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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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