Cohomology of regular spherical actions
Abstract
We study regular spherical actions, namely, actions of a connected compact Lie group G on a manifold M with equidimensional orbits, some of which are spheres. These spherical orbits are diffeomorphic to homogeneous spaces G/H, and there is a list of nine possible families of pairs (G, H). The normalizer N of H in G plays an important role in our calculations, as does the Weyl group W = N/H. In fact, there are three possibilities for W, namely: Z_2, S^1 and S^3. The goal of this paper is to relate the cohomology of M to that of the orbit space M/G. We obtain a particularly simple relationship in the first case, W = Z_2. For the remaining two cases, W = S^1 and W = S^3, it is already known that the cohomology of M and that of the orbit space M/G are related by the Gysin sequence ... \to H^k(M/G) \to H^k(M ) \to H^{k-d}(M/G) \to H^{k+1}(M/G) \to ... In this work, we provide a direct construction of this sequence, avoiding the use of spectral sequences. Moreover, we give an explicit description of the connecting homomorphism as multiplication by a power of the Euler class associated with the action of the Weyl group W = S^l on the fixed-point set M^H.
Related
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BibTeX
@misc{prieto2026cohomology,
title = {{Cohomology of regular spherical actions}},
author = {José Ignacio Royo Prieto and Martintxo Saralegi-Aranguren and Robert Wolak},
howpublished = {HAL},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02111},
eprint = {2611.02111},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05849852v1},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Cohomology of regular spherical actions
AU - José Ignacio Royo Prieto
AU - Martintxo Saralegi-Aranguren
AU - Robert Wolak
PY - 2026
DA - 2026/11/01
PB - HAL
DO - 10.48550/arxiv.2611.02111
UR - https://hal.science/hal-05849852v1
AB - We study regular spherical actions, namely, actions of a connected compact Lie group G on a manifold M with equidimensional orbits, some of which are spheres. These spherical orbits are diffeomorphic to homogeneous spaces G/H, and there is a list of nine possible families of pairs (G, H). The normalizer N of H in G plays an important role in our calculations, as does the Weyl group W = N/H. In fact, there are three possibilities for W, namely: Z_2, S^1 and S^3. The goal of this paper is to relate the cohomology of M to that of the orbit space M/G. We obtain a particularly simple relationship in the first case, W = Z_2. For the remaining two cases, W = S^1 and W = S^3, it is already known that the cohomology of M and that of the orbit space M/G are related by the Gysin sequence ... \to H^k(M/G) \to H^k(M ) \to H^{k-d}(M/G) \to H^{k+1}(M/G) \to ... In this work, we provide a direct construction of this sequence, avoiding the use of spectral sequences. Moreover, we give an explicit description of the connecting homomorphism as multiplication by a power of the Euler class associated with the action of the Weyl group W = S^l on the fixed-point set M^H.
ER -