $d\geq2$, and let $F$ be a finite invariant orbit. The real projective blow-up $\widetilde M=\operatorname{Bl}_F(M)$ has blow-down $b:\widetilde M\to M$ and exceptional set \[ E=\bigsqcup_{p\in F}{\mathbb{P}}(T_pM), \] a closed invariant embedded hypersurface. This is projective blow-up, not an oriented blow-up introducing a sphere boundary. Every action diffeomorphism lifts functorially; on the exceptional fiber over $p$ its action is induced by the derivative at $p$. The lifted maps satisfy the group relations and $b$ is equivariant. Its restriction \[ b:\widetilde M\setminus E\longrightarrow M\setminus F \] is a smooth conjugacy on invariant open dense complements.

For every invariant probability measure $\mu$ on $M$ with $\mu(F)=0$, the inverse map on these complements gives a unique probability lift $\widetilde\mu$ with $b_*\widetilde\mu=\mu$. It is invariant and $\widetilde\mu(E)=0$, and $b$ gives an isomorphism of the probability actions modulo null sets. Uniqueness here is uniqueness of the lift through this blow-down and its specified complement identification, not uniqueness of every possible abstract measurable conjugacy. The lifted measure need not be a smooth everywhere-positive invariant volume form, so this construction does not automatically preserve all smooth volume-rigidity hypotheses.

If the original action has no closed invariant topologically embedded hypersurface, the lifted action cannot be globally topologically conjugate to it: a conjugacy would send $E$ to such an invariant hypersurface in $M$. This is a sufficient conditional obstruction, not a claim that every finite-orbit blow-up changes the global conjugacy class. It illustrates coexistence of exact open-dense conjugacy and measure equivalence with a possible global obstruction. It does not supply the missing Conjecture 5 or Question 6.

Short English statement

This entry is survey context between two separate problems. Its attached aid uses finite-orbit projective blow-up to distinguish classification levels: the lifted action agrees smoothly on open dense complements and preserves zero-orbit-mass probability systems, yet an exceptional invariant hypersurface can obstruct a global topological conjugacy under an additional hypothesis.

Sources

• Local UnsolvedMath registry, permanent ID $20003272$, source code catalogue entry; canonical location aim-other-notes.json, notes[78].
• AIM report Emerging applications of measure rigidity, connective prose following Conjecture 5 and preceding subsection 2.4 and Question 6, https://aimath.org/WWN/measrigid/measrigid.pdf. The supplied assessment restores "Partial results on" and identifies line-break/page-header extraction errors.
• Supplied source labels [85,94,95,158,72,131] for partial results; [94,59] for nonstandard lattice actions; [209] for the classification program; [117,130] for transitivity/rigid-structure formulations; [123,124] for connected-group arithmetic quotients; [56,57] for lattices; and [58] for a survey. Full bibliography entries are absent and are not guessed.
• Research attempt 1 is marked context-only/not-a-problem, proved, and low novelty confidence, with an explicit machine-generated, non-peer-reviewed research-aid caveat requiring independent verification before citation. These source labels differ from the registry open/conjecture tags.