Involutions on homotopy spheres and their gluing diffeomorphisms

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".