Degree of symmetry of a homotopy real projective space

Ku, H. T.; Mann, L. N.; Sicks, J. L.; Su, J. C.

doi:10.1090/S0002-9947-1971-0282377-X

Degree of symmetry of a homotopy real projective space
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by H. T. Ku, L. N. Mann, J. L. Sicks and J. C. Su

Trans. Amer. Math. Soc. 161 (1971), 51-61

DOI: https://doi.org/10.1090/S0002-9947-1971-0282377-X

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Abstract:

The degree of symmetry $N(M)$ of a compact connected differentiable manifold M is the maximum of the dimensions of the compact Lie groups which can act differentiably and effectively on it. It is well known that $N(M) \leqq \dim \; SO(m + 1)$, for an m-dimensional manifold, and that equality holds only for the standard m-sphere and the standard real projective m-space. W. Y. Hsiang has shown that for a high dimensional exotic m-sphere M, $N(M) < {m^2}/8 + 1 < \left ( {\frac {1}{4}} \right )\dim SO(m + 1)$, and that $N(M) = {m^2}/8 + 7/8$ for some exotic m-spheres. It is shown here that the same results are true for exotic real projective spaces.

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