A commutative diagram and an application to differentiable transformation groups

Curtis, W. D.

doi:10.1090/S0002-9939-1972-0290389-1

A commutative diagram and an application to differentiable transformation groups
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by W. D. Curtis

Proc. Amer. Math. Soc. 31 (1972), 260-264

DOI: https://doi.org/10.1090/S0002-9939-1972-0290389-1

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Abstract:

A commutative diagram is presented which relates the groups of concordance classes of diffeomorphisms $\Gamma ({S^{2n}}),\Gamma (C{P^n})$ and $\Gamma ({S^{2n + 1}})$. This diagram is applied to show that every equivariant diffeomorphism of ${S^7}$ is concordant to the identity. It follows that the exotic $8$-sphere, ${\Sigma ^8}$, admits no smooth semifree ${S^1}$-action with exactly two fixed points.

References

Glen E. Bredon, A $\Pi _\ast$-module structure for $\Theta _\ast$ and applications to transformation groups, Ann. of Math. (2) 86 (1967), 434–448. MR 221518, DOI 10.2307/1970609
Michel A. Kervaire and John W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR 148075, DOI 10.1090/S0273-0979-2015-01504-1
Ronnie Lee, Differentiable classification of some topologically linear actions, Bull. Amer. Math. Soc. 75 (1969), 441–444. MR 239623, DOI 10.1090/S0002-9904-1969-12212-3

Bibliographic Information

Journal: Proc. Amer. Math. Soc. 31 (1972), 260-264
DOI: https://doi.org/10.1090/S0002-9939-1972-0290389-1
MathSciNet review: 0290389