Path-connectedness of the CMC vacuum and vacuum initial-data spaces

Determine whether the space of constant-mean-curvature vacuum initial data sets and the full space of vacuum initial data sets are path-connected under the hypotheses of the main theorem.

Background

The main theorem constructs nontrivial homotopy classes based at a particular CMC vacuum initial data set. The authors explicitly note that their argument does not establish whether the relevant spaces have a single path component, so the theorem is necessarily formulated relative to the base point produced by the proof.

References

Let us point out that, under the assumptions of the theorem, we do not know whether the space of CMC vacuum initial data sets, or (M) itself, is path-connected, see however.

— On the topology of the space of vacuum initial data sets  (2609.39987 - Gicquaud et al., 30 Sep 2026) in Section 1, immediately after the proof of Theorem 1.1

It is therefore natural to ask whether this contractibility carries over to (M) and (M) when M is a closed 3-manifold admitting a metric of positive scalar curvature.

— On the topology of the space of vacuum initial data sets  (2609.39987 - Gicquaud et al., 30 Sep 2026) in Section 1, paragraph beginning “We view this article as a first step”

Much less is known about the topology of (M) in dimensions 4 and 5, where non-trivial topology of (M) has been detected by Seiberg-Witten theory and by eta invariants, and the corresponding questions for initial data sets are open.

— On the topology of the space of vacuum initial data sets  (2609.39987 - Gicquaud et al., 30 Sep 2026) in Section 1, paragraph beginning “Much less is known about the topology”