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Exotic structures on hyperbolic manifolds via the EO-theory of Projective Spaces

  • Samik Basu
  • Akashdwip Biswas
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Abstract

Farrell and Jones showed that negatively curved manifolds in dimension ≥ 5 are topologically rigid in the sense that homotopy equivalence implies homeomorphism. In the smooth category, the result does not hold even for hyperbolic manifolds, and there are negatively curved manifolds N homotopy equivalent to a given hyperbolic manifold M but not diffeomorphic. For complex hyperbolic manifolds, the analogous construction is demonstrated only in dimensions of the form 8n+2. In this paper, we prove this result in many other dimensions. The main approach involves the use of EO-theory of complex projective spaces to construct suitable examples of exotic spheres. These theories may be viewed as a version of real K-theory, and are defined at each prime p. The computation also yields positive results about the existence of free smooth S1 and S3 actions on exotic spheres which do not bound parallelizable manifolds.

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Cite

BibTeXRIS
BibTeX
@misc{basu2026exotic,
  title = {{Exotic structures on hyperbolic manifolds via the EO-theory of Projective Spaces}},
  author = {Samik Basu and Akashdwip Biswas},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10070},
  eprint = {2610.10070},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10070},
}
RIS
TY  - UNPB
TI  - Exotic structures on hyperbolic manifolds via the EO-theory of Projective Spaces
AU  - Samik Basu
AU  - Akashdwip Biswas
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.10070
UR  - https://arxiv.org/abs/2610.10070
AB  - Farrell and Jones showed that negatively curved manifolds in dimension ≥ 5 are topologically rigid in the sense that homotopy equivalence implies homeomorphism. In the smooth category, the result does not hold even for hyperbolic manifolds, and there are negatively curved manifolds N homotopy equivalent to a given hyperbolic manifold M but not diffeomorphic. For complex hyperbolic manifolds, the analogous construction is demonstrated only in dimensions of the form 8n+2. In this paper, we prove this result in many other dimensions. The main approach involves the use of EO-theory of complex projective spaces to construct suitable examples of exotic spheres. These theories may be viewed as a version of real K-theory, and are defined at each prime p. The computation also yields positive results about the existence of free smooth S^1 and S^3 actions on exotic spheres which do not bound parallelizable manifolds.
ER  -