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Inertia groups of high-dimensional complex projective spaces. (English) Zbl 1382.57013
Let $$I(M)$$ denote the inertia group of a closed smooth manifold $$M$$. Theorem A. The group $$I(\mathbb C\mathbb P^9)$$ is isomorphic to either $$\mathbb Z/2$$ or $$\mathbb Z/4$$, the group $$I(\mathbb C\mathbb P^{13})$$ contains $$\mathbb Z/2$$. Theorem B. There are infinitely many values of $$n$$ for which there exist nontrivial elements in the inertia group of $$\mathbb C\mathbb P^{4n+1}$$.
Note the following nice applications. Theorem 4.3: There exist three homotopy 18-dimensional spheres $$\Sigma_i, i=1,2,3$$ such that the following is true: (i) The four manifolds $$\mathbb C\mathbb P^9$$ and $$\mathbb C\mathbb P^9\# \Sigma_i, i=1,2,3$$ are pairwise nondiffeomorphic. (ii) The manifolds $$\mathbb C\mathbb P^9\#\Sigma_2$$ and $$\mathbb C\mathbb P^9\#\Sigma_3$$ do not admit a metric of nonnegative scalar curvature but $$\mathbb C\mathbb P^9\#\Sigma_1$$ does. Theorem 4.4: Let $$\Sigma_i$$ be as in Theorem 4.3. Given a positive real number $$\epsilon$$ , there exists a closed complex hyperbolic manifold $$M$$ of complex dimension 9 such that the following is true: (i) The manifolds $$M$$ and $$M\#\Sigma_i, i=1,2,3$$ are pairwise nondiffeomorphic. (ii) Each of the manifolds $$M\#\Sigma_i, i=1,2,3$$ supports a negatively curved Riemannian metric whose sectional curvatures all lie in the closed interval $$[-4-\epsilon,-1+\epsilon]$$.
##### MSC:
 57R55 Differentiable structures in differential topology 57R60 Homotopy spheres, Poincaré conjecture 55P25 Spanier-Whitehead duality 55P42 Stable homotopy theory, spectra
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