Landfall Geometry
Abstract
We study the negative gradient flow of analytic maps f:X→R defined on compact analytic Riemannian manifolds without imposing any nondegeneracy or transversality assumptions. We construct tight isolating blocks around the connected components of the critical set Crit(f) and record only bounded-time contacts made after an orbit exits one block and before it makes landfall within the interior of another. These contacts assemble into a strict topological category cL -- called the landfall category -- whose classifying space BcL is shown to be simple homotopy equivalent to X naturally over all down-sets of the flow partial order. The filtered homotopy type is independent of the isolating blocks. Passing to a finite subanalytic triangulation yields a flow-induced acyclic partial matching, and hence, an explicit finite chain complex generated by the non-exit simplices of the blocks which computes the homology of X. The finite model also computes the Morse--Conley spectral sequence over arbitrary coefficients; over a field, homological perturbation further produces a connection matrix whose entries are finite sums over descending contact itineraries.
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BibTeX
@misc{kirwan2026landfall,
title = {{Landfall Geometry}},
author = {Frances Kirwan and Vidit Nanda},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30993},
eprint = {2610.30993},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30993},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Landfall Geometry
AU - Frances Kirwan
AU - Vidit Nanda
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30993
UR - https://arxiv.org/abs/2610.30993
AB - We study the negative gradient flow of analytic maps f:X→R defined on compact analytic Riemannian manifolds without imposing any nondegeneracy or transversality assumptions. We construct tight isolating blocks around the connected components of the critical set Crit(f) and record only bounded-time contacts made after an orbit exits one block and before it makes landfall within the interior of another. These contacts assemble into a strict topological category cL -- called the landfall category -- whose classifying space BcL is shown to be simple homotopy equivalent to X naturally over all down-sets of the flow partial order. The filtered homotopy type is independent of the isolating blocks. Passing to a finite subanalytic triangulation yields a flow-induced acyclic partial matching, and hence, an explicit finite chain complex generated by the non-exit simplices of the blocks which computes the homology of X. The finite model also computes the Morse--Conley spectral sequence over arbitrary coefficients; over a field, homological perturbation further produces a connection matrix whose entries are finite sums over descending contact itineraries.
ER -