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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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##### References:
 [1] 10.1016/0040-9383(66)90004-8 · Zbl 0145.19902 [2] ; Browder, Differential and combinatorial topology (a symposium in honor of Marston Morse), 23, (1965) [3] 10.2307/2373232 · Zbl 0178.26401 [4] 10.1007/BF02566851 · Zbl 0222.57021 [5] ; Brumfiel, Algebraic topology, 73, (1971) [6] ; Elmendorf, Rings, modules, and algebras in stable homotopy theory. Mathematical Surveys and Monographs, 47, (1997) [7] 10.1007/BF01232234 · Zbl 0804.53055 [8] 10.2307/1971103 · Zbl 0463.53025 [9] 10.1016/0001-8708(74)90021-8 · Zbl 0284.58016 [10] 10.1090/S0894-0347-99-00320-3 · Zbl 0931.55006 [11] ; Kasilingam, Forum Math., 27, 3005, (2015) [12] 10.1007/s12044-016-0269-4 · Zbl 1337.57063 [13] 10.3792/pja/1195520972 · Zbl 0172.25302 [14] 10.4310/jdg/1214432678 · Zbl 0296.53037 [15] 10.2307/1970128 · Zbl 0115.40505 [16] ; Kirby, Foundational essays on topological manifolds, smoothings, and triangulations. Annals of Mathematics Studies, 88, (1977) · Zbl 0361.57004 [17] ; Kochman, Bordism, stable homotopy and Adams spectral sequences. Fields Institute Monographs, 7, (1996) · Zbl 0861.55001 [18] 10.2307/2373121 · Zbl 0172.25303 [19] 10.1007/BF02621865 [20] 10.2307/2373505 · Zbl 0207.53901 [21] 10.2307/1969983 · Zbl 0072.18402 [22] 10.1016/0040-9383(68)90026-8 · Zbl 0172.25103 [23] ; Novikov, Izv. Akad. Nauk SSSR Ser. Mat., 28, 365, (1964) [24] ; Ravenel, Complex cobordism and stable homotopy groups of spheres. AMS Chelsea, 347, (2004) · Zbl 1073.55001 [25] 10.1090/S0002-9947-1971-0275453-9 [26] ; Tamura, C. R. Acad. Sci. Paris, 255, 3104, (1962) [27] ; Toda, Composition methods in homotopy groups of spheres. Annals of Mathematics Studies, 49, (1962) · Zbl 0101.40703 [28] 10.2307/1970425 · Zbl 0218.57022 [29] ; Winkelnkemper, Trans. Amer. Math. Soc., 206, 339, (1975)
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