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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Involutions on homotopy spheres and their gluing diffeomorphisms

Author: Chao Chu Liang
Journal: Trans. Amer. Math. Soc. 215 (1976), 363-391
MSC: Primary 57D65; Secondary 57E25
Erratum: Trans. Amer. Math. Soc. 222 (1976), 405.
MathSciNet review: 0431213
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Abstract: Let $ hS({P^{2n + 1}})$ denote the set of equivalence classes of smooth fixed-point free involutions on $ (2n + 1)$-dimensional homotopy spheres. Browder and Livesay defined an invariant $ \sigma ({\Sigma ^{2n + 1}},T)$ for each $ ({\Sigma ^{2n + 1}},T) \in hS({P^{2n + 1}})$, where $ \sigma \in Z$ if n is odd, $ \sigma \in {Z_2}$ if n is even. They showed that for $ n \geqslant 3,\sigma ({\Sigma ^{2n + 1}},T) = 0$ if and only if $ ({\Sigma ^{2n + 1}},T)$ admits a codim 1 invariant sphere. For any $ ({\Sigma ^{2n + 1}},T)$, there exists an A-equivariant diffeomorphism f of $ {S^n} \times {S^n}$ such that $ ({\Sigma ^{2n + 1}},T) = ({S^n} \times {D^{n + 1}},A){ \cup _f}({D^{n + 1}} \times {S^n},A)$, where A denotes the antipodal map. Let $ \beta (f) = \sigma ({\Sigma ^{2n + 1}},T)$. In the case n is odd, we can show that the Browder-Livesay invariant is additive: $ \beta (fg) = \beta (f) + \beta (g)$. But if n is even, then there exists f and g such that $ \beta (gf) = \beta (g) + \beta (f) \ne \beta (fg)$. Let $ {D_0}({S^n} \times {S^n},A)$ be the group of concordance classes of A-equivariant diffeomorphisms which are homotopic to the identity map of $ {S^n} \times {S^n}$. We can prove that ``For $ n \equiv 0,1,2 \bmod 4, hS({P^{2n + 1}})$ is in 1-1 correspondence with a subgroup of $ {D_0}({S^n} \times {S^n},A)$. As an application of these theorems, we demonstrated that ``Let $ \Sigma _0^{8k + 3}$ denote the generator of $ b{P_{8k + 4}}$. Then the number of $ (\Sigma _0^{8k + 3},T)$'s with $ \sigma (\Sigma _0^{8k + 3},T) = 0$ is either 0 or equal to the number of $ ({S^{8k + 3}},T)$'s with $ \sigma ({S^{8k + 3}},T) = 0$, where $ {S^{8k + 3}}$ denotes the standard sphere".

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Keywords: Free differentiable involutions, Browder-Livesay invariant, equivariant diffeomorphism, concordance group of diffeomorphisms, curious involutions
Article copyright: © Copyright 1976 American Mathematical Society

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