On the topology of the space of vacuum initial data sets
Abstract
We show that the space of vacuum initial data sets on a closed manifold often has many non-trivial homotopy groups. The starting point is a result of the second named author, which constructs non-trivial elements in the homotopy groups of the space of initial data sets satisfying the strict dominant energy condition, together with a later result in joint work with Bernd Ammann showing that these elements often persist when the strictness assumption is dropped. In this work, we use a parametrized version of the conformal method to show that these elements may also be represented by maps into the space of vacuum initial data sets. This requires two results that may be of independent interest: metrics admitting conformal Killing vectors can be removed from the space of metrics without changing its weak homotopy type, and over the remaining metrics the York decomposition can be carried out in families, the TT-tensors forming a trivial Hilbert bundle. To our knowledge, this is the first result on the global topology of this space.
Related
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BibTeX
@misc{gicquaud2026topology,
title = {{On the topology of the space of vacuum initial data sets}},
author = {Romain Gicquaud and Jonathan Glöckle},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39987},
eprint = {2609.39987},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05773417v1},
}
RIS
TY - UNPB
TI - On the topology of the space of vacuum initial data sets
AU - Romain Gicquaud
AU - Jonathan Glöckle
PY - 2026
DA - 2026/09/30
PB - HAL
DO - 10.48550/arxiv.2609.39987
UR - https://hal.science/hal-05773417v1
AB - We show that the space of vacuum initial data sets on a closed manifold often has many non-trivial homotopy groups. The starting point is a result of the second named author, which constructs non-trivial elements in the homotopy groups of the space of initial data sets satisfying the strict dominant energy condition, together with a later result in joint work with Bernd Ammann showing that these elements often persist when the strictness assumption is dropped. In this work, we use a parametrized version of the conformal method to show that these elements may also be represented by maps into the space of vacuum initial data sets. This requires two results that may be of independent interest: metrics admitting conformal Killing vectors can be removed from the space of metrics without changing its weak homotopy type, and over the remaining metrics the York decomposition can be carried out in families, the TT-tensors forming a trivial Hilbert bundle. To our knowledge, this is the first result on the global topology of this space.
ER -