Exotic involutions of low-dimensional spheres and the eta-invariant
It is known that there are precisely 4 homotopy projective spaces in dimensions 5 and 6 [\textit{S. López de Medrano}, Involutions on manifolds, Ergeb. Math. Grenzgeb. 59 (1971; Zbl 0214.22501)]. The author constructs explicitly certain involutions \(T_i: S^n \to S^n\), \(i=0,1,2,3\), \(n=5,6\) and proves that every involution \(S^n \to S^n, n=5,6\), is equivalent to one of \(T_i\). Furthermore, he proves that \(\eta(S^6/T_i)\equiv (2i+1)\mod 16\) for every \(Pin\)-structure on \(S^6 /T_i\). The author also shows that the suspension over the exotic Fintushel-Stern involution \(S^4 \to S^4\) is equivalent to the author's \(T_3\). As a corollary, the author proves that two involutions \(F,G: S^4 \to S^4\) are stably equivalent iff the spaces \(S^4/F\) and \(S^4/G\) are stably diffeomorphic (i.e., become diffeomorphic after forming the connected sum with sufficiently many copies with \(S^2 \times S^2)\).
scientific article; zbMATH DE number 1533109
scientific article