PreprintCC BY 4.0arXiv
On smooth and bundle structures on topological manifolds homotopy equivalent to S4k−1-bundles over S4k
Abstract
We first verify the expected calculations of the topological structure set, in the sense of surgery theory, for the total spaces of S4k−1-bundles over S4k for k≥3, in which manifolds homotopy equivalent to the total spaces are organized. Next, we investigate the question of which of the elements in these structure sets can be realized as such bundles. Moreover, we study the forgetful map from the smooth version to the topological version of the structure set; we determine its image. All elements in the image turn out to have the underlying manifold such a bundle.
Related
- Smooth manifolds homotopy equivalent to products of spheres
- Compact quotients of homogeneous spaces and homotopy theory of sphere bundles
- Simple homotopy types of smooth four-manifolds with finite fundamental group
- Homotopy rigidity for quasitoric manifolds over a product of simplices
- Rational homotopy theory of flag manifolds
BibTeX
@misc{raj2026smooth,
title = {{On smooth and bundle structures on topological manifolds homotopy equivalent to $S^4k-1$-bundles over $S^4k$}},
author = {Ajay Raj and Tibor Macko},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.16307},
eprint = {2609.16307},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16307},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On smooth and bundle structures on topological manifolds homotopy equivalent to S^4k−1-bundles over S^4k
AU - Ajay Raj
AU - Tibor Macko
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2609.16307
UR - https://arxiv.org/abs/2609.16307
AB - We first verify the expected calculations of the topological structure set, in the sense of surgery theory, for the total spaces of S^4k−1-bundles over S^4k for k≥3, in which manifolds homotopy equivalent to the total spaces are organized. Next, we investigate the question of which of the elements in these structure sets can be realized as such bundles. Moreover, we study the forgetful map from the smooth version to the topological version of the structure set; we determine its image. All elements in the image turn out to have the underlying manifold such a bundle.
ER -